130765606-Statistics-for-Economics-for-Class-11-N-M-Shah.doc

PREFACE TO THE THIRD REVISED EDITIONPeriodic updating and review accommodate new knowledge as well as adds freshness even as it allows for continuity. This third revised edition is an effort to that end. Statistics for Economics for class XI in its revised format brings forth the changed mode by the Central Board of Secondary Education, New Delhi in 2005-06. The revision includes sufficient exercises keeping in mind learning tools of Statistics in the context of the study of Economics. This volume incorporates extensive and colourful diagrams and illustrations to enhance a better and friendlier understanding of concepts of Statistics. Answer to the numerical questions in the exercise of unit 3 are also provided so that the students can verify the solutions. It is hoped that this revised edition will be of great help to both teachers and students. — N.M. SHAH PREFACE Stamtics has become an anchor for social, economic and scientific studies Stat,st,cal methods are widely used in several disciplines, be i, planning bul ' management, psephology (study of voting patterns), psycholo^ oT adve Sn; steps. A hst of formulae has been provided at the end of each chapter of unit 3 and rJr rTT —• ^ong years of teLh ng thrsubta I have to acknowledge that in the wriring of this volume I have got immense help rom my frtends and relation. The pubUshers have been very cooperative aTrelp ll 1 Trr r "tT- -derstanding w'fe bes d s tsp rin ' me and looktng after the household, has done ind.spensable work for the volume Z M.M. Shah, himself a scholar and teacher of economics, and who retired as dean of acuity of Commerce, Nagpur University and Prmc.pal, G.S. College of clt rct Shri Ram College of Commerce, Delhi April, 2002 —N.M. SHAH SYLLABUS STATISTICS FOR ECONOMICS-XI One Paper 3 Hrs. 100 marks 104 Periods/50 xMarks 5 Periods/3 Marks 25 Periods/12 Marks 64 Periods/30 Marks 10 Periods/5 Marks PART-A STATISTICS FOR ECONOMICS 1. Introduction 2. Collection, Organisation and Presentation of Data 3. Statistical Tools and Interpretation 4. Developing Projects in Economics Unit 1: Introduction What is Economics? ^^ Meanmg scope and importance of statistics m Economics Unit 2: Collection, Organisation and Presentation of Data SoToflot^tm^^^^^^^ W imrr^ -« and National Sam;:;! sT^.y^lZZT ^^^^ ^^ p,,, Organisation of Data: Meaning and types of variables; Frequency Distribution. Unit 3: Statistical Tools and Interpretation interpretation for the rite d^ived) P'-d' deviat.o„,_Lore„z curvLMeaJ^aX ^^^ --Karl Pearson's method Oirrelation-meaning, scatter diagram-Measures of correlation-(two variables ungrouped data), Spearman's rank correlation "n^ inL'"on types-wholesale price index, consumer price "umLrs production, uses of index numbers; inflation and iLx Unit 4: Developing Projects in Economics Tu . J ■ Periods in n • j latlo^ both TaL tT^rV" data, secondary of L eimnl. f I' organisations outlets may also be encouraged. Some of the examples of the projects are as follows (they are not mandatory but suggestivT) (t) A report on demographic amongst households; suggestive). («) Consumer awareness amongst households (»/) Changing prices of a few vegetables in your market (w) Study of a cooperative institution—milk cooperatives ^eJ^t to enable the students to develop the ways and u1 CONTENTS UNIT 1 : Introduction 1. What is Economics 2. Introduction—Meaning and Scope . UNIT 2 : Collection and Organisation of Data 3. Collection of Primary and Secondary Data 4. Organisation of Data Presentation of Data 5. Tabular Presentation 6. Diagrammatic Presentation 7. Graphic Presentation UNIT 3 : Statistical Tools and Interpretation 8. Measures of Central Tendency 9. Positional Average and Partition Values 10. Measures of Dispersion 11. Measures of Correlation 12. Introduction to Index Numbers UNIT 4 : Developing Projects in Economics 13. Preparation of a Project Report 1 10 22 52 76 87 108 138 178 232 313 354 394 WHAT IS STATISTICS? S:T:A:T:I:S:T:I:C:S: Scientific Methodology Theory of Figures Aggregate of Facts Tables and Calculation for Analysis Investigation Systematic Collection Tabulation and Organisation Interpretation Comparison Systematic Presentation tc fc ea dc in lif wl his in ( UNIT 1 — HmODUCTIOr^ m Whui is ficonoinifs? ■ Jlieaiiiiig, Scope and Imporianct^ of l§)tatislics in Economms ■■ Chapter 1 what is economics? 1- Introduction 2. Activity 3. Definition of Economics 4. Nature of Economics Economics as a Science (i^ Economics as an Art introduction If each of us possessed 'Aladdin's magic lamp, which we had merely to rub in order to get our desires fulfilled immediately, there would be no economic problem and no need for a science of economics. In real life we are not lucky as Aladdin, we have to work to earn our livelihood. All people in this world work to satisfy their unlimited wants and desires. Every one requires food to eat, clothes to wear and house to live in. Besides these in daily life. They need television, mobile phone, motor bike, car etc., to lead a comfortable life. The person visits the market and enquires about the varieties and prices of the item which he wants to purchase. Thinking about his source and alternative choices, he uses his sense of economy and decides to buy that item. This is economics. So, A customer is a person who buys goods to satisfy his wants. A Producer is a person who produces or manufactures goods. A Service holder is a person who is in a job to earn either wages or salary to buy goods. A Service provider is a person who provides services to society to earn money, e g doctors, scooter drivers, lawyers, bankers, transporters, etc. All above persons are busy in different activities to earn, called economic activity in ordinary business of life. They face in their life the problem of scarcity of income z Statistics for Economics-XI »purcha. Thus, and^J^Ttu' "" ^r^'^ff'^^l^dge with economic activities relating to earning ar^ spending the wealth and tncome. Economics is the study of how human beinJZa^ tZ^ tn":^-' T""" unlimited wLts in sZZ Z^LTZ ^omm maxnntse thetr satisfaction, producers can maximise their profits and society can maxtmtse its social welfare'. ^^ infn^W mT publication of Adam Smith's "An Inquiry into the Nature and Causes of Wealth of Nations", in the year 1776 At its Mxth Z name of economics was 'PoHtical Economy'. Some of the suggested names — Catallactics or the science of exchange. — Plutology or the science of wealth. — Chrematistics or the science of money-making ~ "y't to^om- s''.'^' -- ^ (HoItVoTr has its origin into two Greek words : Oikos E'^gli^h ' '^o'^tical (Household) and nomos (to manage). Thus, the word economics was used to mean home management with limited funds available in the most possible economLal mTn2 activity Iife.?herlf " ^ day 1. Non-economic Activities 2. Economic Activities or Activities. These activities are those which have no economic aspect or are not concerned with money or wealth, viz. ~ bkll^S '' 8«-together, attending Dirtnday parties or marriages etc. " wSrCo?'''!'''^'"'^ '' o, & ^"^"dwara, mosque or church to worship God, attending mass prayer (Satsang), etc. Political activities such as various activities performed by different political parties namely by Bhartiya Janata Party (BJP), Congress Party etc ~ ^'ds or helping - Parental activities, such as love and affection towards their children —^ ^l^ey involve any nf Activities. Different types of activities are performed bV different types of people (doctors, teachers, businessmen, industrialists, lawyers etc.) so as taelT^ What is Economics? 3 living. Every one is concerned with one or the other type of activity to earn money or wealth to meet their wants. An economic activity means that activity which is based on or related to the use of scarce resources for the satisfaction of human wants. Economic activities are classified as under : the ECONOMIC ACTIVITIES i nt >ution kos ome I day ispect nding to lies ilping any [types I their Production : Production is that economic activity which is concerned with increasing the utility or value of goods and services. Manufacturing shirt with the help of cloth (raw material) and tailoring (labour) etc. is an act of production. Transporting sand from river bank to a town, where it is needed, is also an act of production. Here utility is created through transportation of goods to the person who needs it. Consumption : Consumption is that economic activity which is concerned with the use of goods and services for the direct satisfaction of individual and collective wants. Consumption activity is the base of all production activities. There would have been no production if there would have been no consumption. For example, eating bread, drinking water or milk, wearing shirt, services of lawyer or doctor etc. are consumption activities. Investment : Investment is that economic activity which is concerned with production of capital goods for further production of goods and services. Investment indirectly satisfies human wants. For example, the production of printing press machines to print newspapers, books, magazines etc. or investment in computers to provide Internet, banking and related services. Exchange : Exchange is that economic activity which is concerned with sale and purchase of commodities. This buying and seUing is mostly done in terms of money or price. So, it is also called ""Product Pricing'' which relates to determination of the price of the product under different conditions of the market, viz., perfect competition, imperfect competition, monopoly etc. Distribution : Distribution is that economic activity which deals with determination of price of factors of production (land, labour, capital and enterprise). This is known as the 'Factor Pricing', e.g., price of land is rent, that of labour is wage, that of capital is interest and price of entrepreneur is profit. Distribution is the study to know how the national income or total income arising from what has been produced in the country (called Gross Domestic Product or GDP) is distributed through salaries, wages, profits and interest. "Economics is that branch of knowledge that studies consumption, production, exchange and distribution of wealth". —Chapman 10 Statistics for Economics-XI definition of economics Economics has been defined by many economists m different ways The set of cat"go^s^'"'" '' ^^ mto the folwL; W 1. Wealth definition—Adam Smith 2. Material welfare definition—Alfred Marshall 3. Scarcity definition—Lionel Robbins 4. GroAvth definition—Paul A. Samuelson 1. Wealth Definition (/) Adam Sm^th the father of modem economics, m his book 'An Inquiry mto the Nature and Causes of Wealth of Nations' in 1976 defined thatproduction and expansion of wealth as the subject matter of («) According to J.B. Say, Economics as "the science which deals wtth wealth" ofTaTth! " ^^^ consumption {in) Ricardo shifted the emphasis from production of wealth to distribution of wealth Criticism : This definition is not a precise definition. It gives importance to wealth rather than production of human and social welfare.importance to wealth The wealth definition of economics was discarded towards the end of the 19th century. 2. Material Welfare Definition "Economics is a sUuiy of mankind i„ the ordinary business of life it examines that What is Economics? 5 Criticism : {a) In economics, we study immaterial things also. (b) Welfare cannot be measured in terms of money. (c) Welfare definition makes economics a purely social science. II h "Tf "" " - d^ff-nt times. Then basic difference between Adam Smith's and Marshall's definition is that Ad.m r-iti* sr. ™ -—hiri^3. Scarcity Definition There are three important aspects in this definition. They are • Icf of Tir m" human wants which is the tact of Me. When one warn gets satisfied, another want crops up. fr" '' are scarce in relation coal IS used m factories, m running railways and in thermal stations for electric generation and by households, etc. electric In short, according to Robbins, Economics is a science of choice It deals with how Crit^sm : Sfet™s Scarcity definition of economics has been criticised on the following grounds ■ (0 The defimtion is impractical and difficult. It is narrow and restricted in scope It ^ development. It has notS^^^ Hi) The definition makes economics a human science. 4. Growth Definition Paul A. Samuelson defines— to p^u^ vanous commod,Ues overtime and distribute them for consumptionZ^ or 12 Statistics for Economics-XI The definition combines the essential elements of the definitions by Marshall and Robbms. Accordmgly, economics is concerned with the efficient allocation and use of scarce means as a result of which economic growth is increased and social welfare is promoted. The definition has been accepted universally. In short, the growth definition of economics is most comprehensive of all the earUer definitions. iture of economics ^ Nature of economics—as a science or art. It is science and art as well. NATURE OF ECONOMICS ics as Art A. ECONOMICS AS A SCIENCE Science can be divided into : (a) Natural science, and (b) Social science : Sciences like Physics Biology and Chemistry are natural or physical sciences, where experiments can be conducted in the laboratory under controlled conditions. Relationships can be decided between cause and effect, which are based on facts. Observations can be made and used to prove or disprove theories. The results apply universally. Economics is a social science because it is systematic study of economic activities of human beings. Economics is a science as it is a branch of knowledge where various facts have been systematically collected, classified and analysed. The following arguments are given in favour of economics as a science. (/) Systematised Study : The study of economics is systematically divided into consumption, production, exchange and distribution of wealth and finance which have their own laws and theories. Economics as social science which is a systematic study of human behaviour concentrating on maximum satisfaction to households maximum profit to producers and maximum social welfare to the society as a whole. ^ Hi) Scientific Laws : Economics is a science because its laws are universally true Different laws m economics namely, law of demand, law of supply, law of dimimshing marginal utility, law of returns, Gresham's law etc. are applicable to all types of economies, whether capitaKstic, socialistic or mixed economy y ae. of ; to What is Economics? y (m) Cause and Effect Relationship : Economic laws establish cause and effect relationship like the laws in other sciences. For example, the law of demand shows the relationship between change in price and change in demand..It shows that mcrease in price of a commodity (the cause) will decrease its demand (the effect) establishing the negative or inverse relationship between price and quantity demanded. The law of supply shows that the increase in price of a commodity (cause) will increase its supply (the effect) establishing the positive relationship between price and supply of quantity of commodity. (iv) Verification of Laws : Like other sciences economic laws are also open to verification. These economic laws can be verified through any empirical investigation. On the basis of the arguments given above, we can say economics is a science—not exacriy natural or physical science but social science that studies economic problems and policies in a scientific manner. Economics—A Positive or Normative Science (a) Economics as a Positive Science A positive science is one which makes a real description of an activity. It only answers what ts} what was! It has nothing to suggest about facts, positive economics deals with what IS or how the economic problems facing a society are actually solved. Prof. Robbins held that economics was purely a positive science. According to him, economics should be neutral or silent between ends; /.e., there should be no desire to learn about ethics of economic decisions. Thus, in positive economics we study human decisions as facts which can be verified with actual data. Some exampi es of Economics as a positive science are : {i) India is second largest populated country of the world. (k) Prices have been rising in India. (m) Increase in real per capita income increases the standard of living of people. (iv) The targeted growth rate of the tenth five-year plan is 8 per cent per annum. {v) Fall in the price of commodity leads to rise in its quantity demanded. Alfred Marshall and Pigou have considered the normative aspect of economics. earthquakes.(vi) Minimum wage law increases unemployment. In reality economics has developed along. {v) Effective steps should be taken to reduce income-inequalities in India. The normative statements. (vii) The share of the primary sectors in the national income of India has been declining. {viii) Ordinary business of life is affected enormously by tsunami. {vii) Government should stop minimum support price to the farmers. etc. in fact. The role of economist is not only to explain and explore as positive aspect but also to admire and condemn as negative aspect which is essential for healthy and rapid growth of economy. F^smvc Some examples of Economics as a normative science are : (/) Minimum wages should be guaranteed by the government in all economic activities. (b) Economics as a Normative Science A normative science is that science which refers to what ought to be} what ought to have happened} Normative economics deals with what ought to be or how the economic problems should be solved. Thus. [iv) Free education should be given to the poors. [Hi) Rich people should be taxed more. The statements which make assessment of activity and offer suggestions are called 14 Statistics for Economics-XI normative statements. That part of economics which deals with normative statements is called Normative Economics. Economics as positive science and normative science is inseparable. (vi) India should spend more money on defence. economics is both positive and normative science. Normative statements cannot be empirically verified. the bird flue. (vtii) Our education system should produce sufficient qualified and trained persons to the economy. . droughts. It makes an assessment of an activity and offers suggestions for that. both positive and normative lines. are the opinions of different persons relating to tightness or wrongness of a particular thing or policy. as it prescribes that cause of action which is desirable and necessary to achieve social goals. (//) India should not take loans from foreign countries. therefore government should check rise in prices. Define economics in the words of Alfred Marshall. Economics is. a science as well as an art. In order to achieve the objective of full availability of oil m India. What is Economics? y exercises i Explain the origin of word 'Economics'.In the followii^ examples first part of statement is positive giving facts and second i part IS normative based on value judgements. thus. It helps in solution of practical problems Art is the practical application of scientific principles. H) Indian economy is a developing economy. What are your reasons for studying Economics? How will you choose the wants to be satisfied? Give Adam Smith's definition of economics. i What is economic activity? Distinguish between non-economic and economic activities. ' We all know that there is oil shortage in India. the act should be honestly implemented. Economics is an art as it gives us practical guidance in solution to various economic problems. Sc ence lays down princip es while art puts these principles into practice. Make a Ust of economic activities that constitute the ordinary business of life. Explain welfare definition of economics.sposmve sconce We also know the govermnent aims at removing' oil shortage X information supplied by economics is normative science. (ii) A rise in the price of a commodity leads to a fall in demand of quantity of commodity. the government should make development through correct and proper planning. therefore. Which is the most accepted definition of economics? Give the definition. "Economics is the science of choice. (iti) Rent Control Act provides accommodation to the needy peoples. The information given by economics . the govermnent has followed the path of oil plaLng The path of planmng is an art as it implies practical application of knowledge with a view to achieve some specific objectives." Explain. So. B. . ECONOMICS AS AN ART Art IS practical application of knowledge for achieving some definite aim. we can say that economics is an art. 4.. technology etc) neLTnWr' new machines have been devdoS'tLt I f ff because man-whether Indtn ' ^^"^'^es of energy. plannW and X ^^^^ ^^ ^ progJsst tcTs . 6. machinery etc." Explain. Give argimient in favour of economics as a science. T? -^f^^ble. We ntw ^^ ^^e irrigation system. All this is possible thinking and reasoning which had evo^ given us civiIisation4he wtef ^P^^^^^es. systems. Discuss the nature of economics as a science. Chapter 2 introduction-meaning and scope Introduction What is Statistics? Functions of Statistics Importance of Statistics Limitations of Statistics Misuse of Statistics i introduction e—."Economics is about making choices in the presence of scarcity. Is economics a positive science or a normative science or both? Explain. '""I' ^^ ^^"^^^e-is gifted with f ^^ has electricity. 5. Is economics an art? Give reasons. better organisations for the comX bul^^^^^^^ All thic h^c .developments and chemistry. lin 3. medicine. "-"'npiex Dusiness and administration today . Is Economics a science or an art? Explain with reasons. How scarcity and choice go together? What is meant by economics? Economics is a science? Give reasons. Similarly. In Physics.."''' observes from his daily ex^elre whT'> ^^^ ^hich shop. makes very precise statement—he would say that roses have a sweet smell. He has to do it carefully since a wrong judgement can completely ruin him. he collects data (information) systematically. tasting and smelling—are taken to be reliable and then recorded (noted). In this respect. the television etc. This might be a conclusion based on impression people get from the newspaper reports of cases of theft and robbery. gets it organised in some logical or systematic way. The empirical methodolog^ consists Jf la^^^^ mformation. only those things that can be observed by our senses—seeing.^'e made Whether a common man buying vegetabks he looks t dS^lr^u^^^^^^^^^^ decision-makmg. Chemistry and Botany. This impression may or may not be true. How do we reach that conclusion? We all like its colour.demand. or ^rks out wLT. A shopkeeper decides to stock these . We can find out whether it is true or not only by comparing the number of cases of theft and robbery reported during one year with . the Introduction—Meaning and Scope \ 11 demand and supply. it is not a subjective or personal conclusion. this would be a subjective statement. While and then mentally calculates. analysing the information Th^ ^observations and collecting conclusions by fu^er^bserta^TsZs W al ZtZ' ^^^^ ^ knows it or not. hearing.wayfoftalSmnrp^^^^^^^ ^PP^'ed itself in'findmg and scientific man'ner. A scientist however. touching. he uses this method tof'""'. ^^ops P. But I say that I like the rose most of all the flowers. We all agree that the rose is beautiful. shape and above all its smell. analyses this data according to certain principles and draws conclusions. a'nd the pattern of demand and manufactures larle o^T "^^^ufacturer also observes or manufactures new items acco^SnSr^rthTZL^ ^^^ demand radio. it is possiWe trcXr T f "^^^^"^edia-the newspaper. people might say that theft and robbery have increased these days. Quantitative Data and Qualitative Data : An empirical investigation is an investigation where facts are collected through observation. A meZdo^X b'n^ things. The interview board discusses the comparative merit of the candidates and ranks them for final selection. internet or newspapers. the number of people in a state who are strict vegetarians. are known as 'Quantitative data'. heights or weights of students. We will also consider later the functions and hmitations of statistics. number of people in country are really poor-rich-middle class. When information or observations are recorded in numbers or quantity. prices of wheat during this week.the number of cases reported in other years. such as economics. income of individuals. everyday temperature. This judgement is not quantifiable.. For example. Statistics can be defined in two ways : (a) In plural sense. etc. intelligence of students. An investigator would collect such information from police records. we say we have quantified information. we will see that certain conditions must be fulfilled for a quantitative statement to be called statistics. music etc. The systematic h 12 vw 11 ^^'^tistics for Economics-Xl treatment of quantitative expression is known as 'Statistics'. channels. Supposing a selection for a post is to be made. e. number of people are illiterate who will not get jobs.g. management etc. do not always deal with what we call inherently measurable or quantifiable facts. number of highly educated and will have best job opportunities. This is necessary for preciseness of statement. appreciation of art. Not all quantitative expressions are statistics. It may be collected through questionnaire or opinion poll using landline or mobile telephone. preference of people viewing TV. beauty. candidates are interviewed. First. .. However. is smistics ? > It is necessary to have quantitative measurements even for things which are not basically quantifiable. It is not possible in certain cases to measure or quantify information. For example. Non-quantifiable/qualitative items can however be measured in percentages. sociology. some questions are put to them and their qualifications are taken into consideration. percentage of people watching TV. Social sciences. This information obtained in percentages is called 'Qualitative data'. not all information can be numerically expressed. let us understand what comes under the name Statistics. news in English or Hindi or other regional languages. it is based on impression. it is a group of observations. 70 and 30 are Statistics? Figures are innocent and do not speak anythmg. 300. 55. But "pocket expenses of Anil. Prakash. 80 and 70 respectively" are statistics. we find in newspapers ^ statistics of scores in a cricket match. 700 etc have a ' of statistical meaning.. "pocket expenses of Anil during a month is Rs 50" is not statistics." The above definition covers the following main points about statistics as numerical presentation of facts (Plural sense). i sin statistics of export and import etc. But when they refer to some place. 80. Let us look at the table given below : Students in Two Schools (2005-2006) Kendria Vidyalaya Govt. they are called statistics. i^nn'W^nir"* 'consider whether figures 1600. statistics of price. in this context the figures 1600. Similarly.g. Senior Secondary School Students Number Percentage Number Boys Girls 1600 400 80 20 700 300 Total 2000 100 Percentage 70 30 1000 100 The above table gives a numerical description of students in Kendria Vidyalaya and Govt Senior Secondary School. j ^ " martT^^'^Vl Statistics we mean aggregates of facts affected to ^ ^ marked extent by muhtphctty of causes numerically expressed. e.(b) In singular sense. we call this statistics of students. Statistics are aggregates of facts : A single observation is not statistics. enumerated or estimated according to reasonable standards of accuracy. statistics of agricultural production. 400. . Students are grouped as boys and girls and percentage is st^ calculated for each group. time etc. 700. Sunil and Suresh during a month are Rs 50.. collected in a systematic manner for a predetermmed purpose and placed in relation to each other. 20. Now. 400. person. They must be collected in a systematic manner. where the area of statistical enquiry is large. In such cases. For example. regionwise. singular sense. analyse and interpret them. electricity bills are affected by consumption and rate of electricity (c) Statistics are numencally expressed : Qualitative statements are not statistics unless A they are supported by numbers. they should be comparable periodwii>c. or influenced by a number of phenomena. experts make estimations on the basis of whatever data is available. there arises a need to organise. "all statistics are numerical statements of facts bui all numerical statements of facts are not statistics. if we want to collect statistics of agricultural production. When the above characteristics are not present numerical data cannot be called statistics. 30 second division. Statistical methods deal with these stages : PRESENTATION . it is not a statistical statement. 20 third division and 10 failed out of 100 students. we must decide before hand the regions.g." Statistics defined in singular sense (as a statistical method) : Statistics in its second. But sometimes. if we say that the students of a class colle. For example. if) Statistics are collected for a pre-determined purpose : Unless statistics are collected for a specific purpose they would be more or less useless. e. commoditywise etc. (d) Statistics are enumerated or estimated according to reasonable standard of accuracy: Enumeration means a precise and accurate numerical statement.{b) Statistics are affected to a marked extent by multiplicity of causes : Statistics are generally not isolated facts they are dependant on. refers to the methods adopted for scientific empirical studies. But when a statement reads as 40 students got first division. Therefore. (g) Statistics are placed in relation to each other : Statistical data J»re often required for comparisons. accurate enumeration may not be possible.. present. Introduction—Meaning and Scope 13 are very good in studies. commodities and periods for which they are required. Whenever a large amount of numerical data are collected. The degree of accuracy of estimates depends on the nature of enquiry. (e) Statistics are collected in a systematic manner : Statistics collected without any order and system are unreliable and inaccurate. it is a statistical statement expressed numerically. Thus. "Statistics may be defined as a science of collection. diagrams etc. analysis and interpretation of numerical data. First stage — collection of data refers to gather some statistical facts by different methods. graphs. On the basis of this conclusion certain decisions can be taken. ' (e) Interpretation of data : Interpretation of data implies the drawing of conclusions on the basis of the data analysed in the earlier stage. (c) Presentation of data : Data collected and organised are presented in some systematic manner to make statistical analysis easien The organised data can be presented : with the help of tables. classifying and tabulating. Stages of Statistical Study According to the figure. One has to go through the four stages to arrive at the final stage. The second stage is to organise the data so that collected information is easily intelligible.I Interpretation Statistics as Methodology According to Croxton and Cowden. Third stage of statistical study is presentation of data After collection and organisation the data are to be reproduced by various . (d) are Analysis of data : The next stage is the analysis of the presented data. presentation and analysis. This is the arrangement of data in a systematic order after editing. interpretation of data is the last stage in order to draw some conclusion.'" It Statistics for Economics-XI The above definition covers the following statistical tools : (a) Collection of data : This is the first step in a statistical study and is the foundation of statistical analysis. Therefore. organisation. data should be gathered with maximum care by the investigator himself or obtained from reliable pubHshed or unpublished sources (b) Organisation of data : Figures that are collected by an investigator need to be organised by editing. presentation. There large number of methods used for analy sing the data such as averages. they are — collection. dispersion correlation etc. diagrams. of stj Introduction—Meaning and Scope \5 2. For example. 4. analysed and interpreted. percentages etc Stati I resea Mars quan. conditions etc. periods. ^ In brief statistics is a method of taking decisions on the basis of numerical data properly collected. comparison of data of different regions. organised. are used for comparison. etc. so that different characteristics of data can easily be understood on the basis of their quality and uniformity. ratios. Statistics presents the facts in a definite form : This definiteness is achieved by stating conclusions in a numerical or quantitative form. 3.. graphs.niethods of presentation. percentages etc. presented. Fourth stage of statistical study is the analysis of data. relationship between sugarcane prices and sugar. . is helpful for drawing economic conclusions. Calculation of a value by different methods and tools for various purposes is made to arrive at the last stage of study viz interpretation of data. in i scie drai relai (( functions of statistics Following are the functions of statistics : 1. Statistics studies relationship : Correlation analysis is used to discover functional relationship between different phenomena. namely tables. for example. Statistics simplifies complex data : With the help of statistical methods a mass of data can be presented in such a manner that they become easy to understand. averages. the complex data may be presented as totals. For example.. relationship between supply and demand. Some of the statistical tools like averages. Statistics provides a technique of comparison : Comparison is an important function of statistics. whether demand increase affects the price.' on the basis of available statistics of past and present. Dcsai. production etc. Statistics help in finding the association between two or more attributes.C. such as Botany. importance of statistics The use of statistical method is so widespread that it has become a very important tool in affairs of the world. Cournot. !H If . Mahalanobis. Statistics helps in formulating policies : Many policies such as that of import.L.relationship between advertisement and sale. Medicine etc. Sociology. Dr V. Prof. For example. A. statistical data and techniques are useful. Bowley etc. Leon Walras. iiave cgntrmuieu aTOtWtne aeveiopmeiu ui lucuiclk-^t-----' of statistics. Some laws such as Malthus' theorj^ of population and Engel's law of family expenditure are based on statistics. Famous economists (like Augustin. 7. For example. are formed on the basis of statistics. association between literacy and unemployment. he might be interested to know the impact of today's investment on the national income in future which is possible with the knowledge of statistics. Vilfredo Pareto. Let us examine the importance of statistics in some fields relating to economics and business : {a) Statistics and Economics (b) Statistics and Economic Planning (c) Statistics and Business (d) Statistics and Government Statistics and Economics A number of economists have given a practical shape to statistical tools for economic research. ctc. association between innoculation and infection etc.C. It IS indispensable to fields of investigations especially in the sciences.) evolved a number of economic laws by quantitative and mathematical studies^ In India.. Alfred Marshall. It helps particularly in drawing research conclusions. P. 5. export. Economics. 6. Statistics helps in forecasting : Statistics also helps to predict the future behaviour of phenomena such as market situation for the future is predict^).K R V Rao R. for example. Statistics helps to test and formulate theories : When some theory is to be tested. can be tested by collecting and comparing the relevant data. wages. whether cigarette smoking causes cancer. Economist might be interested in predicting the changes in one economic factor due to the changes ir another factor. Edgeworth. It helps m formulation of economic policies. but there is no end to his desires and demands. Engel's law of family expenditure. the inductive method of economics is dependent upon statistical methods. As a result. A number of economic problems can easily be understood by the useTf tatistical tools.ril—T tools and the importance of ZnomV: rr' f" "T™" ^^^^^ . m the same way as the doctor uses stethoscope for diagnosis of a patient. New things are being invented today in all the sciences becauTo7the econoT^b^""" T^ • T ^^^^ ^ new laws in Statistical methods have made a contribution to the development of empirical side of economics.. No sooner does he consume one thing. which depend on his income.. then on comforts and luxuries.^ Let us unLstand the importance of statistics keeping in view the various parts of economics oTrh!" consumption : Every individual needs a certain number 7 K' necessities. We discover how staS^" T"^' consumption. economics is TT'' are the tools and appliances of his laboratory.16 Statistics for Economics-XI . Malthus theory of population etc. a new science has evolved which is called Econometrics the Z" ^r' have evolved due to statistical analysis in the f eld of economics. The Td/vT r T rt^ ^^e taxaWe liability of individuals and their standard of living (b) Statistics and the study of production : The progress of production every year can .^ relanonLp amo^ Tecorm^rA ? of mathematics andLtistic! m economics. he desires to obtain the other. It gives statement of facts. The success of a plan is dependent upon sufficient and accurate statistical data available at all these stages.g. There are a number of problems of underdeveloped countries. The comparative study of the prod^Lty of various elements of production (e. lack of industries. have to make bricks. Various problems arise di^I to o7lttfc7datr " ^^ ^ with"hrhelp et/eo. Planning without statistics is a leap in the dark. . The statistics of production are ver^ helpful or ad^stment of demand and supply. statistics is useful in the various fields of economics. Every phase in planning—drawing a plan. evolution of economic laws and helps in economic planning. These problems can be fully viewed and understood only by getting the actual figures for different areas.-^onal and international demand. execution and review is based on statistics. direction to solve problems.g.other econormst. over population. land. ^ wuucuou fn^ttirf = ^^ . P^r ^^^^ ^^ competition afd demand o ^mmodity m a market. both human and natural. lack of agricukural development. The law of price determination and cost price which are : (d) Statistics and the study of distribution : Statistics are helpful in calculation of national income in the field of distribution. The comparison of the stage of development of one country with other is possible only with the availability of statistical data. labour.easily be measured by statistics. these models are helpful in solving economic problems. Statistical methods are used in solving the problem of the dismbution of national income. capital and enterprise) is also done with the help of statistics. Similarly. e. lack of education etc." -Marshall Introduction—Meaning and Scope 1^ Thus. On this basis we can call economics a Science of Human welfare and statistics as an Arithmetic of Human welfare.. Statistics and Economic Planning Statistics is the most important tool in economic planning. Economic laws in the modern economic world are based on mathematics and statistics which help to form econometric models. Economic planning is the best use of national resources.. Every developed country executes L census of production with a view to make a comparative study of various fields of production and economic planning. A producer needs statistics for deciding the cost of . of .ea. planning without statistics is a ship without radar and compass. For .general review of progress in all fields of economic development needs the help of statistical data and statistical methods. Statistics and Business Planning Business activities can be classified as under : f" L BUSINESS 1 I Internal Wholesale Retail International Import Export i of Trade ^-1-1 t V Banking Transport Insurance Wterehousing Packmg Advertisement Mf 18 vw 11 ^^'^tistics for Economics-Xl of statmical method have to be followed The it the producer to f„ the pnces of steps ^o^rd""' method.ta„st. Priorities of expenditure of a national budget can be determined through the comparative study of past performances with the present. '' celling activities. profttable trade he must know what the ZoZ. Thus. the ministry plans and makes policy to import oil in 2010.s required where statistical toTls tt iX-b^^ i hfe e^rrtt^r that is ! what proportion of the.or functioTstfinterpretation of material control. Tfential analys^p tz: p":^-lt:tsere 717 ^ ^^ ^ Statistical tools of collection. etc. trade etc. unemployment. classification L uncertainties.mod.ght fix for the same.. Similarly he ^st t ^-We what amounts h. This . poverty disinvestment.t. which depends on expected oil production by domestic sources and likely demand for oil for the year 2010. depending upon the accuracy of the statistical law. cost control. fmanc aTclro^f T"'' " ^nd so on. taxes. price control.mportant or demand e„po„ for var. so that there is no obstacle for the promotion of economic development. Above all the ministry of planning takes into account statistics of various fields of economy. statistical techniques are used to analyse economic problems of country. data are used in all ma. Sometimes to make plans and policies. planners require the knowledge of future trend. capS decde' i payments of matured policies ^ Proportion kept ready for n>. wealth.es and at a^d he has to forecast when the dLaXodd Ll": °nf "Tf of reserves he must have. This is very .s deposttors. In fact no mod^Xanirr °° they without analysis of the complex faJ^rthrLr"" ^ business analysis statistical tools are afeolt. Thus. otherwise his bak wol fT^oT rh transaction . viz. Such policies may develop the economic status of the people and the nation. personnel management Statistics and Government Introduction—Meaning and Scope \9 like that of crimes.. budgetary control.would last. The policy of family planning can be made effective in controlling the population of country. Keeping in view the increase of global price rise. ' control.o„s co. Statistics is indispensable for all important functions of the ministries of the state. It does not study individuals : Statistics is the study of mass data and deals with aggregates of facts which are ultimately reduced to a single value for analysis. Statistics can be misused : Statistics is liable to be misused..000. ^ • . It is necessary to know the misuses and limitations of statistics. . 20 h (Si. is a statistical statement. f'i umitations of statistics Statistics is very widely used in all sciences but it is not without limitations. 3. etc. Individual values of the observation have no specific importance. This helps the planners to make the production policy for the future. e. In short. For example.trend could be based on the data of past years or recent years. The required data can be obtained by surveys.g. say Rs 1. 2. QuaHtative phenomena.. It does not study the qualitative aspect of a problem : The most important condition of statistical study is that the subject of investigation and inquiry should be capable of being quantitatively measured. honesty. The following are the limitations of statistics. does not convey statistical meaning while the average income of 100 families say Rs 400. statistical tools are of maximum utility in the governance of state and formulation of various economic policies. poverty. The results obtained can be manipulated according to one's own interests and such manipulated results can mislead the community. production poHcy of 2010 depends on the consumption recorded in past years and recent years which decides the expected level of consumption in 2010. The ultimate results obtained by statistical analysis are true under certain circumstances only. 4. intelligence. the income of a family is. For example. They are true on an average because the results are affected by a large number of causes. The Ministry of Finance is responsible for preparing the annual budget of the country for which reliable statistical data of revenue and expenditure is necessary. ^t^tistics for Economics-XI . physics and mathematics. 1. Statistical laws are true only on an average : Laws of statistics are not universally applicable Hke the laws of chemistry. cannot be studied in statistics unless these attributes are expressed in terms of numerals. 4.sh between qualitatrve and quantitative data 3 Dto Tr-characteristics? 4 M rr - above and plural sense obrva'Sn'" cr™"'" -d never with a single llT"" '' counting..D.. the truth with the help of his exercises 1.' " " ^^^ ^^^ tools for grinding their owTa^sf b^ student of commerce and economic ...observations of mass data nXkrT L ^ased on to rectify them. Therefor^ it^eslnreT" " -.support knowledge of statistics..... ■ . 6. hou^ 1 the This generally takes place at the time of sefectinZm^^^^^ a and interpreting analysis of data^ "^^^ing comparisons and^^eS-pJT~ . 7.re m ^ misuse of statistics statistics by deliberately twisting or man pulat^ne^ ^^ ^--e be mterpreted by a lawyer to prove ^^ZTltTl^^. 8.stm8„.ca." Discuss..... 5. states are a fa/. Introduction—Meaning and Scope ... Illustrate with two examples. (ii) . Explain its utiHty in the field of economic planning. Explain briefly : (a) Statistics." In the light ot this statement explain the importance of Statistics as an effective aid to national planning. (b) Statistical methods. comment. J (i) Statistics is of no use to economics without data. administrators and educationists. 19. "Statistical Analysis is of vital importance for successful businessmen. "The Government and poHcy maker use statistical data to formulate suitable policies of economic development".21 16. . Statistical methods are no substitute for common sense.u "Planning without Statistics is a ship without radar and compass. Explain the relationship between Economics and Statistics and discuss how far it is correct to say that the science of economics is becoming statistical in its method." Discuss with illustrations. 20. Write notes on : (a) Importance of statistics in modern economic set up. {b) Statistics in economic analysis. 17. "Statistics in these days is indispensable for dealing with socio-economic problems". How far is this statement true? What is the importance of Statistics in modern economic set up? Explain giving examples. (d) Statistics in economic analysis. (c) Statistical data." Explain this statement in about 200 words. Define Statistics. economists. "Statistical thinking is as necessary for efficient citizenship as the ability to read and write. 18. Discuss with illustration the importance of Statistics in the solution of social and economic problems. Mark the following statements as true or false. i. knowledge or information. statisti s-XI Collection of Primary and Secondary Data what is a statistical enquiry ? Enquiry means a search for truth. in other words the first and the foremost condition for the answer to the questions in statistical enquiry should be quantitative. Statistical enquiry therefore means a search conducted by statistical methods. The application of a statistical technique is possible when the questions are answerable in figures (quantity). for instance : .Statistics can only deal with quantitative data. There are different subjects on this earth. (Hi) Statistics solves Economic problems. some are described by the degree of expression (quality) and some by the degree of figures or magnitudes (quantity). UNIT 2 }1: If K ^lecton and organisation of data 12 « CoUecAion of Primary and S p Organisation of Uata « Presentation of Data Chapter 3 collection of primary and secondary data What is a Statistical Enquiry? Sources of Data g Primary and Secondary Data Drafting the Questionnaire Methods of Collecting Primary Data Census and Sample Surveys Sample Surveys Methods of Sampling Random Sampling Non-Random Sampling Advantages of Sampling Reliability of Sample Data How Secondary Data is Collected? Some Important Sources of Secondary Data Census of India National Sample Survey Organisation (NSSO) COI cer are coIJ Sourct w other. 5 2003-04 36.3 2001-02 31. statistical enquiry means statistical investigation or statistical survey. Weight of students measured in kg. Economic Survey 2004-05. Income of families measured in rupees.. is different from one year to other.5 2000-01 30. Let us observe the following table.Profit of firms measured in rupees. and respondents are those from whom the statistical information is collected. viz.1 2002-03 34. one who conducts this type of enquiry is called an investigator. Questions that can be answered in quantity lies within the purview of statistics. which cannot be answered through statistical methods. Intelligence measured in marks obtained by students in a particular test.8 1990-91 13. there are questions like—How great was Jawaharlal Nehru? How brave was Bhagat Singh? etc. They are not same.0 1980-81 6. But. they are known as enumerators. We observe that the production of finished steel in India. TABLE 1 Production of Finished Steel in India (in Million Tons) fc^ year Production 1950-51 1. Survey is a method of collecting information from individuals.. They varies from year to year. They are called Variables . What is the average production of rice per acre in India? What is the total population of India? How many students are there in a class? Thus.9 Source : Government of India. The investigator needs the help of certain persons to collect information. Age of students measured in years. The finished steel production in ti i-. What is the source of data? 2. Statistics for Economics-XI V V tu f by x-variable and the production of finished From the followmg text. primary data is collected originally and secondary data is collected through other sources. The same data is primary in one hand and secondary in the other. and record the information from the salary register or.g. sources of data BefoT^n' Z" ^^ '^^ta. for the factory it is a primary source. you may have the other choice that of visiting the factory accounts department. Y or Z variables. Primary data is first hand information for a particular statistical enquiry while the same data is second hand information for an another enquiry. This is secondary source for an investigator but. Thus. By which method of survey is data collected . any Government publication is first hand (Primary) for Government and second hand (Secondary) for . e. How do we collect data? 3.in statistics which is represented as X. may gather this information from the published report of the factory about the payment of wages.. 24 . This is the first stage in statistics! SOURCES OF DATA dary ^ Governmern departments RaiZr^cf ' ^^ ^reparmg^at • cantr^ine??""'"'" external data whij ^^ary and secondary ' Collection of Primary and Secondary Data 25 But. we will understand : 1. Sometimes the questionnaire is managed by the respondents also. PIHMARY DATA-* PUBLISHED How Primary Data is Collected The most popular and common tool is questionnaire/interview schedule to collect the primary data. lesser the possibiUty of response.7en questionnaire. Therefore. a website which saves time and cost.26 Statistics for Economics-XI . investigators. researchers or trained. MIMTim Following are the basic principles of drafting questionnaire : (1) Covering letter : The person conducting the survey must introduce himself and make the aims and objectives of the enquiry clear to the informant. More the number of questions. The informant should be taken into confidence. The questionnaire is managed by the enumerator. for example.nstr„c.. secondary data can be obtained either from published sources or from any other source. normally \l (I ■• I-J Ni m. | investigator. A selfaddressed and stamped envelope should be enclosed for the convenience of the informant to return the questionnaire. numbered for the convenience of the informant and the ''' sraetsr^s r— They .a research worker. (2) Number of questions : The informant should be made comfortable by asking minimum number of questions based on the objectives and scope of enquiry. Thus. He should be assured that his answers will be kept confidential and he will not be solicited after he fills up the questionnaire. A personal letter can be enclosed indicating the purposes and aims of enquiry.ons about units of measutement shol b^ g. (Pu. such questi-ons should bet Smlt ^^l^rrt^^ ^^^^^^^^^^ Collection of Primary and Secondary Data A SPECIMEN QUESTIONNAIRE 27 H.. a cross) (.)Vrd>x n (f) Any other q Sd fnTu^f: 'ir^'s'ts^. are answerable m'Yes' or 'No' or 'Right' or Are you married? Yes/No Are you employed? Yes/No should start from general These questionsleXle aS In which class do you read? In which subject you are more interested? 'rr ^-ooses.."''ject.^:' ^^^-omd be ■Wrong'.g. S.^'"dents in Universi^- W How will you solve the wage problem in your mdustry? work.g.the mformant should ^ abrtr^ve Ae aZ^f "" 'TT' ■".I Hit: 28 1 Statistics for Economics-xi Example : ml" "" " "f P".) English p M Punjabi □ iit.. e.ve. For this the blank space. e. Such . ® ^y "smg a tick-mark in WUch of the folloJng languages you use most for uniting. 8. We^wmg are the methods of primary data collection which a« in common use \ Collection of Primary and Secondary Data COLLECTION OF DAW of 29 r lARY SECONDARY ►Direct Personal Interview ►Indirect Personal Interview ►Telephone Interview •^Information from Correspondents ►Mailed Questionnaires ►Questionnaires Filled by Enumerators 1 Published Sources -1 Unpublished Bourns .(a) Which brand of tea do you take? (b) Why do you prefer it? (16) T .ssn'r btk" (b) Do you love your children? (c) Do you beat your wife? t iiethods of collectiiig priiiiiary data etc. enq in a not Mer 7. pleasing and should not be biased. It cannot be used when field of enquiry is wide. Market Reviews and Reports etc and collect the desired information. 2. accurate. Original data are collected by this method.»Government Publications ► Publications of Internal Bodies ►Semi-official Publications Report of Committees and Commissions —> Private Publications (a) Journals and Newspapers (b) Research Institutions (c) Professional Trade Bodies (d) Annua! Reports of Joint Stock Companies (e) Articles. Merits : 1. .. This method can be used if the field of enquiry is small. 3 The required information can be properly obtained. 8'. In the same way one can think of personal Imryof collection of information regarding family budget and living conto Ta group area. tactful. Investigators can use the language of communication according to the educational standard and attitude of the informant. 7 Informants' reactions to questions can be properly studied.. There is uniformity in collection of data. 4 There is flexibility in the enquiry as the investigator is personally present. Since the enquiry is intensive and m person. J'" SSf m '30 . The investigator must be skilled.. 5" Information can be obtained easily from the informants by a personal interview. the results obtained are normally reliable and accurate. 6. Limitations : 1 j u „ 1. exit poll. banks etc. television channels. Various researchers. Personal bias can give wrong results ""erwise 5. This method is lengthy and complex ifSifMlSi m-smrnrn Merits : Kiai^ obtained from the third party. (m) Telephone interview : The investigator asks questions over landhne telephone. These days online surveys through Short Message Service. the data collected by one method with the other. Even sometimes website or internet are used for obtaining statistical data. e. it is more or less free froJ 3 of the investigator and the inforr^ant I 3.Statistics for Economics-X 2. newspapers. mobile service providers. 3.. For this reason the choice of the method depends on the nature of enquiry and sometimes we balance the demerits of one method by '. It is costly method and consume more time.' Limitations : ar in to oh G( in( wt Mt Lin s-XI Collection of Primary and Secondary Data Thus. This way we can counter chec'. we find that both the above methods—direct and indirect personal interviews — have certain plus and minus points. SMS has become popular.. time and money H iLtJ™"'"" """ ' i of P'oHems can properl. political or economical opinions.e. Merits : . i.tsing the other method also for the same investigation. It saves labour. music or dance performance opinion etc. mobile telephone and even through website.g. use telephone service to get information from different people.. It can cover a wide area under investigation. They collect the information according to their own judgements and own methods. by this method. wholesale price index number. These agents regularly supply the information to the central office or investigator. Limitations : 1. Telephone interviews are cheaper than personal interviews. 2. Sometimes respondents are reluctant to answer some questions in personal interviews. local agents or correspondents are appointed in different parts of the investigation area.1. thefts. It gives results easily and promptly. It is suitable when the information is to be obtained from a wide area and where a high degree of accuracy is not required. This method is comparatively cheap. accidents etc. 2. but it sometimes becomes helpful in obtaining information from respondents. It is adopted by Government departments to get estimates of agricultural crops and the. 3. Radio and newspaper agencies generally obtain information about strikes. 32 Statistics for Economics-XI f C r fij . 3. Limitations : 1. The investigator can assist the respondent by clarifying the questions. 2. 4. In this method original data is not obtained. Merits : 1. (IV) Information firom correspondents : In this method. Reactions of respondents on certain issues cannot be judged. Telephone interviews are better in such cases. It gives approximate and rough results. It can be conducted in a shorter period of time. 2. Information cannot be obtained from people who do not have their own telephones. his personal b^as may affect the accuracy of the information sent. .Jess than the cost 3. ^ ^ regularly to the (b) This method can be successful when the informants are educated. ^ nn^'^'TT correspondents and agents may mcrease errors. misinterpret or may not understand'some There may be delays in getting replies to the questionnaires 4. o^p^eLri 1 ~ ^^ ^^ - -. As the correspondent uses his own judgement. There may be loss of questionnaires in mail.. This method is suitable for the following situations • cTmpef LrS? '' " questionnaire. We can obtain original data by this method Limitations : faift?"" ^^^ ^^^ informants.3. inf( The be j org. They may ques^r 3. understood and answered 1 ' possibility of getting wrong results due to partial responses. and the high Mer Limit 1 3. Government agencies compel bank and companies etc. ms method can be used only when the informants are educated or hterate so that ^ they return the questionnaires duly read. the mo " " ^^^ ^^^^ kept con^'entkllt ' Merits : ^^ ^his method in cases where informants are spread over a wide geographical area. and those IrmrnTre^uir ^^ ^^ ^ ^^ ^^^ ^^e splcm: 6. to supply information regularlv to the Government in a prescribed form. rmants with standardised questionnaires wl.ich are to be fiUed^jn ^e im^estigator helps the informants in recording their answers. -ui (b) Informants should not be required to spend for posting the questionnaires back therefore. Generally. (VI) Ouestionnaire filled by enumerators : Mailed questionnaire method poses a tanber oi difficulties in collection of data. inadequate and unrepresentative. prepaid postage stamp should be affixed. S The second alternative approach is to send trained investigators^or enumeratoi^m M. This is the most common method used by research iSons. Merits : . Thus. (e) The language of the schedule should be polite and should not hurt the sentiments of the informants. ic) This method should be used in a large sample or wide universe. these filled questionnaires received to incomplete. to get Proper answers to Ac questions put to them. (d) This method is preferred in such enquiries where it is compulsory by law to till the schedule. 'Ki 33 Collection of Primary and Secondary Data Following are some suggestions for making this method more effective and successful. (a) Questions should be simple and easy so that the informants may not find it a m burden to answer them.4. there is little risk of non-response. The invest^a^rs shoidd i honest tactful and painstaking. They train investigators properly specifically for the purpose of an enqu^ ^d also tram them in dealing with different persons tactfu ly. 5. The statistical information collected under this method is highly reliable. Without good interview and proper traming. P 34 Klot Suiwey or Pre-Te« • p ^f^^stics for Economics-Xi a Pre-test or a guidihg survl. Limitations : ^ n■^ 1. most ■ of the collected information is vague and may lead to wrong conclusions. . 4.. It needs a good battery of investigators to cover the wide area of universe and therefore it can be used by bigger organisations. " ^ -^uc. As the information is collected by trained and experienced enumerators. 5. it is reasonably accurate and reUable. If. Personal presence of investigator assured complete response and respondents can be persuaded to give the answers to the questionnaire. 5 This method can be adopted in those cases also where the informants are illiterate. It is an expensive method as compared to other methods of primary collection of data.u 3' True and reliable answer to difficult questions can be obtained through ■ establishment of personal contact between the enumerator and the informant. . . 4" Enumerators need to be trained. 2. This is done to try out the auetlr before starting th. as the enumerators are required to be paid. 2 The results are not affected by personal bias. 3 This method needs the supervision of investigators and enumerators.1. This method is time consuming since the enumerator is required to visit people spread out over a wide area.-. 6'. It can cover a wide area. mam survey. I. ght willbe CX)VERNMENT SENIOR SECONDARY Ii f( P O £ S< UI gro' Sun 20 Source s-XI Collection of Primary and Secondary Data .enr „ (w) Training of field staff " rX" -- ^ namr. SupposeTe'^are 500 1 "T'" <>' School.the general mformation about L po^Ja"^^^^ -thods for obtaL the pilot survey helps in : ^ ^e sampled.hVlorplZL ekcon termed as sampling.proje.e^VS^td^^.. 'he ei™ needed for or fan. 0-t.—^ " — Have a Cear n„dersra„d.n. [i) Estimating the eosr of ^ ■ avaihbiii.. A par. The information supplied b.he ten.ons and a. of . If we want to know the average wekta Secondary W'll get the mformation abont all the fLrhnnd f each girl and obtamed by dividing the total weigte of he tn' " we.„ .so .He . Population and Sample o all .® of fc "" rr. casus MID SAIWnE SURVEVS ia) Census method/Census Survey. and (b) Sample method/Sample Survey . „. 41 27. collecting weights of all the 500 girls in Senior Secondary School is census method of collection where no student is left over.80 22.. life expectancy and composition of population etc.89 12. In above example.8 86.85 13. Let us review the following census data in the following Table no.1 1911 2..8 1931 3.35 24. Most recently population census in India was carried out in February.54 27...9 89... The data relating to estimation of the total area under principal crops in India are obtained by using village records maintained regularly by Patwari. 2 regarding relative growth of Urban and Rural Population in India obtained from Reports and Economic Survey 2002-2003.58 21. 2. 2001 by house to house enquiry to cover all households in India. In this method every element of population is included in the investigation. as each student is a unit..62 25. literacy. TABLE2 Relative Growth of Urban and Rural Population in India r-..83 10..2 Rumi PvpuUition .. In this case weight of 50 girls is the sample.32 25..3 89.12 11.9 1941 4.44 31. The average of 50 girls reasonably be representative of average weight of 500 girls. when we make a complete enumeration of all items in population.. The population census is carried put once in every ten years in India.25 23.. Census Surveys The objective of a census method or complete enumeration is to collect information for each and every unit of the population/universe.1 87.7 1921 2... it is known as 'Census Method" or 'Method of Complete Enumeration'. Year i f r UrhaiP Popuiatinn {tn itorpi) Rural PopttUuioti (m rmrei) Total Ptipuldtion (m Lrine») As Perceraage of Total Popukttidn Urban popuhtion 1901 2. are published by Registrar General of India.by taking only 50 girls out of 500 and obtain the average of this part of the total population..2 88.21 10.59 22. work force. 3. Demographic data obtained by census method on death rates and birth rates. Thus. Following are few examples of census : 1. 11 17.11 68.50 74.63 25.91 18.76 62.3 82.89 43.3 1991 21.8 Source : Census Reports and Economic Survey 2002-2003.^ mcrease of 2 1 per cent in the growth rate of urban population m the decade ending 2001 over the decade^nding .87 36. if:.or uian areas T ' Population of around 24 crores lived in urban areas The urban population formed about 11 per cent and rural population 89 per r^om I urban population had gone up to around 28 per cen n 2001 while still over 72 per cent people lived m rural areas. 2001 28.ea.7 76. In 2001 74 2 crore persons.74 crore persons.2- 102.7 27.9 80.24 29.7 crore total population lived in around 5 5 lakhs' i: ifo^lVt 2 r urban a. The decadal growth at frru^d in1he rlr" T -.02 43. out of about 102.0 1971 10.2 72. iti I7 I (1 36 Statistics for Economics-XI 74 " 'rr ^^ Of India's population.89 36. The above table show the relative growth of rural and urban population m India since 1901.1951 6.33 23.22 52.87 84.0 82. Th - -.1 1981 16.93 54.82 19. The net addition to rural population between 1991-2001 was 1133 crore while urban population increased by 6.7 1961 7.7 74:3 . SAMPLE SURVEYS We may study a sample drawn from the large population and if that sample is adequate representative of the population. MiTHODS OF SAMPLING i Random Sampling (a) -1 .• {a) We look at a handful of gram to evaluate the quality of wheat. and (b) Non-Random Sampling. various methods of sampling can be grouped under mam (a) Random Sampling.^ — — ^^lected from^ a ~ methods of sampling ^^^Broadly speaking. In above example. collecting the weights of 50 girls out of 500 girls m Semor Secondary School is sample method of collectiol In this method ew students as sample considered for our study.-.) Th^^^elevision network provides election coverage by exit polls and prediction is nnnT"'' T ""V^^^'^^ical termmology population or universe does not mean the total numbe of people m an area. metnod tew Following are a few common examples of samplin. it means the total number of observations or terns fn r::att. Collection of Primary and Secondary Data 37 Let us discuss now the various samphng methods which are popularly used in practice. rice or pulses. etc A ^^^ '^^^ ^P^" ^1-tric bulbs out of each lot" [c) A drop of blood is tested for diseases like malaria or typhoid etc ^ ^ fudtrnfof^'tC ^^^ » P-^-ion for final (. we should be able to arrive at val 7corcSn Method of collecting of data. Non-Random Sampling ib) Simple or Unrestricted Random Sampling Restricted Random Sampling (f) Stratified Sampling (//) Systematic Sampling or Quasi Random Sampling (f/f) Cluster Sampling or Multi-stage Sampling (a) (to) (c) Judgement Sampling Quota Sampling Convenience Sampling random sampling Random Sampling is one where the individual units (samples) are selected at random. It is called as probability sampling. Random sampling does not mean unsystematic selection of units. It means the chances of each item of the universe being included in the sample is equal. The term 'Random Sampling' here is not used to describe the data in the sample but it refers to the process used for selecting the sample. Following are the methods of random sampling. Simple or Unrestricted Random Sampling This method is also known as simple random sampling. In this method the selection of item is not determined by the investigator but the process used to select the terms of the sample decides the chances of selection. Each item of the universe has an equal chance of being included in the sample. It is free from discrimination and human judgement. Random sampling is the scientific procedure of obtaining a sample from the given population. It depends on the law of probability which decides the inclusion of items in a sample. To ensure randomness, mechanical devices are used. There are t^vo methods ot obtaining the simple random sample. They are : (a) Lottery Method, and (b) Table of Random Numbers. (a) Lottery Method : A random sample can generally be selected by this simple and popular method. All the items of the universe are numbered and these numbers are written on identical pieces of paper (slip). They are mixed in a bowl and then there starts the selection by draw one by one by shaking the bowl before every draw The numbers are picked out blind folded. All slips must be identical in size, shape and colour to avoid the biased selection. IMH' 38 Statistics for Economics-XI metal pieces on which nuZ^tT Th! d "" "--^en or device and each time one piece comesrotated by a mechanical of digits, for instance if the numbe'Ts ZZ This^i^ IS us. m drawi::^^-^^:; u '^^^ ""^ber -- -ry large the above procedures if the disks, balls or slips L not XrouThnf' ^^ T^^^^^ " been a marked tendency to usetSroTrindT^^^^ > T T' " ^^^^ ^as samples. A table of random digkst simply a the purpose of drawing such , by a random process. The follLing of Som ^gS tt^^^^^^^^^^^^ ~ (.) Tippet. Random Sampling Numbers. There are 10^00 numbe^t^anged 4 digits MG. Kendall and Babington Smith's Random Sampling Numbers, having 1 lakh ic) Rand Corporation's a million random digits (d) Snedecor's 10000 random numbers. ie) Fisher and Yates Table having 15000 digits Rc 2952 3170 7203 3408 0S60 Tippett Numbers 6641 3992 9792 5624 4167 9524 • 5356 1300 2693 2762 3563 1069 5246 1112 9025 ho Th hoi ave strs ;san [peo 7969 5911 1545 1396 2370 7483 5913 7691 6608 8126 college. We will first nuX aVMOo Tui" f ''"t"'' ^ students, now we will cons J a pag^of ?fp i'" ^^^ ""'"''"ing the 15 successive number either horLLhy or ^^a^ "" ^^ Merits ' w a universe. There are less selected. ^ bas equal chance of being Regularity begin to operate ^^^ ^aw of Statistical ( [Meri 1. 2. 3. 4. 5. 6. s-XI Collection of Primary and Secondary Data 3. This method is economical as it saves time, money and labour in investigating a population. 4. The theory of probability is applicable, if the sample is random. 5. Sampling error can be measured. Demerits 1. This requires complete list of population but up-to-date lists are not available in many enquiries. 2. If the size of the sample is small, then it will not be a representative of a population. 3. When the distribution between items is very large, this method cannot be used. 4. The numbering of units and the preparation of the slips is quite time consuming and not economical particularly if the population is large. t Restricted Random Sampling They are as follows : (t) Stratified random sampling : In this method the universe is divided into strata or homogeneous groups and an equal sample is drawn from each stratum or layer at random. This method is therefore useful when the population of the universe is not fully homogeneous. For example, suppose we want to know how much pocket money an average university student gets every month will be taken equal sample from various strata, namely : B.A. students, M.A. students and Ph.D. students etc. Stratified random sampling is widely used in market research and opinion polls, it is fairly easy to classify people into occupational, economic, social, religious and other strata. There are different types of stratified sampling {a) Proportional stratified sampling is one in which the items are taken from each stratum in the proportion of the units of the stratum to the total population. (b) Disproportionate stratified sampling is one in which units in equal numbers are taken from each stratum irrespective of its size. (c) Stratified weighted sampling is one where units are taken in equal number from each stratum, but weights are given to different strata on* the basis of their size. Merits 1. The sample taken under this method is more representative of the universe as it has been taken from different groups of universe. 2. It ensures greater accuracy as each group (stratum) is so formed that it consists of uniform or homogeneous items. 3. It is easy to administer as universe is sub-divided. 4. Greater geographical concentration reduces the time and expenses. 5. For non-homogeneous population, it is more reliable. 6. When original population is not normal (skewed), this method is appropriate. Statistics for Economics-XI V\L N m. 40 Demerits 1. Stratified sampHng is not possible unless some mformation concerning ti population and its strata is available. concerning u 2. If proper stratification is not done the sample will have an effect of bias. If differ, strata of population overlap such a sample will not be a representative one («) Systematic sampling or quasi-random sampling : Systematic sampling is a simo by preparing this list m some random order, for example, alphabetical order SMnlr U the list, « stands for any numl Suppose we have a universe of 10,000 items and we want a sample of 1000, then ^^^ « - 10 The method of selecting the first item from the list is to decide at random f^^t ?hen th "r "'Tu' Suppose we pTck up Z Z t Then the other items will be 15th, 25th, 35th, and so on unSl we have got oVr fuH sal fullv rw'' T I u" " that the list of the univers! fully random and that there are no inherent periodicities in the list. Merits 1. It ^yystematic, very simple, convenient and checking can also be done quickl] 2. In this method time and work is reduced much. 3. The results are also found to be generally satisfactory. Demerits random will not be a determming factor in the selection of a sample. 2. It IS feasible only if the units are systematically managed 3. If the universe is arranged in wrong manner, the results will be misleading | to divL and sub " ^^ to divide and sub-divide a universe according to its characteristics. Thus if a survev ki be conducted in a country it will first be divided into zones or states l region t^^^^^ mailer units cities towns and villages and then into localities and hLseToW; At Jd non-ranoom sampun6 ccessi 3n No. Date.^............ s-XI the :rent tipler on is ieved mber. : take from item, mple. irse is ickly. !on as It m e have y is to ;n into .t each nethoc the hsl of the s, non-ing are Collection of Primary and Secondary Data (a) Judgement or purposive sampHng (b) Quota samphng (c) Convenience sampHng 'n Judgement Sampling \ This is also called purposive or deliberate sampling. In this method individual items of sampling are selected by the investigator consciously using his judgement. Therefore, it requires that the investigator should have a good knowledge of the universe and some experience in the field of investigation. Obviously, the choice of samples will vary from one investigator to another. For example, from a universe of 10,000 ladies who use a particular brand of hairdye, the investigator will select a sample of say, 1,000. His choice of this sample will be such that it is irrespective of the universe. For this an exercise oi judgement is required. In order for the judgement sampling to be reliable, it should be free from individual lies or prejudice. Since the choice of sample is not based on probability it does not guarantee accuracy and it makes detecting of sampling errors difficult. However, this methods is useful in solving a number of kinds of problems in universe and economics. The purposive or judgement sampling is suitable in the following conditions : (a) The number of items in the universe is small to which some items of important characteristics are likely to be left out. (b) When small sized sample is to be drawn. (c) When some known characteristics of the universe are to be intensively studied. (d) It is also appropriate for pilot survey. Quota Sampling It is a method of sampling that saves time and cost and is commonly used m surveys of political, religious and social opinion.- Interviewers are allotted definite quotas of the universe and they are required to interview a certain number from their quota. Quotas are decided on the basis of the proportion of persons in various categories. In other words, the investigator is given instructions about how many interviews should be taken say in a given localitv and what proportion should be from say upper, middle and lower mcome groups, as by some other classification which is predetermined. For example, for a study of truancy (running away) from school in Delhi the investigators are allotted quotas of say 10 schools each out of which two should be public schools (Boys), one public school (Girls), three Boys' Senior Secondary Schools, two Girls' Senior Secondary Schools, two Co-education Schools and from each school he is asked to interview 50 students, taking 10 students each from Classes VIII, IX, X, XI and XII. The interviewer can select any 10 students according to his own judgement. It is a kind of judgement samphng and provides satisfactory results only when interviewers are carefully trained and personal prejudice is kept out of the-process of selection. ' ' hi !i< 42 , Convenience Sampling Statistics for Economics-XI; P0.1:;: f'^intfri:- l^essr ~ -ce ^ example, for the study of truanrvTr a school or schools in I neiS^ ^^e basis of convenience F. ^^^^^ ^^e invesdgat™selec schools. This method is used wLn^e " .V ^^ ^^ convenient for hL to g^trthe not clear or complete source hst is t^a^lbl" T ^^e sample unit i easily available lists, such as teleDhLTW may be obtained W results obtained by' this m^aTntl^^^^^^ unsatisfactory. ^^^ ^ruly representatives of the universe and are ^^HIAGESOF (I • of dae. bee. Jo, getting quick results. ^ therefore, sampling is very useful in sxhris fre^ir" —a, . for fc cV^ "^dr m - « method. 2 in some ways more reliable than cenLs aliow a samplmg mefcd t « fc o„,, pos.b,e or E ~ or bote, ma„ufaeLd"« fcTar:^;-,,*^ ^dl"' sampling method. '' P^^^'^le due to the scientific nature of » appropriate «e,d . neceiary^Srfctr:?"^^^^^^^^ le otl: Thi the larf Statistic The means a Collection of Primary and Secondary Data 43 ^^i^lity of sample data ; The main purpose of sampling is to collect maximum information with minimum ^nditure of money, time and labour and yet achieve a high degree of -^curacy and Ability. For ensuring reliability certain principles must be followed. In samphng method : is presumed that whatever conclusions are drawn from a sample are also true for the lole population. This presumption is based mainly on the followmg two laws : (a) The Law of Statistical Regularity, and (b) The Law of Inertia of Large Numbers. .'r u (a) Law of statistical regularity : The law of statistical regularity is derived from the mathematical theory of probability. It says that a comparatively small group of items chosen at random from a very large group will, on the characteristics of the large group. Basically, it applied to rWom se^lection. Thus so in the process of sampling each unit of the universe has an equal chance of being selected. Therefore, the selected items can be said to be representative of the universe. Although the law is not as accurate as a scientific law is, it does insure a reasonable degree of accuracy. Since there is a certam regularity m natural phenomena, we assume a certain uniformity in nature A random samphng is said to follow the law of statistical regularity because of this basic uniformity m a universe. r , -t- r lb) Law of inertia of large numbers : This law is also called the law of stability of mass data. It is based on the law of statistical regularity. Basica ly, it states that if the numbers involved are very large, the change in a sample is likely to be very small in other words, the individual units of a universe very continually but the total universe changes slowly. That is, large aggregates are most stable than «tnaU Because of the slow change in the nature of total universe this law is called the law of inertia (laziness) of large numbers. For example, sugar production of factory will vary significantly from year to year but Ac sugar production of a country as a whole will remain comparatively s able. Or a g eat Inge may take place in the male-female ratio of family may appreciably -bange ove a short period, but the male-female ratio of a country as a whole will ^^ the period, ^o take another example, if a. coin is tossed 6 times we may get heaj^s f^r ^ Js and tails two times. But if a coin is tossed 5^0 times^ there is a high p^^i^ of getting heads and tails 2,500 times each. This happens due to ^^^ I oplation of this law. That is, when one part of large group is changing m one direction the other moves in the opposite direction. Thus, reliability of sampling depends mainly on randomness of selection of data and the large size of universe, expressed by the above two laws. Statistical Errors There is a great difference in the meaning of mistake and error in statistics. Mistake imeans a wronfcalculation or use of inappropriate method in the collection or analysis 44 Statistics for Economics-X. other words, the difference between the approximated (estimated value) and the actual value (true value) is called statistical error m a technical sense. For examl we make a' estimation that in a particular meeting, 1,000 persons are there. But we clnt persons It may be wrongly counted, as 1,030. There is a difference of 30 between the estimate value and counted va ue. This difference is called '...or' in statistics. But w^en weTak* aTS''^r VrThey arl knowi as mistake . For example, there is a meeting, we sent a person to count the audience Sources of Errors Following errors are likely to occur in collection of data : ur'l^! origin arise on accoum of inappropriate definitions of statistical unit scale, or defective questionnaire etc. For example, wrong scale to measun meLl -'I height to nearest of inch or approximatrTh differences may also occur due to differences in measuring tapes due tc manufacturing defect. In Physics or Chemistry such errors of mLsurementrwlI occur while taking readings on various instruments. nZZ incomplete data, madequat. crsdonna " sample, non-response of respondent, incomplete answers i questionnaire, misinterpretation of questions in questionnaire, careless oi unqualified investigators, etc. '-diciesb oi| dr^o I'f f^ -"hmetic calculatio due to clerical errors, arithmetic slips etc. by omitting some figure consideri wrong value, making wrong totals etc. by respondent L investigator thrjlta^"''''^'"''^''''" ' statisticians for misinterpret! Types of Errors (a) Absolute and relative errors : Absolute error is the difference between the actua true value and estimated approximate value while relative error is the raTo o absoS error to the approximated value.absolut Absolute error = Actual value - Estimated value Symbolically, Ue = U' -U wr the enu mei wh( faul Relative error = Actual value - Estimated value Symbolically, Estimated value e= U'-U U Sec furthei obtain alreadj are inv b) unf s-XI Collection of Primary and Secondary Data Here, Ue = Absolute error e = Relative error U' = Actual value U = Approximate value niustration. Sales of commodity approximated Rs 497 and actual sale Rs 500. Absolute error (Ue) = 500 - 497 = 3 500-497 3 and Relative error (e) = .006 i" how secondary data is collected Secondary data are those which are collected by some other agency and are used for i^her studies. The errors mainly arising at the stages of ascertainment and processing 'of data. Unbiased errors are generally compensating. To avoid these errors. or reports which are ^eady published by them as a part of their routine work. faulty work during the collection of information and faulty method of analysis. the statistician must take proper precaution and care in using itfie correct measuring instrument. It saves cost and time which 'are involved in collection of primary data. Biased errors arise due to fauli^ j^iocess of selection. Statisticians should have none of these errors. (c) Sampling and non-sampling errors : The errors arising on account of drawing inferences about the population on the basis of few observations (sampling) are called sampling errors. These errors are cumulative in . Unbiased errors can be removed with proper planning of statistical investigations. then it is called biased error. They are common both in census enumeration and sample surveys. We can obtain the required statistical information from other institutions.ir re and increase when the sample size also increases. Secondary data may be either (a) published or (fc) unpublished. 500 Relative error is generally used in statistical calculations because absolute error gives wrong or misleading calculations.500 500 Relative error (e) can also be represented in percentage X 100 = 0. They are those which arise acccidently just on account of chance in the normal course of investigation.6%. He must see that the enumerators are also not biased. are called non-sampling errors. It is not necessary to conduct special surveys and investigations. 46 . Unbiased errors are not the result of any prejudice or bias. (h) Biased and unbiased errors : Biased errors arise due to some prejudice or bias in the mind of investigator or the informant or any measurement instrument. Suppose the Hiumerator used the deliberate sampling method in place of simple random sampling method. (Hi) Semi-official publications : Local bodies such as Municipal Corporations. publish statistics relating to trade and commerce. Distri Boards etc: publish periodical reports which give factual information about heal sanitation. Indian Trade Journal. and newspapers. collect and regularly puWish the data on different fields ( economics. (b) R^earch institutions. Federation of Indian Chamb of Commerce. World Bank. the Patel Committee Report e (v) Private publications : (a) Journal and new^apers. («) Publications of international organisations : We can obtain valuable internation s atistics from official publication of different international organisations. (iv) Reports of committees and commissions : Various Committees and Commission are appointed by the Central and State Governments for some special study an recommendations. Report of the Tariff Commission. Journals like Eastern Economists. . Journal of Industr and Trade. deaths etc. The reports of . like Financial Expres Economic Times.uch committees and commissions contai valuable data^ Some of the reports are : Report of National Agricultu Commission. Research scholars at ti university level also contribute significandy to the availabihties of secondai (c) Professional trade bodies. births. International Labour Organisation (ILO International Monetary Fund (IMF). Published Soiu-ces The various sources of pubhshed data are as under : (/■) Gove^ent pubUcations : Different ministries and departments of Central ar State Governments publish regularly current information along with statistical da on a number of subjects. like. This information is quite reliable for related studies.Statistics for Economics-XI Ji hm. Institute of Chartered Accountants. Labour Gaze Agriculture Statistics of India. Sugar Mills Associatio Bombay Mill . There are a number of institutions doing research o allied subjects This is the most importarn source of obtaining secondary dat The National Council of Applied Economic Research and Foundation ( !>cientihc and ^onomic Research are such institutions. commerce and trade. etc. etc. Monthly Statistics of Trade. Chambers of Commerce and Trade Associatio. ^ examp es of such publications are: Annual Survey of Industries. ti United Nations Organisation (UNO). Sf. research workers and scholars do collect data but they normally do not pubHsh it. 6. 5. 2. etc. 4. They may have been influenced by the biased investigation or personal prejudices.Owners Association. (d) Annual reports of joint stock companies are also useful for obtaining statistic information. The information which was collected on a particular base may not be suitable and relevant to an enquiry. Precautions in the Use of Secondary Data The investigator should consider the following points before using th j secondary data : (a) Are the data reliable? {b) Are the data suitable for the purpose of investigation? (c) Are the data adequate? (d) Are the data collected by proper method? (e) From which source were the data collected? if) Who has collected the data? . universities. provide valuable data for reseat Collection of Primary and Secondary Data Unpublished Data Research institutions. trade associations. 3. Apart from the above sources we can get the information from records and files of government and private offices. pubhsh statistical data. They may not have been collected by proper procedure. labour bureaus. ^^^ ^md^'' s-XI also. Stock Exchanges. They may not satisfy a reasonable standard of accuracy. They may not be suitable for a required purpose. They may be out of date and not suitable to the present period. Bank and Cooperath Societies. These are pubKshed by companies every year. Trade Unions. They may not cover the full period of investigation. Limitations of Secondary Data One should use the secondary data with care and full precaution and should not accept them at their face value as they may be suffering from the following limitations: 1. the secondary data should not be used at its face value. Central Statistical Organisation (CSO).^nd tabulate statisticar data. importance of statistics (data) in the country has become great. 48 Statistics for Economics-XI This can be studied >finder following sections : (A) Agriculture-Statistics (B) National Income and Social Accounting (C) Population Statistics (D) National Sample Survey (E) Price Statistics (F) Industrial Statistics (G) Trade Statistics (H) Financial Statistics (I) Labour Statistics There are some agencies both at the national and state level. Labour Bureau. It is risky to use such statistics collected by others unless they have been properly scrutinised and found reliable. Statistics are necessary for framing and judging the progress of economic planning. In the context of economic planning. which collect. ■ ■ ijl^ofrlant sources of secondary dali|> of india and national survey organisations) There are various sources and organisations through which statistical data are being compiled in India. great and rapid strides have been made in the field of collection of data. etc. In 1869 Hunter was appointed Director General of Statistical Surveys. Registrar General of India (RGI).(g) Are the data biased? . The study of Indian statistics is made under following heads : I. Some important major agencies at the national level are ^ensus of ^dia. Statistical Organisation of India (CSO) II. process . Thus. Indian Statistical Material. Narionai Sample Survey Organisation (NSSO). Director General of Commercial Intelligence and Statistics (DGCIS). census of india unique experience of undertaking the biggest census in the world in 1981 and has also an unbroken record of more than hundred years of decadal censuses Ihe Indian census is universally acknowledged as most authentic and comprehensive source of information about our land and people. That later followed into . He not only elaborated the statistical system but also assisted the statistical surveys of districts and provinces. Since India achieved Iiidependence. suitable and adequate. 772 496.028. This is a mirror of a fair ^presentation of the socio-economic and demographic condition of our people which constitute about one-sixth of the human population on this planet. age-composition. Census of India 2001 gives information ot population m India as : TABLE 3 Persons Males Females Sex Ratio 1.328 —^.^ _" 532. Information of demographic characteristics include birth and death. The study of population is important for several reasons in overall study of economic development.536 933 Collection of Primary and Secondary Data 49 national sample survey organisation (ii5s0) . He advised m conducting of census of India which undertook explanatory surveys from 1869 to 1872 and thereafter matured into a decennial census which ever since contmued without interruption.001. migration and literacy etc. sex ratio. The census statistics s useful for assessing the^impact of the developmental programmes and identify new thrust areasTor focussing the efforts on improving the quality of life in our country Basic population data fmm Primary Census Abstract.famous Gazetteers.453. fertility.610.156. The economic Characteristics of ppulation are manifested through workers' participation m economic classification of workers m various occupations. employment The data generated by the Census of India 2001 provide benchmark statistics on the people of India at the beginning of the next millennium. After 1872 the next census was taken in 1881 and ^nce then it has ^become a regular feature of holding census every ten years uninterruptedR The Census of India provides the most complete and continuous demographic record of T Independecne was held in 1951 and latest one completed m . The Governing Council is empowered to take all technical decisions in respect of survey work. Mahalanobis to fill up data gap for socio-economic planning and policy making through sample surveys. A Deputy Director General heads each division except FOD. rural labour and consumer expenditure. {d) Employment and unemployment.The National Sample Survey (NSS). maternity child care. An Additional Director General heads FOD. debt and investment. The data collected by NSSO surveys on different socio-economic subjects are released tiirough reports and its quarterly journal 'Sarvekshana\ The data comprises different iocio-economic subjects like employment. It was established on the basis of a proposal from Prof. from planning of survey to release of survey results. The programme conducts periodical surveys on : {a) Demography. and (e) Self employment in non-agricultural eflterprises. . The NSSO headed by a Director General and Chief Executive Officer. large scale continuous survey operation conducted in the form of successive rounds. on continuous basis in a comprehensive manner for whole country. On march 1970. The NSSO took a forward view of the data requirements to planners. production of small scale household enterprises consumption etc. (c) Land holdings and livestock enterprises. [ii) Collection of data relating to the organised industrial sector of the country. Data Processing Division (DPD) and Coordination Publication Division (CPD). is a nationwide. P. processing and publication of the NSS data. health and family planning. has four divisions namely. The Governing Council consists of 18 experts from within and outside Government and is headed by an eminent economist/statistician and the member-secretary of the council is Director General and Chief Executive Officer of NSSO. Functions of NSSO The functions of National Sample Survey Organisation are : (i) Collection of data on socio-economic conditions. Survey Design and Research Division (SDRD). A major objective of NSS has been to provide data required to fill up the gaps in information needed for estimation of national income. Field Operation Division (FOD). {Hi) Supervision of surveys conducted by states in agricultural sector through their own agencies and also giving guidance to them for analysing and coordinating the results of these surveys. {b) Assets. the NSS was recognised and all aspects of its work were brought under a single Government organisation namely the National Sample Survey Organisation (NSSO) under the overall direction of a Governing Council to impart objectivity and autonomy in the matter of collection.C. unemployment literacy. initiated in the year 1950. research workers and other users and draw up a long term programme. 3. . 14.e mdex numbers NSSO also undertakes field work of Annual S^ ^dust^ conducts crop estimation surveys. T 2004)-was on morbidity and head care. 7. 15. 9. i-XT What do you understand by Statistical Enquiry? Explain d~ft~f ^ merits anc Discuss the comparatwe merits of various methods of collecting primary data. Apart from collection of rural and urban retail prices for compilation of consume pn. ^ maustries an exercises 2. 11. utilisation of educational of services etc Th car Aoar. fTolfll . 8. 6. 5. 10.■1tr 50 Statistics for Economics-XI utUisat^n of public distribution system. 4. 12. SrsTable^^^^^^^^^^^ " ' i-tigations-. 13. How would you organise a survey for this purpose? . What precaution! must be taken while preparing questionnaire? precautionf Write short notes on : (a) Census of India {b) National Sample Survey Organisation (NSSO) mat IS Secondary^data? Discuss the various sources of collecting secondary data. s-XI Collection of Primary and Secondary Data 16. Name the methods of selecting a sample. Distinguish between census and sample surveys. 19.What are the similarities and dissimilarities between the two methods-l questionnaires to be filled in by informants and schedules to be fild in h enumerators? Explain with examples. mat precaution should be taken before using secondary data? Explain iTZn 1 constructing interview schedules and questionnaires F ame at least four appropriate multiple choice options for following questions (t) How often do you use computers? («) What is the monthly income of your family? (m) Rise in petrol price is justified : (iv) Which of the newspaper do you read regularly ? Jv) Which of the following most important when you buy a new dress' l<rame two way questions (with 'Yes' or 'No') Following statements-true or false. (m) Telephone survey is the most suitable method of collection of data when the population is literate and spread over a large area. * mat is a questionnaire? Give a specimen of a questionnaire. 17. List four important types of sampling : methods. Distinguish between population and sample.with merits and demerits. Describe the method of stratified sampling . Explain the reasons for preferring sample surveys in the collection of data. (/) Data collected by investigator is called secondary data. Describe the questionnaire method of collecting primary data. («) There are many sources of data. The Education Ministry is interested in determining the level of education of unmarried girls in the country. 23. Discuss briefly the following : (a) Law of inertia and large numbers (b) Law of statistical regularity 25. Do samples provide better resuhs than surveys? Give reasons for your answer. Distinguish between random sampling and systematic sampling. Explain briefly the different methods of sampling. 30. Give suitable examples. What do you mean by 'Sample' investigation? Explain its suitability with illustrations. 24. 31. 29. Which of the following methods gives better result and why? (a) Census (6) Sample Chapter 4 organisation of data ¥ (b) . 26. Examine the important types of sampling methods. Write a critical note of random sampling method. What do you understand by 'Census' investigation? Explain its suitability with illustrations. population and variable. 27. 22. Give illustrations.20. 21. :—^ (c) Sampling and non-sampling errors Give two examples each of sample. Does the lottery method always give you random sample? Explain. 28. Discuss briefly the following : (a) Biased and unbiased errors (b) Absolute errors and relative errors p. How would you distinguish convenience sampling with judgement (deliberate) sampling? Explain. What do you mean by statistical errors? 32. Give a comparative study of stratified sampling and multi-stage sampling. literacy.00 per kg. Individuals may be ranked according to quality of attributes. in a group of different sections. e. Definition 2. in a group of students. Methods of Classification Statistical Series 1. Income of family D is Rs 10. deafness. height or weight of students.g. The characteristics which are not capable of being measured quantitatively are called attributes. Height of Rajesh is 60". Objects of Classification 3.g. sickness.000 per month. time in different watches.45 a. black and blue eyed. Variate can also be called 'variable' or 'magnitude' or 'observation' or 'item' or 'measure' or value'. m a group of different watches. aptitude for art and music.Classification 1. The collected data (either by primary or secondary method) are always in an unorganised form in schedules or questionnaires or another written form. students in different sections. in a group of families. income of families. intelligence. The ranks are sometimes used as their numerical values for purposes of statistical analysis. They always differ from one to another. Characteristics of Classification 4. e. The . prices of vegetables. Types of Series 3.. A single item out of all the observations of group as numerical may be called variate or variable. Frequency Distribution (a) classification The quantitative information collected in any field of society or science is never uniform. price and incomes.m. Definition 2. in a group of vegetable prices. blindness.. They cannot be measured numerically in the same way as heights and weights. Time in HA/IT watch is 10. For example.. tall and short. etc. Price of potato is Rs 10. or. Students in Section A are 50. The classified information presented in precise and systematic tables is called tabulation. viz. students are classified according to division they secured in certain examination. varied and unorganised data. Classification is grouping of data according to their identity. or according to age. DEFINITION According to Professor Connor.collected data in unorganised form is called RAW DATA. letters in the post office are sorted out in groups of cities and towns of destination. "-Classification is the process of arranging things (either actually or notionally) in the groups according to their resemblances and affinities. it is necessary to make them available for comparison. The process of grouping into different classes or subclasses according to characteristics is called classification. (2) The basis of classification is unity in diversity. (3) The classification may be either actual or notional. students in a school may be grouped as boys and girls. Agra. The classified information arranged in a logical and systematic order in a particular sequence is called seriation or statistical series. in library the books and periodicals are classified and arranged according to subjects. Delhi. analysis and appreciation by proper and suitable grouping and arrangement in condensed form. Similarly. similarity. Because of the limitation of human mind Organisation of Data 53 to understand such a complex." According to this definition.. In other words. animals or plants may be grouped according to origin or structure etc. or resemblances For example. classification is for division of data. seriation is for arrangement of data in a systematic order and tabulation is for presentation of data in a table.. the chief features of classifications are : (1) The facts are classified into homogeneous groups by the process of classification All the units having similar characteristics are placed in one class or group. Chennai. and give expression to the unity of attributes that may subsist amongst a diversity of individuals. OBJECTS OF CLASSIFICATION The chief objects of classification are : . (4) The classification may be according to either attributes or characteristics or measurements.Chandigarh etc. will make such comparison easy. uneducated^ employed. occupation etc. simple. then the structure and nature of the population can easily be understood. To present the facts in a simple form : Classification process eliminates unnecessary details and makes the mass of complex data. third and failure classes on the basis of marks obtained by them. For example. unemployed etc. logical and understandable.1. Com. To present a mental picture : The process of classification enables one to form a mental picture of objects of perception and conception. brief. Summarised data can easily be understood and remembered. To bring out clearly points of similarity and dissimilarity : Classification brings out clearly the points of similarity and dissimilarity of the data so that they can be Statistics for Economics-XI easily grasped. such as educated. CHARACTERISTICS OF CLASSIFICATION It is important that the classification should possess following characteristics : 1. But classification of students into first. To facilitate comparison : Classification of data enables one to make comparison. marital status. To bring out relationship : Classification helps in finding out cause-effect relationship. second. It is necessary that various classes should be so defined that there is no room for doubt and confiision and must have a class for each item of data in one of the classes. the data collected in a population census is so huge and fragmented that it is not possible to draw any conclusion from them. To prepare the basis for tabulation : Classification prepared the basis for tabulation and statistical analysis of the data. When these massive figures are classified according to sex. Unclassified data cannot be presented in tables. education. Classification should be unambiguous : Classification is meant for removing ambiguity. if there is any in the data. For example. data of small-pox patients can help m finding out whether small-pox cases occurred more on vaccinated or unvaccinated population. 4. 6. . 5. draw mferences and locate facts. students in two colleges are given. Facts having similar-characteristics are placed in a class. 2. 3.. This is not possible in an unorganised and unclassified data. no comparison can be made of their intelligence level. If marks obtained by B. O .8 1971 54. The classes must not overlap : Each item of data must find its place in one class and one class only There must be no item which can find its way into more than one class. 4. » 6. If the investigation is carried on to enquire into the economic conditions of labourers. should tally with the total of the universe. (Qualitative Classification) 4 According to magnitudes or variables (Quantitative Classification) For example. . and that principle should be maintained throughout. can be . should have arithmetical accuracy : The total of items included in different classes. ch—s. Classification.7 1961 43. Classification should be suitable to enquiry i-Classification should be according to the objects of enquiry. The data would not be fit for comparison. Classification should be flexible : It should be flexible and should have the capacity of adjustment to new situations and circumstances.6 1981 68. 55 ■ganisation of Data . some classes became obsolete and have to be dropped and fresh classes have also to be added. Population of India 1951 35. 3. (Geographical Classification) 3 According to attributes. 5. With change in time. According to time. then it will be useless to classify them on the basis of their religion. (Chronological Classification) 2 According to area.onped the following bases : 1.e.4 . Classification should be stable : If classification is not stable and if each time an enquiry is conducted it has to be changed. Therefore.2. the classification must proceed at every stage in accordance with one principle. cal order in example (b).7 OR 2001 102. For example. ' 2.8 1981 68.6 1961 43. appear in an alphabe. wheat in ..8 2001 102. 56 Two-fold classification .1991 81.25 kg (Per acre) 1925 127 (b) 439 862 893 Descending order (figures of the second row above) ^ Country " America China India France Denmark Brazil Yield of wheat in ^^^ ^^^ ^^^ 225 127 Note 4«rLes of Ae conntrTes do no.8 1951 35.4 1971 54.7 Year Population (in crores) Year Population (in crores) (a) Alphabetical order (names of the countries) Country America Brazil China Denmark France India Yield of .7 1991 81. POPULATION Statistics for Economics-XI r Males co to 2ooE CO = iS « « su Employed Females Married (1) I Unemployed Employed Unmarried Married Unmarried (2) (3) (4) Married (5) i Unemployed Unmarried Married (6) (7) 1 Unmarrit (8) Second and AM Iges of cLsffaLT " "" confusion in classification Th. ' cieariy dehned to avoid . .. .Q in i-u^ u . sales..s s^^r^^ :""" "' the division of the data I tt^ flTrr' " .': """"" 57 Organisation of Data STATISTICAL SERIES Jiasis of bharacter Z Times Series (on the basis of time) 2.« ISa^r tr ^ ^^ „ reference ro sonre . Condition ^ries (on the basis of condition) r Series of Individual ^servation 2.:. Continuous Series Frequency distribution Series on die Basis of General Chat«c.t'Tts"7rf ^ ^^ formation of searisrical feries "7'classification . sate. cry. ' '1 (b) statistical series DEHNinON other.n.. Discrete Series ■ 3. the resu.gh.weeU.ry." Tu 1 -10 27 58 72 Thus..we. .he collected and classified" ir^ rprl^^eJrr.e nn.^coun.ear.. or da. Spatial Series (on the basis of space) 3.s made in tl. . ^^^ ^ ^^ ^^ Sugar Production of a Factory ^^^ ^^^^ ^^ 2006) of . For example : ^^^^^ ^^ ^^^^^^^ Country Per Capita .onrH. town. Por example.. : "r. village or colony.he universe under srudy ^^ j. there are 15 workers in the income group of Rs 100 to 199 77 7 • mcome group of Rs 200-299 and so on. weight..999 35 1000-1499 25 1500-1999 15 2000-2499 20 2500-2999 5 Series on the Basis of Construction After collection and classification of data it is the most important job now to construct the data in an arranged order that is the formation of series for further study of presentation. {b) Discrete Series. Data are presented with reference to some condition. {Hi) Descending order. This arrangement can be done in three ways : {a) Series of Individual Observation. Series of Individual Observation Mass data in its original form is called raw data or unorganised data which can be arranged in any of the following ways : (/■) Serial order of alphabetical order. {b) and (c) are with reference to frequency distribution. height. The mass data when put in ascending or descending order of magnitude is called an array. They are not listed in groups. analysis and interpretation.100 500 City 3» Delhi Mumbai Chennai Kolkata Bangalore 792 649 573 532 459 58 Statistics for Economics-XI 3. (c) Continuous Series.900 2. of Workers 500. .100 3. age. A series of individual observations is a series where items are listed singly after collection. income etc.USA France Japan Canada India 5.800 2. (ii) Ascending order. viz. For example : Weekly Income of 100 Workers Income (Rs. Condition series : A series of values of some variable made according to a condition is called condition series.) No. . which is in unorganised form (Raw Data) as shown in Table 1. Discrete Series - 3. Spatial Series (on the basis of space) 3. Data are presented with reference to some time unit. month. Continuous Series Frequency distribution Series on the Basis of General Character 1 Time series. A series of values of some variable according to successive points in time is called time series. Times Series (on the basis of time) 2. TABLE 1 Daily Wages Paid to Workers (in Rupees) 60 102 61 101 92 80 87 72 86 73 96 101 92 56 90 58 85 74 83 63 84 62 92 100 56 84 90 86 67 72 Organisation of Data TYPES OF SERIES Statistical series can be clas^i^ed in the following way : STATISTICAL SERIES 57 ^J^sls of "Character 1. or day. Condition Series (on the basis of condition) 1. Series of Individual observation - 2.Suppose an investigator has obtained the following information from a factory about the payment of daily wages of 30 workers. year. week. viz. For example : . sate.892 Tues. 2. city.Sale in Super Bazar Sugar Production of a Factory Year Production (in WO tons) 1999 78 2000 75 2001 94 2002 86 2003 89 2004 92 2005 95 (1st week of Jan. 2. 2.822 2 Spatial series. town. 3. 1.650 Fri..592 ■ Satur. viz. For example : Per Capita Income Number of Schools . A series of values of some variable according to geographical division of the universe under study is called a spatial series or geographical series. village or colony. country. Data are presented with reference to some geographical division. 2006) Day Sale (Rs) Men. 3.090 Thurs.757 Wednes. For example : Weekly Income of 100 Workers Income (Rs. of Schools Delhi 792 Mumbai 649 Chennai 573 Kolkata 532 Bangalore 459 58 Statistics for Economics-XI j 3.100 India 500 City No. Data are presented with reference to some condition.Country USA Per Capi^ 5.. income etc.900 Japan 2. Condition series : A series of values of some variable made according to a condition IS called condition series.800 Canada 2. age. height. viz.100 France 3. Series on the Basis of Construction .) No. of Workers 500-999 35 1000-1499 25 1500-1999 15 2000-2499 20 2500-2999 5 i. weight. 1 21 60 11 61 92 S. No. A series of individual observations is a series where items are listed singly after collection. ^^ y -.No. Series of Individual Observation Mass data in its original form is called raw data or unorganised data which can be arranged in any of the following ways : (/•) Serial order of alphabetical order. Wages (Rs. analysis and interpretation. This arrangement can be done in three ways : (a) Series of Individual Observation. They are not listed in groups. . which is in unorganised form (Raw Data) as shown in Table 1.V (Rs) S.No. (ii) Ascending order."" . Suppose an investigator has obtained the following information from a factory about the payment of daily wages of 30 workers. (Hi) Descending order.After collection and classification of data it is the most important job now to construct the data in an arranged order that is the formation of series for further study of presentation. (b) Discrete Series. Wages IRS. (c) Continuous Series. {b) and (c) are with reference to frequency distribution. TABLE 1 Daily Wages Paid to Workers (in Rupees) 60 102 61 101 92 80 87 72 86 73 96 101 92 56 90 58 85 74 83 63 84 62 92 100 56 84 90 86 67 72 Organisation of Data 59 The above raw data can be arranged either in serial order (Table 2) or ascending order (Table 3) or descending order (Table 4) as given below : TABLE 2 Arranged in Serial Order 5. The mass data when put in ascending or descending order of magnitude is called an array. .90 84 73 62 56 FREQUENCY DISTRffiUTION Before discussing anything about frequency distribution it is advisable to know the following important terms of frequency distribution under which the two types of distributions are grouped. The two types are : (a) Discrete Frequency Distribution. (b) Continuous Frequency Distribution.2 87 12 86 22 96 3 92 13 90 23 85 4 83 14 84 24 92 ■ 5 56 15 90 25 67 6 102 16 101 26 80 7 72 ■ 17 73 27 101 8 56 18 58 28 74 9 63 19 62 29 100 10 84 20 86 30 72 TABLE 3 Arranged in Ascending Order (Wages in Rupees) 56 62 73 84 90 96 56 63 74 85 90 100 58 67 80 86 92 101 60 72 83 86 92 101 61 72 84 87 92 102 TABLE 4 Arranged in Descending Order (Wages in Rupees) 102 92 87 84 72 61 101 92 86 83 72 60 101 92 86 80 67 58 100 90 85 74 63 56 96 . frequency of families havmg 4 children is 8.the total frequency. ^ Set I.. of Students (f) 56-58 12 58-60 16 60-62 62-64 4 64-66 10 Total 57 diffe^r^t' i' arrangement of items mto a particular order or sequence m sfuZZ '' ' " Height "f Frequency : The number of times given value in an observation appears is the frequencv four chiljen ^ 10 students m the group of 64" to 66" and 16 students m group of 58 to 60 etc. class^'lrau^nr"'^ ^ Tu^"/^ ^^^ quantitative classes is called the class frequency^. of families (f) 01234 25 45 37 15 8 Total 130 Height in Inches No. So the frequency of famihes having no child is 25. There is no instance of a class in Set I "^SiT't^TlT ^ J ^^ . . Children in Families t Children No. frequency of students m the group of 6^" to 66•^s 10 and frequency m the group of 58" to 60" is 16. out of the five classes of Set II students in a group of 58" to 60" to 60 IS 16 and of 62" to 64" is 4.60 Statistics for Economics-XI Terminology of Frequency Distribution Examine the following two sets of illustrations to clearly understand the basic termmology of frequency distribution.^. which boundaries of a d^ss. cJJaT^^ distribution : The distribution of observations over the several values is n ^^ Set n T" ' ^^ children m tamilies.g. No. Wages (Rs) S.'Thu's m thi fir" andTht hf H / f magnitudes 56. are known as the upper and lower limits. Wages (Rs) S. 58. No. 100-200. No.g.limit.. 10-19 4-8 form fhe b'ndar^T I = ^^^ magnitudes. Class. It is a decided group of magnitudes. Wages (Rs) 1 60 11 61 21 92 2 87 12 86 22 96 3 92 13 90 23 85 4 83 14 84 24 .e. 60. e. the total 130 and 57 in our set I and set II. respectivdy For clTn'^f Set^f^^^^^^^^ '' " '^T " upper. and Set II is the frequency distribution of heights of students. 92 5 56 15 90 25 67 6 102 16 101 26 80 7 72 • 17 73 27 101 8 56 18 58 28 74 9 63 19 62 29 100 10 84 20 86 30 72 TABLE 3 Arranged in Ascending Order (Wages in Rupees) . 62. and 64 are the lower limits respec^v classes ''' Organisation of Data ^^ ^^^ ^^^ ^mits of their ^^ The above raw data can be arranged either in serial order (Table 2) or ascending order (Table 3) or descending order (Table 4) as given below : TABLE 2 Arranged in Serial Order S.. 56"-58". (a) Discrete Frequency Distribution. The two types are : . (b) Continuous Frequency Distribution.90 84 73 62 56 FREQUENCY DISTRIBUTION Before discussing anything about frequency distribution it is advisable to know the following important terms of frequency distribution under which the two types of distributions are grouped. Children in Families Children 0 25 1 45 No.56 62 73 84 ' 90 96 56 63 74 85 90 100 58 67 80 86 92 101 60 72 83 86 92 101 61 72 84 87 92 102 TABLE 4 Arranged in Descending Order (Wages in Rupees) 102 92 87 84 72 61 101 92 86 83 72 60 101 92 86 80 67 58 100 90 85 74 63 56 96 . Set I. 60 Statistics for Economics-XI Terminology of Frequency Distribution Examine the following two sets of illustrations to clearly understand the basic terminology of frequency distribution. of families (f) . So the frequency of families having no child is 25. Frequency: The number of times given value in an observation appears is the frequency For example.g. Class frequency : The number of values in each of the quantitative classes is called the class frequency.2 37 3 15 4 8 Total 130 Height in Inches No. of Students (f) 56-58 12 58-60 16 60-62 15 62-64 4 64-66 10 Total 57 Series is a systematic arrangement of items into a particular order or sequence in ffe. Total frequency : The sum (total) of the frequencies is known as the total frequency. and 10 students in the group of 64" to 66" and 16 students in group of 58" to 60" etc. the total 130 and 57 in our set I and set II. . different classified categories. so the class frequency of the class 58" to 60" is 16 and of 62" to 64" is 4. frequency of families having 4 children is 8. There is no instance of a class in Set I. in the above sets there are 25 families having no child and 8 families having four children.g. as Set I for Children in Families and Set II for Height of Students. out of the five classes of Set II students in a group of 58" to 60" are 16 and students in a group of 62" p 64" are 4. and frequency in the group of 58" to 60" is 16. Frequency distribution : The distribution of observations over the several values is called frequency distribution. e. frequency of students in the group of 64" to 66" is 10. e.. and Set II is the frequency distribution of heights of students.. Set I is the frequency distribution of children m families. For example. the class interval is upper limit (l^) . weight. Thus in the first column of Set II. children in a family can be either 2 or 3. It is the span or width of a class which can be obtained by finding the difference between the upper and lower limits of the class. Discrete and Continuous Variables Discrete and Discontinuous Variables are those which are exaci or finite and are not normally fractions. respectively For 1 example. 7-13 etc. and 64 are the lower limits! and right hand side magnitudes 58.. 58. students in a class are all the examples . l^ is the lower limit and is theupper limit. 64 and 66 are the upper limits of their] respective classes.Organisation of Data 6\ Cias. are known as the upper and lower limits. but appear by limited gradations.e. Variable : A quantity which varies from one individual to another is known as a variable or variate. 60.. It is a descrete variable which is not expressed in a fraction. Mid-point : The mid-value which lies half way between the lower and upper class limits is known as mid-point. It is a decided group of magnitudes. number of units sold etc. 100-200. fans installed in an auditorium. i. height.l^) = 66-64 = 2.7. as being the representatives of the classes. Thus.lower limit (/j). 62.mterval : The magnitude spread between the lower and upper class limits is called class interval. but cannot be 2. for class 64"-66". 62. 4-8. 10-19. in a class of 62 "-64" the mid-point is upper class limit + lower class limit 2 or /2+/1 64+62 = 63 22 Calculated mid-points are the most important values. 60.2. and are taken for use in further statistical calculations. Upper and lower limits of the classes : The lowest and the highest magnitudes.Class. 62 is lower limit and 64 is upper limit. The class interval in this case is 2. 56"-58". e. left hand side magnitudes 56. rooms in a house. For example. In the same way test scores of a cricket match. For example. Quantitative characteristics such as income.g.. 2. (l^ . which form the boundaries of a class. are variables. A variable may be either discrete or continuous. for a class of 62-64. workers in a factory. They cannot manifest every conceivable fractional value.8 or 2. of discrete variables.50. 4 are discrete variables. 1. .. the car must pass through'all the infinitely small gradations of distance between 0 km to 5 km. Heights of students from 56" to 58" in our set II for example.. 3. weights. All the fractional values are continuous variables. 3.e. 0. For instance covering a distance on a road.. Prepare frequency table of ages of 25 students of XI Class in your Solution.e. cover all the fractional values falling within the limit of 56 and 58. second for 'Tallv magnitude to another may preferably be tL satne® column ^^^lustration .50.50. or 2 to 3 km. Frequency Distribution of Ages of 25 Students Age 15 . 3. They are capable of manifesting every conceivable fractional value {i. 4.. They fall in any numerical value within a certain range. In other words. in decimals) within the range of possibilities. Thus the variable is said to be of a discrete type when there are gaps between one value and the next. 5. in set I. and so on. Even fractional values are discrete or discontinuous variables provided there is an uniform difference from one variable to the other variable.50. 2. incomes. rainfall etc. by car. Conslruclion of Discrete Frequency Distribution 1. The occurrence of the observation will be integers. say from 0 kilometre to 5 kilometres one never jumps from 0 to 1 km. e.. and so on.. and (b) Continuous frequency distribution. 1. 2. II 62 Statistics for Economics-XI j (a) Discrete frequency distribution. 6. 1. 2. 3. 2. 1. i. For example.g. heights. but every fraction of distance from 0 km to 5 km is touched. 1 to 2 km. For example. Continuous variables are those that one in units of measurement which can be broken down into infinite gradations. if wage rate per unit is 50 paise then workers of a factory may get wages in rupees as : 0. Prepare a table with three columns-first for variable under study. in a class of 62"-64" the mid-point is upper class limit + lower class limit 2 or l2±k 2 64+62 = 63 Calculated mid-points are the most importam values. height. number of units sold etc. It is the span or width of a class which can be obtained by finding the difference between the upper and lower limits of the class. and are taken for use in further statistical calculations.or c^ass 64"-66". (l^-/. i. Mid-point : The mid-value which lies half way between the lower and upper class limits is known as mid-point. 9 mi 5 III 3 1 1 Total 4 25 Organisation of Data Class interval : The magnitude spread between the lower and upper class limits is called class interval. . is the lower limit and is the-upper limit.e.-b4 . Quantitative characteristics such as income.lower limit (/.. The class interval in this case is 2. Variable : A quantity which varies from one individual to another is known as a variable or variate. Thus. as being the representatives of the classes. . the class interval is upper limit (Z^) .) .16 17 18 19 Tally bars mill Total (/) 7 mi nil . weight. A variable may be either discrete or continuous..).2. For example. are variables.6t. children in a family can be either 2 or 3.Discrete and Continuous Variables Discrete and Discontinuous Variables are those which are exac. or finite and are not normally fractions. 1 to 2 km. 2. 2. 1. 4 are discrete variables. by car. The occurrence of the observation will be integers. cover all the fractional values falling within the limit of 56 and 58. All the fractional values are continuous variables Heights of students from 56" to 58" in our set II for example.. They fall in any numerical value within a certain range.50.50. In other words. Set II of the distribution of heights of students is a continuous series. 3. heights. 4.. Even fractional values are discrete or discontinuous variables provided there is an uniform difference from one variable to the other variable.g. rainfall etc. Set I of the distribution of children in families is a discrete series. For example. 3. 3. say from 0 kilometre to 5 kilometres one never jumps from 0 to 1 km. rooms in a house. 0. 6.. e. 3. 1. weights. For mstance covering a distance on a road. students in a class are all the examples of discrete variables.g. and so on. For example. and so on. but continuous variables cannot be presented in a discrete . but cannot be 2 2 2 8 or 2 7. but appear by limited gradations. in decimals) within the range ot possibilities. 2. Z. 1. ihus the variable is said to be of a discrete type when there are gaps between one value and the next. They cannot manifest every conceivable fractional value. It is a descrete variable which is not expressed in a traction. e.5U. fans mstaUed in an auditorium.. Continuous series : Any series described by continuous variables is called continuous series. incomes.g. 62 Statistics for Economics-XI j Discrete series : Any series represented by discrete variahdes is called a discrete series e.e. . i.. Continuous variables are those that one in units of measurement which can be broken down into infinite gradations.e.50. They are capable of manifesting every conceivable fractional value {i.. In the same way test scores of a cricket match. if wage rate per unit is 50 paise then workers of a factory may get wages in rupees as : 0. the car must pass through'all the infinitely small gradations of distance between 0 km to 5 km. in set I.. It is to be noted that a discrete variable series can be presented in a continuous type of series also. or 2 to 3 km. but every fraction of distance from 0 km to 5 km is touched. workers in a factory. 1. 5. For example. ly. and (b) Continuous frequency distribution. one can have the choice of a continuous frequency distribution. lo. The gap between one magnitude to another may preferably be the same. Illustration 1. 15. now individual observations can be constructed and condensed in two ways : (a) Discrete frequency distribution. Prepare a table with three columns—first for variable under study. 16. Construction of Discrete Frequency Distribution 1. Place all the values of the variables in the first column in ascending orderbeginning with the lowest and giving to the highest. Considering discrete and continuous series. . Blocks of five bars or mi or W are prepared and some space IS left between each block of bars. Prepare frequency table of ages of 25 students of XI Class in your school. 3. 15. This column IS for facility in counting. representing corresponding frequency to each value or size of the variable. Whenever the range of values in a discrete series is too wide. 16. second for 'Tally bars' and the third for the total. 16. 9 17 m^ 5 18 III 3 19 1 1 7 .series. 2. 15 Solution. Put bars (vertical lines) in front of the values accordingly in the second column keeping in view the number of items a particular value repeats itself. Count the number of bars in respect of each value in the variable and place it in the third column made for total or frequency. Va 16. 17. 4. Frequency Distribution of Ages of 25 Students Age Tally hens ■ Total if) 15 mill 16 mi nil . 17. ^ 3 2 2 2 2 1 2. 1 2 2 .^ ^ 3 3324 Solution. 22 37 24 22 based on their replies as recorded below.3 3 3 6 l" 6 2 1 5 1 5 4 3 4 2 0 3 1 4 Frequency Array of Domestic Appliances Used by 45 FamiUes Number of Appliances 012 3 4 5 67 Tally bars mill mi M M miM II M Total 1 7 15 12 5 2 2 1 . 2 1 2 2. Prepare a frequency array------------------. In a aty 45 famUies were surveyed for the number of domestic apphances they used.Total 4 25 63 Organisation of Data Illustratxon 2. {b) By Exclusive method : (i) Lower limit excluded : Marks : 5-10.e.99. . 25-30 etc.9 and so on.«s „e e^cludea. and is iiiciuu J <: QQ 10-14 9 15-19. 15-20. There are various methods by which class intervals can be designated. uppe. 10-15. if the students' obtained marks are grouped as 5-10.45 I »| appliances.99. If the marks of a student are 10. 20-25. 15-19. 25-30 These are to be These are to be . But if a student gets 5 marks. we will have to prepare a group 0-5 to include. 20-24. 15-19. 25-29 10-14.. 64 Statistics for Economics-XI Sometimes lower limits are excluded from their respective classes. - (^i) Inclusive Method. There are two methods of classifying the data according to class intervals. They are: {a) By Inclusive method : 10-14.99.99. 10 to 15. i. For example. 20-24. he is included in the first group.99 5-9. Marks : 25-29. or Prices in (Rs) : 5-9. 15-20. we include in the first group the students whose marks are above 5 and up to 10.. is the lower o^ f ^ cfass^ second. 20-25. "H-e ''' orrinterva. Construction of Continuous Frequency Distribution "ations are divided mto groups havmg class mtervals. -15. 25. By mid-points of class inter\'al : Marks: 7..5-2. 15.5 These mid-points are required to be converted into class intervals.e. 10-15. 10. -20. 20-25. 20-25 25-30. values are put on the basis of construction of series. Prepare a frequency array based on their replies as recorded below. 10-15. Say for first midpoint (12.5 = 10). 25-30. However.5 27. We get thus the class interval 5-10. («) Upper limit excluded : Marks : 5-10. Lower limits 5. read as 10-.5 + 2. i.25-30 Upper limits 10. In the above series '5' in place of 'below' and '40' in place of 'above' may be put. it is always presumed that upper limits are excluded in absence of any specific instructions. (c) By mentioning lower limits (followed by a dash) : Marks : 520-. 10-15. 25-. 15-20. Thus making the classes as : Marks 0-10 10-15 15-20 20-25 25-30 30-35 35^0 Organisation of Data 63 Illustration 2.e. i. (7. -30.5-7.20-25. if the class intervals are given as 5-10. 10-15..5) and divide the difference by 2.5 17. The quotient is added and subtracted to first mid-point we get. In the same way intervals of all the mid-points can be obtained. -25. . In a city 45 families were surveyed for the number of domestic appliances they used. 15-20. 15-20. 30 of their respective groups are excluded. 30 of their respective groups are excluded. By mentioning upper limits (preceded by a dash): Marks : -10.5 22. 15-20. (5/2). 20.10-15.5 12. 20.15-20. 5-10. 5-10. read as (d) (e) 15-. 15. 20-25 etc. 20-25 and 25-30. 10-15.5 = 5) and (7. (f) 'Open-end' class intervals In certain frequency distributions 'open-end' class intervals are given as we find in the example given below : Marks Below 10 10-15 15-20 20-25 25-30 30-35 35 and above Total Frequency (f) 7 10 13 18 8 5 3 64 In such cases. 25.. 13 2 2 2 2 1 2.. in the group 5-9. 2 using 6 appliances and only 1 family using 7 domestic appliances.e. and [b) Exclusive Method.1 2 2. 15 using 2 appliances.9 and so on. If there are no whole numbers. I Frequency Array of Domestic Appliances Used by 45 Famihes Number of Appliances 0 1 1 1 Mil 7 2 MMM 15 3 MMii 12 4 M 5 5 II 2 6 II 2 7 1 1 Tally bars Frequency (f) Total 45 Thus. Construction of Continuous Frequency Distribution Observations are divided into groups having class intervals. 10-14. 5 using 4 appliances. 15-19. 20-24. For example. 10-14.. 25-29 etc. from the above table it is clear that out of 45 families 1 is not using any domestic appliance. [a) Inclusive Method. If the marks of a student are 10 he is included in the next class. 2 using 5. . the classes can be made 5-9. 12 using 3 appliances. 10 to 14. There are two methods of classifying the data according to class intervals.3 3 ^ 3 33 2 3 2 2 6 1 6 215 1 5 3 242 7 42434 2 03143 Solution..9. 7 using 1 appliance. 15-19. i. we include in first group students whose marks are between 5 and 9. (a) Inclusive Method : Under this method upper class limits of classes are included in respective classes.9. if the students obtained marks are grouped as 5-9. If the marks of a students are 10 he is not included in the first group but in the second. 10 to 15. 10152025readas 5-10. 25. or Prices in (Rs) : 5-9. 20-25. 15-20. he is included in the first group. 20-25 and 25-30. There are various methods by which class intervals can (a) By Inclusive method : 10-14. 10-15. 10. . (ii) Upper limit excluded : 5-10. 15-20. if the students obtamed marks are grouped as 5-10. 25. 25-29 10-14. 25-29.99. 15. They are: 20-24. 20-25 etc it is always presumed that upper limits are excluded in absence of any specific instructions. 15-19. 15-19. (&) By Exclusive method : (/) Lower limit excluded : Marks : 5_io. 10-15.20-25. (c) By mentioning lower limits (followed by a dash) : ^'^'ks : 5-. 78 Statistics for Economics-XI Sometimes lower limits are excluded from their respective classes. For example. 15-20. For example. 10-15. if the marks obtained by the students are grouped as 5-10.e.. we include in first group of students whose marks are 5 or more but under 10. 15-20. Lower limits 5. 20. 30 of their respective groups are excluded. 25-30 These are to be These are to be 10-15.20-25. 20-24. 20-25.. However. we will have to prepare a group 0-5 to include. 10-15.99. 10-15.99. i. 20. 15.15-20.(b) Exclusive method : Under this method upper limits are excluded.15-20.25-30 Upper limits 10. Marks : be designated.99 S-9. if the class intervals are given as 5-10. But if a student gets 5 marks. The upper limit of class interval is the lower limit of the next class.99. 25-30 etc. 25-30 etc we mclude m the first group the students whose marks are above 5 and up to 10 If the marks of a student are 10. 30 of their respective groups are excluded. 20—25. The choice for the class intervals should be either 5 or a multiple of 5. . 5-10. 9.5 22. It simplifies our further statistical calculations.5) and divide the difference by 2. 27. 25—30.5 17. We get thus the class '"t^al 5 10 In jhe same way intervals of all the mid-points can be obtained. read as (e) -15. 11. 20-25 25-30.{d) By mentioning upper limits (preceded by a dash): Marks : -10. lU 15. In the above series 5 m place of below and '40' in place of 'above' may be put.5 27. [b) Odd figures for example 3. (J/2). Thus making the classes as • Marks 0-10 10-15 15-20 20-25 25-30 30-35 35-40 ganisation of Data 65 ciples of Grouping There is no hard and fast rule for grouping the data. should be avoided for class intervals. 15—20.5 + 2. the magnitude of the figure and possibility of simplified calculations of further statistical studies.5 = 10). Say for first midpoint (12 5-7. -20. The quotiem is added and subtracted to first mid-point we get. By mid-points of class interval : Marks : 7. -30.5 = 5) and (7. -25. 10-15. 15-20. 7. (7.5 12. 33 etc.5 These mid-points are required to be converted into class intervals.5-2. (f) 'Open-end' class intervals In certain frequency distributions 'open-end' class intervals are given as we find in the example given below : Marks Below 10 10-15 15-20 20-25 25-30 30-35 35 and above Total Frequency (f) 7 10 13 18 8 5 3 64 In such cases values are put on the basis of construction of series. The preference for the total number of classes depends on the numbers and figures to be grouped. but following general principles ay be kept in mind for satisfactory and meaningful classification of data : [a) It is advisable to have total number of classes between 5 and 15. Marks Below 5 5-10 10-15 15-20 Above 20 The first and the last classes are open-end classes.■ Continuous Type I Marks Marks No. [e) The class interval should be equal for all classes. of Students No. Disi^^ti'^pe . For example. we prepare a table having three columns—first for variables. For statistical calculations the open-ends should be closed. the first is open at the lower-end and last at the upper end. See the following tration : Simple Series r . should be 0 or a multiple of 5. Simple Series and Cumulative Series : We have seen in the above illustrations the erns of simple series of discrete type and continuous type (Using inclusive and exclusive [lods of class intervals). In simple series the frequency is shown against each value or in cumulative series the frequencies are progressively totalled. of Students 10 4 0-10 20 8 10-20 8 30 15 20-30 15 40 20 30-40 20 50 13 40-50 13 66 Statistics for Economic Cumulative Series Less than 4 . V . Maintaining the regularity of the class intervals we can close these groups as 0-5 and 20-25.[c) Lower limit of the class as far as possible. second for 'Tally bars" and the third for the total representing corresponding frequency to each class. (/) As far as possible open-end classes should be avoided. [d) For maintaining continuity and correct classes exclusive method of preparing classes is adopted. (g) For frequency distribution. (c) Find the number of households whose monthly expenditure on food is (/■) less than Rs 2000 (ii) more than Rs 3000 Monthly Household Expenditure in (Rupees) on Food of 50 Households (c . From the following table given below of monthly household expenditi (m Rs) on food of 50 households. less than 30 marks are 27 and so on. more than 10 mai are 56.f} 12 (4 + 8) 27 (4 + 8 + 15) 47 (4 + 8 + 15 + 20) 60 (4 + 8 + 15 + 20 + 13) More than Marks i No. (a) Obtain the range of monthly household expenditure on food.Marks Less than 10 Less than 20 Less than 30 Less than 40 Less than 50 No. less than 20 marks 12. Illustration 3. of Students (i More than 0 i 60 More than 10 56 (60^) More than 20 48 (60-12) More than 30 33 (60-27) More than 40 13 (60-47) Now. In the same way the students getting more than 0 mark are 60. we can read students getting less than 10 marks are 4. more than 20 marks are 48 and so on. (b) Divide the range into appropriate number of class intervals and obtain the frequei distribution of expenditure. of Students {c. 1904 1559 3473 1735 2760 2041 1612 1753 1855 4439 5090 1085 1823 2346 1523 1211 1360 1110 2152 1183 1218 1315 1105 2628 2712 4228 1812 1264 1183 1171 1007 1180 1953 1137 2048 2025 1583 1324 2621 3676 1397 1832 1962 2177 2575 1293 1365 1146 3222 1396 (g) Sit )atten nethoi lass.(e (. It .r. the magnitude of the figure and possibility of simplified calculations of further statistical studies.S ■ rj^ . Range = L . 65 msatton of Uata . The preference for the total number of classes depends on the numbers and figures to be grouped. 7... The choice for the class intervals should be either 5 or a multiple ot 5. (a) Finding the highest and lowest expenditure on food of 50 households to get range by the following formula. (b) Odd figures for example 3. j (lustra Solution. 27. 33 etc. but following general principles ly be kept in mind for satisfactory and meaningful classification of data : la) It is advisable to have total number of classes between 5 and 15. should be avoided for class intervals.viples of Grouping There is no hard and fast rule for grouping the data. _ . 11. 9. See the following ration : Simple Series ■ Discrete Type % Marks 10 20 30 40 50 Continue No. (d) For maintaining continuity and correct classes exclusive method of preparing classes is adopted. ic) Lower limit of the class as far as possible. of Students Marks No. In simple series the frequency is shown against each value or I. Maintaining the regularity of the class intervals we can close these groups as 0-5 and 20-25. the first is open at the lower-end and last at the upper end. (e) The class interval should be equal for all classes. should be 0 or a multiple of 5. Marks Below 5 5-10 10-15 15-20 Above 20 The first and the last classes are open-end classes. Simple Series and Cumulative Series : We have seen in the above illustrations the terns of simple series of discrete type and continuous type (Using inclusive and exclusive lods of class intervals). of Students 4 8 15 20 13 0-10 10-20 20-30 30-40 40-50 66 Statistics for Economics-XI j Cumulative Series Less than Marks Less than Less than Less than Less than Less than 10 20 30 40 50 4 8 15 20 13 . (f) As far as possible open-end classes should be avoided. we prepare a table having three columns—first tor variables. second for 'Tally bars" and the third for the total representing corresponding frequency to each class. . For statistical calculations the open-ends should be closed.simplifies our further statistical calculations. For example. rf (P) For frequency distribution. in cumulative series the frequencies are progressively totalled. (c) Find the number of households whose monthly expenditure on food is (/■) less than Rs 2000 (ii) more than Rs 3000 Monthly Household Expenditure in (Rupees) on Food of 50 Households 1904 1559 3473 1735 2760 2041 1612 1753 1855 4439 . From the following table given below of monthly household expendit (m Rs) on food of 50 households. more than 20 marks are 48 and so on. of Students {c. Illustration 3.f. In the same way the students getting more than 0 mark are 60. less than 20 marks 12. of Students {t \ 60 56 (60-4) 48 (60-12) 33 (60-27) 13 (60-47) Now we can read students getting less than 10 marks are 4.No.} 12 (4 + 8) 27 (4 + 8 + 15) 47 (4 + 8 + 15 + 20) 60 (4 + 8 + 15 + 20 + 13) More than Marks More than 0 More than 10 More than 20 More than 30 More than 40 i No. (a) Obtain the range of monthly household expenditure on food. more than 10 ma are 56. less than 30 marks are 27 and so on. (b) Divide the range into appropriate number of class intervals and obtain the frequeu distribution of expenditure. L = Longest value. Range = L . L = 5090 and S = 1007 Range = 5090 .S '^ganisation of Data where.5090 1085 1823 2346 1523 1211 1360 1110 2152 1183 1218 1315 1105 2628 2712 4228 1812 1264 1183 1171 1007 1180 1953 1137 2048 2025 1583 1324 2621 3676 1397 1832 1962 2177 2575 1293 1365 1146 3222 1396 Solution. we decide 9 classes to include all the given values preparing a continuous frequency distribution by exclusive method (excluding upper limit). (a) Finding the highest and lowest expenditure on food of 50 households to get I range by the following formula. and S = Smallest value Here. Frequency (Excluding upper limit) Household Expenditure (Rs) 1000-1500 miMTHiTHl 1500-2000 MM III 13 2000-2500 Ml 6 20 Tally bars Frequency (f) .1007 = Rs 4083 (b) Dividing the class interval of Rs 500. we get 4083 67 500 = 8.166 Now. e.2500-3000 M 5 3000-3500 II 2 3500-4000 1 1 4000-4500 II 2 4500-5000 5000-5500 0 1" 1 Total 50 (c) (i) Number of households whose monthly expenditure is less than Rs 2000 (i. obtain class boundries and mid-values. 3000 . Illustration 4. Statistics for Economics-M O Frequency Distribution Class interval 10-13 14-17 18-21 22-25 26-29 30-33 34-37 38-41 Tally bars mini mini mill mi Frequency (f) 875421 .e.5500) = 2 + 1+ 2 + 0 + 1 = 6 Households 1..2000) = 20 + 13 = 33 Households (ii) Number of households whose monthly expenditure is more than 3000 (i.13).. 1000 . Form a frequency distribution from the following data by inclusive rthod taking 4 as the magnitude of class intervals taking the lowest class as (10 . 31 23 19 29 22 20 16 10 13 34 38 33 28 21 15 18 36 24 18 15 12 30 27 23 20 17 14 32 26 25 18 29 24 19 16 11 22 15 17 10 68 Solution. Total 40 Oass Boundaries In above illustration 10-13 14-17 io 91 ^c of inclusive method of construction of coin nor f ' ^^^ I™ or discontinuity between upper uZ S Xs ' w I ^e find 'gap 1 between the upper limit of the fct dasfia and rZT ^^ we find a '. and S = Smallest value Here.v.1007 = Rs 4083 (b) Dividing the class interval of Rs 500.13 = 1 2. Divide the difference by 2 1 Mid-point = 2 (c) m. L = Longest value. classified data is obtained Steps """"" class and ehe „pp„ 14 . The contmuity of tie vaLtk ^^^^^^^ adjustment m the class interval. = li+h nil [ methoc Also ol 31 38 12 18 Organisation of Data where.' L = 5090 and S = 1007 ' Range = 5090 . we get 4083 .gap' of 1. Frequency (Excluding upper limit) Household Expenditure (Rs) 1000-1500 MM MM 1500-2000 MM III 13 2000-2500 Ml 6 2500-3000 M 5 3000-3500 II 2 3500-4000 1 1 4000-4500 II 2i„ 4500-5000 5000-5500 Tally bars Frequency (f) 20 10 1" 1 Total 50 (c) (i) Number of households whose monthly expenditure is less than Rs 2000 [i.. Form a frequency distribution from the following data by inclusive bod taking 4 as the magnitude of class intervals taking the lowest class as (10 . we decide 9 classes to include all the given values preparing a continuous frequency distribution by exclusive method (excluding upper limit). 31 23 19 29 22 20 16 10 13 34 38 33 28 21 15 18 36 24 18 15 12 30 27 23 20 17 14 32 26 25 18 29 24 19 16 11 22 15 17 10 . 1000 .e.e.166 Now. obtain class boundries and mid-values.67 500 = 8. 3000 .13)..2000) = 20 + 13 = 33 Households («■) Number of households whose monthly expenditure is more than 3000 (i.5500) = 2 + 1+ 2 + 0 + 1 = 6 Households Illustration 4. 26-29 and so on are class Um of mclusxve method of construction of contmuous frequencv distribution.Iass Boundaries In above illustration 10-13.—^ " ^^ Steps 14 . 14-17. 1 Divide the difference by 2 . Frequency Distribution Statistics for Economics-XI j Class interval 10-13 M 5 14-17 mini 8 18-21 mini 8 1 Tally bars Frequency (f) 22-25 m^ii 7 26-29 m^ 5 30-33 nil i 4 34-37 n 1 7 38^1 1 i i 1 Total 40 f^.68 Solution. We S 'Z or discontinmty between upper limit of a class and lower limit of next class Fo elS JTnT: trotftr' f ^^^ ifL dat Ii. 18-21.13 = 1 2. 22-25. 5-41.5 1 39.5 5 11.5-37.5 37.5-17.5 8 15.5 7 23.5 5 27.5 21. d by lit of Class ii^erval frequency (f)mid values (m.5 3..5-21. 9.5 2 35.5 25.5-33. Subtract the value obtained from lower limits of all the classes (.5 4 31.v.5 15 22 Total 40 28 17 29 21 23 27 18 12 7 2 .5 13.5-25. Prepare a frequency distribution by inclusive method taking class interval of 7 from tbe following data : imits japs' nple.v.).5-29. = Ink 2 lelativ It i factual 1 ■ganisation of Data Now we get 69 Illustration 5. Add the value obtained to upper limits of all classes (+ 0 5) are itri^Ld'^"'""" " ^^ each cL Mid-point = Upper class limit+Lower class limit 2 m.5 17.5 33.5 29.=0.5 8 19.0 5) 4. s 14.5-13. 9 4 6 1 8 3 10 5 20 16 12 8 4 33 27 21 15 9 3 36 27 18 9 2 4 6 32 31 29 18 14 13 15 11 9 7 1 5 37 32 28 26 24 20 19 25 19 20 Solution. If actual frequencies I expressed as per cent of the total number of observations. Frequency Distribution (Inclusive Method) class Tally bars Frequency (f) 0-7 miMmi 15 8-15 miMTHi 15 16-23 mm nil 14 24-31 mimi 1 11 32-39 M 5 Total 60 ^tive Frequency Distribution It is sometimes required to show the relative frequency of occurrences rather than ual number of occurrences in each class of frequency distribution. relative frequencies are ained. 70 Individum ] 2 3 4 . 5 Money (Rs) 114 108 100 98 106 Individual 6 7 89 10 Money (Rs) Individual 109 11 131 117 12 136 119 13 143 121 14 156 126 15 169 Money (Rs) Individual ■"-'igclllisc frequencies. 16 17 18 19 20 182 195 207 219 235 Money (Rs) 75-100 100-125 125-150 150-175 175-200 200-225 225-250 Frequency Distribution (Excluding upper limit) . Solution. .. We take a .35.OJ lof 7 : Tally bars Frequency (f) Mil Total 2742221 Relative frequency (%) 20 10 35 20 10 10 10 5 100 Soli (Assuming the class interval of Rs 25) of Frequency Distribution of Money Money (Rs) 50-100 100-150 150-200 200-250 Tally bars MMii Total Frequency (f) 1 12 4 3 Relative frequency (%) 5 60 20 15 dative ] It is ! (ictual nuj exprei obtained. .e. 5 29.) Total 40 fflustration 5.5-37.5-17.5 21.5-41.v.5 8 15.5 mid lvalues {m.5 5 11.5-13.5 17.anisation of Data 69 Class interval Frequency (f) 9.5-21.5 33.5 8 19.5 4 31.5-25.5 1 39.5 13.5-29.5-33.5 2 35. 17 15 22 29 21 23 27 18 12 7 2 4 6 1 8 3 10 5 20 16 12 8 33 27 21 15 9 3 36 27 18 9 2 6 32 31 29 18 14 13 15 11 9 7 5 37 32 28 26 24 20 19 25 19 20 Frequency Distribution (Inclusive Method) Class Tally bars 0-7 mimm15 8-15 miMM 15 16-23 mm 111! 24-31 MM 1 11 Frequency (f) 14 .5 25.5 5 27.5 7 23. Prepare a frequency distribution by inclusive method taking class interval 28 9 4 4 1 Solution.5 37. relative frequencies are ained. Solution. 90 Statistics for Economics-XI Illustration 6 In a hypothetical sample of 20 individuals the amounts of money them were found to be : Individual Money Individual (Rs) Individual (Rs) Money Individual (Rs) 1 114 6 109 11 131 16 182 2 108 7 117 12 136 17 195 3 100 8 119 13 143 18 207 4 98 9 121 14 156 19 219 5 106 10 126 15 169 20 235 frequencies.32-39 M 5 Total 60 lative Frequency Distribution It is sometimes required to show the relative frequency of occurrences rather than Illmber of occuLnces in each class of frequency distribution If actual frequencies ^pressed as per cent of the total number of observations. Frequency Disttibution (Excluding upper limit) Money {Rs) Tally bars Frequency If)Relative frequency (%) 75-100 II 2 10 100-125 Mil 7 35 125-150 nil 4 20 150-175 II 2 10 175-200 II 2 10 200-225 II 2 10 225-250 1 1 5 Money . For example : ative 1 Set I Set II Set III pfass Frequency Class Frequency Class Frequency 0-5 X 2 X 2-A X 5-10 Y 5 Y 2-6 X+Y 10-20 Z 7 Z 2-8 X+Y+Z 20-30 A 7-20 A 8-10 A 30-50 B 20-40 B 10-12 B 50-75 C 40-60 C 12-14 C x)ss of Information Raw data is grouped by making equal or unequal class frequency distribution. 5-7.Total 20 100 (Assuming the class interval of Rs 25) Frequency Distribution of Money Money (Rs) Tally bars 50-100 1 1 c 100-150 150-200 Mmiii nil 200-250 3 III Total 20 12 4 J 60 20 15 100 -XI vith ganisation of Data ^^ quency Distribution with Unequal Classes Data are sometimes given in unequal class intervals. 12-20 and so on. say 15. Such series are used when there f great fluctuation in data. By making such classes there is . 7-12. 5-10. 10-15 or 0-5. example. the statistical analysis is based on die mid-points of these classes without giving any importance to individual observation. 24 26 27 25 28 24 27 28 25 26 17 18 19 17 20 18 18 19 18 19 25 26 27 25 27 26 25 26 26 26 17 18 19 19 20 19 17 20 17 18 72 r . and (ii) age of wives. livariate Frequency Distribution We have so far studied above frequency distributions involving single variable only. Further. -I i '1 i. ages of husbands and wives for group of couples. uch frequency distributions are called univariate frequency distributions. the significance of individual observation is lost. ^ such. etc.loss of information of individual observation. we may study the weights and heights of group of individuals.uiined by a group of students in two different subjects. We are given two variables : (i) age of husbands. A frequency table where two variables have been measured in the ue set of items through cross classification is known as 'bivariate frequency distribution" ntervalB 'two-way frequency distribution'. We Id represent the data in the form of a two-way frequency distribution so that we are to show the ages of husbands and wives simultaneously. lUustration 7. Various values of each variable are grouped into ious classes (not necessarily the same for each variable). Following figures give the ages of 20 newly married couples in year. the marks . This is also called bivariate \cy distribution. Often we come aoss data composed of measurements made on two variables for each individual items. jresent the da ! of husband t of wife {of husband ! of wife Solution.fV Age of husband (years) 24 25 26 . and so on. S.27 28 Total (/) 17 Bivariate Frequenqr Distribution . Age of wife. (years) Statistics for Economics-Xll I (1) III (3) I (1) (1) (1) (3).N. 125 to 135 lbs.. 1 2 3 4 5 6 7 . (1) (1) (2) (2) (1) 20 I (1) (1) (1) Total ( 25742 20 Illustration 8 Tbe data given below relate to the heights and weights of 20 nersc 66" IT class Tnterval 62^ 64 -66 and so on and 115 to 125 lbs. N. Such series are used when there eat fluctuation in data. Height S.) 64—66 62—64 Height -------. 66— 6S—70 70—72 Totali 115-125 125-135 135-145 145-155 155-165 165-175 II (2) i (1) 1 (1) 1 (1) II (2) 1 (1) II (2) 11(2) i (1) 1 (1) 1 (1) 1 (1) Total (/) 3 4 5 4 4 1 (1) 111 (3) 456311i 20 71 anisation of Data ^^uency Distribution with Unequal Classes CData are sometimes given in unequal class intervals. Weight 170 70 11 163 70 135 65 12 139 67 136 ■ 65 13 122 63 137 64 14 134 68 148 69 15 140 67 124 63 16 132 69 117 65 17 120 66 128 70 18 148 68 143 71 19 129 67 129 62 20 152 67 Height Inches lbs.8 9 10 Solution. For example : ions. iO W Set I Setll Set III . 64". Various values of each variable are grouped into ous classes (not necessarily the same for each variable). the marks ained by a group of students in two different subjects. 1 frequency distributions are called univariate frequency distributions. 5-10. 7-12. A frequency table where two variables have been measured in the • set of items through cross classification is known as 'bivariate frequency distribution' i'two-way frequency distribution'.30-50 B 20-40 B 10-12 B J&-75 C 40-60 C 12-14 C Frequency A a (f) KS of Information Raw data is grouped by making equal or unequal class frequency distribution.jiate Frequency Distribution We have so far studied above frequency distributions involving single variable only. By making such classes there is loss of information of individual observation. 12-20 and so on. Following figures give the ages of 20 newly married couples in year. etc. Inhistration 7. . the significance of individual observation is lost. Further. say -5. 10-15 or 0-5. such. the statistical analysis is based on ' mid-points of these classes without giving any importance to individual observation. Often we come uss data composed of measurements made on two variables for each individual items. • example. 5-7.Frequency Class Frequency 1 0-5 X 1 X 2-A X 5-10 Y 5 Y 2-6 X+Y 10-20 Z 7 Z 2-8 X + Y+ Z A 7-20 A 8-10 t20-30 . we may study the weights and heights of group of individuals. ages of husbands and wives for oup of couples. 24 26 27 25 28 24 27 28 25 26 17 18 19 17 20 18 18 19 18 19 25 26 27 25 27 26 25 26 26 26 17 18 19 19 20 19 17 20 17 18 (of husband t of wife ! of husband t of wife . We are given two variables : [i) age of husbands. We lid represent the data in the form of a two-way frequency distribution so that we are to show the ages of husbands and wives simultaneously. and (ii) age of wives. 72 Bivariate Frequency Distribution Statistics for Economics (years) 24 25 26 27 28 17 Total (/) I (1) III (3) I (1) 18 19 I (1) I (1) III (3) I (1) 6 (1) (2) (2) (1) 20 (1) (1) (1) . This is also called bivariate icy distribution.Solution. 1 2 3 4 5 6 7 8 9 10 170 135 136 137 148 124 117 128 143 129 Solution. Height 70 65 65 64 69 63 65 .N.Total i 25742 ______——20 You are required ro nterval 62"S. N.70 71 62 S. 11 12 13 14 15 16 17 18 19 20 Weight 163 139 122 134 140 132 120 148 129 152 Height 70 67 63 68 67 69 66 68 67 67 Bivariate Frequency Distribution Inches lOrganisation of Data 73 _: . by ! inclusive method : 28 17 15 22 29 21 23 27 18 12 7 2 9 4 6 1 8 3 10 5 20 16 12 8 4 33 27 21 15 9 3 36 27 18 9 2 4 6 32 31 29 18 14 13 15 11 9 7 1 5 . Distinguish between discrete and continuous variable. ! What do you understand by classification? Explain the methods of classification of j data giving suitable examples. Explain with illustration the 'inclusive' and 'exclusive' methods used in classification of data. Write short notes on the following : I (a) Classification and series. 11. Define Frequency Distribution. 9. i Define classification. Is there any use in classifying things? Explain with illustrations.exercises uestions : I Distinguish between variable and attribute. . I (/) Equal and unequal class frequency blems : Prepare a statistical table from the following data taking the class width as 7. State the principles required to be observed in its formation. [b) Geographical and chronological classification. 8. Define series and explain the different types of series. Distinguish between univariate and bivariate frequency distribution. 10. Do you agree that classified data is better than raw data? 13. (c) Exclusive and inclusive class-intervals. ^ Explain discrete and continuous variables with examples. I [e) Simple and cumulative frequency. What is loss of information in classified data? 12. i (d) Discrete and continuous series. What is a relative frequency distribution? Illustrate. Explain the objects and characteristics of classification. Explain with examples. of students .37 32 28 26 24 74 I i^v Statistics for Economics-XI j 50 57 58 51 53 62 64 60 61 51 64 55 55 52 60 65 58 60 52 63 56 56 58 64 63 62 60 54 62 54 54 60 65 60 62 59 56 63 52 53 62 53 61 61 59 the following marks in lowest class interval frequency table taking 69 33 91 53 63 69 70 36 80 78 52 51 73 73 92 64 55 49 74 57 95 70 64 57 75 80 42 85 43 29 77 65 73 95 76 53 86 73 40 83 43 76 84 72 75 57 58 59 62 65 67 87 81 84 61 75 85 81 58 81 the 4. 47 69 78 62 72 43 87 61 84 23 Change the following into continuous series and convert the series into 'less than' and more than cumulative series : Marks (mid-values) No. 5 15 25 35 45 55 8 12 15 9 4 2 Marks obtained by 24 students in English and Statistics in a class are given below » S.No. Marks in English Statistics j Marks in Statistics S.8 25 19 20 28 19 9 22 16 21 25 19 10 23 18 22 24 16 11 24 18 23 23 17 12 24 17 24 25 19 I " ^^ Marks in ] ^ganisation of Data tin a survev it was found that 64 famiUes bought milk in the following quantities a parSar Inth. Quantity of milk (in litres) bought by 64 famthes m a month. Marks in English 1 22 16 13 23 16 2 23 16 14 25 17 3 23 18 15 23 17 4 23 16 16 22 17 5 23 16 17 27 15 6 24 17 18 27 16 7 23 16 19 26 18 .5.No. .O 99 9 22 12 39 19 14 23 6 24 16 18 7 i y. p • i I i i iH i i ■■ . « • • . and (b) the class boundaries. 167. I Find out the frequency distribution and 'more than' cumulative fi^quency^^ble .1 Comrert the above data in a frequency distribution making classes of 5-9. Vviii*i/~vn Marksin 10 11 10 11 11 14 12 12 13 10 Marks in Economics 20 13 24 21 12 23 22 11 22 21 12 23 23 10 22 23 14 22 22 14 24 21 12 20 24 13 24 25 10 23 Marks in Statiaics Marks in Eammcs 8 Prepare 'less than' and 'more than' cumulative frequency distributions of the ^ 140-150 15.160 160-170 170-180 180-190 190^200 Ino. 153. find (^i) size of the class intervals. 160. 146. 174. 139. -u1 I: The marks obtained by 20 studends in Statistics and Economics are./ '. of workers : 5 10 20 \. r _____—. 181 pounds. 10-14 and J so on. below : 10 30 40 50 60 Quantity(kg) : 17 22 ^ lociqo If class mid-points in a frequency distribution of a group of persons are : 125. 132. given below. . » ". Parts of a Table Types of Table J" jjji w. (it) Semi-tabular presentation. as we.Introduction 2.' . (ni) Tabular presentation. J^tJZ riLX There are four methods of presentation. (i) Text presentation. and (iv) Pictorial presentation. b. Definition and Objectives of Tabulation 3.. Essentials of a Satisfactory Table 4.PRESENTATION OF DATA g^^lfeftwr^ Prcssentaiion -4«nmaiic Presentation Chapter 5 tabular presentation ' 1. eUH.e da. They are : eral people. There was around ten-fold increase in this sphere between 1991 . (i) Text Presentation \^Jabular Presentation ^^ increased from an extremely low figure of less than 2 lakhs in 1950-51 to over 46 lakhs in 1990-91. headings and notes to make and full meaning of the data and their origin. but is useful when figures are required to be compared along with one or two sentences of explanation.5 crore mobile phones. 2. They facilitate comparison due to proper systematic arrangement of statistical data in different columns. According to H. (iv) Pictorial Presentation Pictorial presentation is visual form of statistical data in diagrams and graphs..and 2004-05 as the number of landline connections increased to 4. In brief. ^ (Hi) Tabular Presentation Tabular presentation is a systematic presentation of numerical data in columns and rows in accordance with some important features or characteristics. A statistical table is an arrangement of I systematic presentation of data in columns and rows.6 in 2001 to 6.42 crore besides 4. the teledensity [viz.'' bjectives of Tabulation Statistical data arranged in a tabulated form have following important objectives: I 1. This method is not often used. 78 It . («) Semi-Tabular Presentation ■ Semi-tabular presentation is both through tables and paragraphs. They simplify complex data and the data presented are easily understood. According to Tuttle. with sufficient explanatory and alifying words. With Wnifold increase in telephone connections.nilar and should be compared". Tabulation is the process of fpresenting in tables. and objectives of tabuutio» c Systematic presentation of data is one of the most important consideration in statistical j work and it is done through the use of tables. Thus the number of telephones stood 9. "Tables are a means of recording in permanent form the alysis that is made through classification and of placing juxtaposition things that are . Secrist. Tabulation is a process and the outcome of which are statistical Itables.7 crore in March ? C05. "A statistical table is the logical listing of related quantitative ta in vertical columns and horizontal rows of numbers. tabulation is a scientific process involving the presentation of classified ata in an orderly manner so as to bring out their essential features and chief iracteristics. phrases and statement in the form of titles. the number of telephone connections per hundred persons) has increased from 3.7 m 2005. proportion. 3. between them easy. According to objective : A table should be according to objective of statistic investigation. 6. If the table is too large becomes confusing to the eyes and there is great difficulty in following the lir and columiis at a glance. ^^ « easily understandable. They leave a lasting impression without any confusion. 4. price m rupees" or "weight in kilograms". etc. then a number of sr tables should be preferred to one big table. If more details are to be given. it should be simple and comp. To It so. should be complete within itself containing all the explanations necessary to mi clear the meanmg to items. etc. comparison is one of the chief objectives of tabulatio Whenever it is necessary. Tabulated data are good for references and they make it easy to present intormation on graphs and diagrams. All the requir . 5. Columns and rows should be numl when It is desired to facilitate reference to specific parts of a table nSffr '' scientifically prepared. correlation etc. Comparable : The facts should be arranged in a table as to make comparis. proportion of columns and rov writing of figures. care should be taken in determining its size. So. ma Attractive : A table should be attractive to draw the attention of readers.. should be given the table to facihtate comparison. They present facts in minimum space and unnecessary. Tabulated data makes easy for summation of various items and errors and omissions can easily be detected. percentage. Manageable size : The size of the table should be neither too big nor too sma loo much of details should not be given in a table. 7. average. 4. ^^^m of a satisfactory ti^^ The following are the essentials or characteristics of a satisfactory table : 1. They facilitate computation of different statistical measures namely averat dispersion. Units of measurement must be clearly stated such. 2. repetition and explanatic are avoided and required figures can be located more quickly. because.Statistics for Economics-XI j 3. proper captions and stubs.) whenever more than or^e table ^s^ prepa ^^^^^ ^^ ^ ^^^^^ ^^^^^^^ either at the centre on the top anove rnc^^^ Sometimes table number the table number is given refers to the chapter or section like 1.) or numbered (say 1. in the future. ^ ^ng a^tference 1 Table number : A table should ^B ^^^ etc. 3. A complete title explams of classification of data. footnotes etc. A table must be codihed ^e A. Captions and stubs (Column he^^ ^^^^ by smbs^ ^^^ . ^ ^^ or a catch title written 2. Title given above of all lettering used m the The lettering of the title should be ^he most pr ^^ ^^^^^ ^^^ ^^e table. of ^ 3.2 and 2. ^ ^ ^ 2. In ^ ^ould mean second table m and second digit to its order. field to which the data are -^^ed jf ^a) bas. 4 etc.4 are also used. ^^the fourth table in second first chapter or section and Table 2.ru es of tabulation should be carefully observed. Title : There may be a V'^^^^Se^ be W. clekr and self explanatory. Large numbers are hard to read and dif to compare therefore. up to the nearest 79 Wlabular Presentation OF ATi A good table . Certain figures which are I be emphasised should be in distinctive type or in a 'circle' or a 'box' or ber^^ thick lines^ A table should have miscellaneous columns for the data which can« be grouped m the classification made. source. they should be approximated e.g.. in few words. an art. A table should have a suitab title.4 wo chapter or section. This table ^^ould be mad^^^^^^^ in view the purpose of arrangement of items b^ excluded. if any) Statistics for Economics-Xll ifcsiEswwaiiisaMJB^ aw . d etc. ^„rce note should be 7. 80 \'L structure of table Number Title (Head note. . a.»o. F«. ^^rtable :lt con^ms je ^^^^ part of the table. : I.7 to sp y f « diei is any or to explain »me . Footnotes can be identified by vanom sys ^^ ^^ 3 „ or signs (say £ etc. to the reader to fi^ and gather additional information. ^ sSnif cance which are to be the table It is a phrase or a statement 6. Source : ta case of the ■'•j'^XalnTst ^taon.headings given to columns a e caUed captio ^ ^^ ^ be numbered Both stubs and captions ^^ould be « ^^^^^^ ^^^ and stubs rfof cSt^ons verticaUy and stubs ..e. b. in columns and rows and : (0 alphabetically W Items in a table may be arranged ^progressively. name of the pubUsher or a°t ti:" o^-. and (vt) geographically.) or ^ Umitattons of data. c.) Footnotes m also necessa. is usefu. (Hi) chronologically. million tons. is placed « "om of tte ta^ ^^ ^ which contains systems and keys like puttmg star(s) table. 00 69.90 2001 71.70 Total 27.---- table 1 Literacy Rates in India Year Rural 1951 19.16 40.40 Source : Economic 45.02 1961 34.38 .70 81.30 1971 48.-—— ■ — _________ Footnote : Source : .96 56.60 1991 57.60 1981 49.80 76.60 66.10 86.40 45. 76 39.30 Total 18.33 10.70 59.86 15.60 64.64.20 67.10 40.40 60.87 urttan 22.00 46.20 73.97 29.50 15.00 44.16 Rural 12.50 27.29 54.35 21.33 .13 75.10 22.59 25.20 Total (Per cent) Persons 8.40 The table clearly shows that • per cent among females.50 48.70 56.10 80.80 21.30 30.85 Rural 'Females 4. This shows Urban 34.90 36.70 73. Represent the above data in a tabular form. STOWN A 100 TOWN B 100 1 Non-Coffee drinkers drinl(ers 35 40 Non-Coffee drinkers drinkers . etc.38 per cent in 2001. Illustration 1.28. girls still get discriminated in the matters like health. This clearly speaks of inadequate facilities of education av^IbL in the rural areas as well as comparatively lower willingness of the conservative rural folk to go to schools for education. while it that there is a general bias Total 81 globular Presentation against female education and in our conservative society. nutrition.30 34. Males non-coffee drinkers were 30% and Females coffee drinkers were 15%. iii) Literacy rate in urban areas was high at 80 per cent in 2001 than rural areas where ^ ^ ^t rs less L 60 per cent. before Solution. Let us calculate the missing percentages of the above information before representing the data in a tabular form.57 52.21 65. the ^tal coffee drinkers were 45% and Males non-coffee were 55%.45 43. education. In a sample study about coffee drinking habits m two towns. 30 25 TABLE 2 Coffee Drinking Habits in Towns A and B (in percentages) Coffee Drinkers Non-Coffee Drinkers 82 Alternative Solution 104 Statistics for Economics-XI TABLE 3 Coffee Drinking Habits in Towns A and B {in percentages]} SBSSsSiePlSlillBJi Toum A Town B ■ ■ ■■ ^ CoffeeNon-Coffee Total CoffeeNon-coffee Total ] Drinkers Drinkers Drinkers Drinkers Males 40 20 60 25 30 55 Females 5 35 40 15 30 Total 45 55 100 40 60 100 . 628 are boys and 440 are science students. 720 are Hindus. Of the 1. The number of Hindu boys is 392. Solution.1 45 Illustration 2. the number of science students among the Hindu boys was 148. TABLE 4 Faculty Boys Girls Hindus Non-Hindus Total Hindus Non-Hindus Total J Total 1 Non-Hindus Total Hindus .125 students studying in a school during 2005-2006. that of boys studying science 205 and that of Hindu students studying science 262. finally. Enter these frequencies in a table and complete the table by obtaining the frequencies of the remaining cells. I Tabular Presentation Solution. From the point of view of purpose : (i) General purpose tables. 83 TABLE 5 Growth of Population in India (figures in crores) Source : Census of India 2001. From the point of view of originality : {/■) Original tables. (ii) Special purpose tables. Census of India 2001 reported that Indian population had risen to 102 crore of which only 49 crore were females against 53 crore males. From the point of view of construction (i) Simple or single tables. urban population an even higher share of non-workers (19 crore) against the workers (9 crore) as comp.Science Arts 148 244 57 179 205 423 262 262 458 178 227 440 j 685 1 114 214 Total 392 405 236 628 328 169 497 720 121 48 235 1125 j niustration 3. B. 74 crore people resid m rural India and only 28 crore lived in towns or cities. Males : 53 crore Females : 49 crore fs of tables Table can broadly by classified as under : A. library . (ii) Derivative tables. (ii) Complex tables. to the rural population where there were 31 crore workers out of 74 crore populatic Represent the above information in a tabular form. While there were 62 crore nc workers Population against 40 crore workers in the entire country. C. . purpose tables.o„ i.• . . . .. Se.^e «ae fom. wMcH co„. and Section .era..„ SsfT f u 84 Simple and Complex Tables . «J. table 8 According .o.. table 7 I---i" a School Mirt. . '^"tsZa^^'J (T-ble Tabulation) dlustration : g'^Is mto Sec A.i «ble provides of » ts^s^SSllt^^^^ it eaiy to make comparisons and clear relationships. and b in our following j r^. in wUch they are origi^lly collected -sSl^ciurX^^^^^^^^ -^a . table 6 of Students in a School Marks 0-10 10-20 20-30 30-40 Total 15 12 28 5 60 Double or WWay Tabte (Double Tabulation) .o Mari.ai„ —. inro™a. .^ \ : ^ ^ ^^^^^^^^ "■" . Statistics for Economics-Xli we^eTrL"""' --op" in . ular Presentation 85 The above table can even be called as manifold table. ' I What is a statistical table? Discuss briefly the essentials of a good table. higher order table or ma^ 2lon Z "lh we can increase the number of charactensttcs... ■ 1 What are the objects of tabulation?... Draw a structure of a table I ExpLn bnefly the main characteristics of a good statistical Whai are the points to be taken into accomit while preparing a table? IxpTarand discuss the various types of tables used in a survey after the data have . " rh'aX"is needed when a number of characteristics are to be simultaneously ii But as more characteristics are included. cloth and wool industries. Explain the objects of tabulation. ^ Discuss briefly the importance of tabulation. . y ^ five 2006.. iron-ore mining... If the field of investigation is not big. ^ hArepare a blank table to show the distribution of population according to sex and ^ four religions in three age groups in Delhi and Mumbai. J ... ^i:^!rt. may be confusing to the reader. EXERCISES tions : I State the advantages of Tabular Presentation of data.■ Fishing. the table becomes more complex.ween tabulation and classification.. ifabulation wiU be more accurate than the manual process. u.he table requirements are varymg. ^ . more sections. i Describe the major functional parts of statistical tables.»w„s following d. coal mining.. i following industries : . the data have not too many future use and thirdly when . and . 2005 and 2006. 30% were coffee drinkers. Tabulate the following • JXaSSe^—^^^ of the total sales during the yeaT "^P^cfvely. . and 26% were male coffee drinkers 55% people were males. and 20% were male coffee drinkers .Preset the following information in a suitable table • oi ^ —— Wong ro a trade un. Under-graduate and Post-graduate classes. Male and Female. Social Sciences.Town A 51% 16% 18% Town B 54% 28% 20% p Males in Total Population ' Smokers Male Smokers . Texnles accounted for 30% pesttt^ •t Ls Town A Town B 60% people were males 40% were coffee drinkers. J Tabulate the above data.™ 20M T "" of which 1290 were men Sn Z h ^ u students according to : (a) Faculty (b) Class (c) Sex id) Years ''' """ ""i •<> 1^80 mformafon regarding the college] 8. Commercial Sciences. rORIAL PRESENT^ r 1 Presentation resentation —► ONE-DIMENSIONA^DIAGRAMS (/■) Simple Bar Diagram-(i7) Sub-divided Bar Diagram (i/i) Multiple Bar Diagram (/i^ Percentage Bar Diagram (v) Broken Bar Diagram (vA Deviation Bar Diagram -►TWO-DIMENSIONAL DIAGRAMS (f) Rectangles (fO Squares '//A Circles and Pie-diagrams -►THREE-DIMENSIONAL DIAGRAMS (!) Cubes (iO Cylinders (I/O Blocks etc.Chapter 6 digrammatic presentation Introduction Importance and Uses of Graptis and General Rules for Constructing Diagrams Types of Diagrams A. —► PICTOGRAM —►CARTOGRAMS OR MAPS ♦ GRAPHS OF FREQUENCY DISTRIBUTION (i) Line Frequency Graph Histogram (Hi) Frequency Polygon (iV) Smoothed Frequency Curve (Frequency Curve) 'Ogive' or Cumulative Frequency Curve GRAPHS OF TIME SERIES (A One Variable Graphs (Ii) Two or more than two Variable Graphs (i/0 Graphs of Different Units fr 88 r =a . Pie Diagrams Limitations of Diagrammatic Presentation Other methods of presentation. One-dimensional Diagrams B. ee. Twf is dSt^^^ ^ Central Tendency of this book. Diagrams play arimorttr'1 " campaigns. becomes easy Thus. information to the common man TW are wideT Particularly to givj and other fields. ^^apter on Measures of] whjes for constructing I dCf .conunon or representing the statisticalT^^^^^ xs the most popular an are appeahng to the mind through the eyL as the^ of Presentatior For the purpose of simplifying an^tter^p'Tas ''h chapter and some iZonZ Znr used m presenting statistical informal ^ ^^^^ W diagrams m this ^ are commonly and uses of graphs and 1. self expLato^ a^d^^^^^^ sub-headings can also be . Uei general rules are observed : " advantageous if following conveys main facts depicted by the Sa^m T^n ^^ but short.Be acu^el through p„„. They are interesting. fairs.e^"^ ^^^^^ -ho is not ^agrams are used for publicity X^a^aX'"' • • TW save time and energy of say without any stram on' mmd a"d knowl^^^^^^^^^ ^^^^ ^^ey warn to data simple and intelligible knowledge of mathematics as they make the) quick comparisons.c presentation of graphs anj^ia^^ SLv I T'^ ^^ ^y Ae! journals. m the modern advertising ^ ^ ^^ an importarn role hke median. mode etc. attractive and impressive • h. diagrams can be used for ] 4. given It must be brief. newspapers. quartiles. board meeting etc D L^T ^^ ^ " exhibitions.I fluctuations of the statistical values bv^n? ' ^^^ ^^^nd and interested in going through tiff^ure's^. They have universal utility • Sinr^. . an index must g for identifying and understanding the diagram. zo. .t far as possible be in even numbers or multiple ot 5. A i^A^^ ■ The 'sc/j/e' of measurement on both X-axis ana i sl'zrrrf ^^ « f U t^aror^rVftX™ ^nJt through different colours. One-dimensional Diagrams B.. . . the source from which data have been obtained. neat and appealing to the eyes. more effective than a complex one. It should be attractive.. Pie Diagram A ONF-DIMENSIONAL DIAGRAMS ™lionaMiagrams are also called ^^^^ JthXiror:! used in practice. lU.for Identification and for purpose TrefereJe ^e used Digrammatic Presentation 89 i . types of diagrams There are various types of geometric forms of diagrams used in practice as shown on following two Geometric forms of diagrams : A. shades. . dotting. luo 'months' on X-axis. ^u v onH Y . 'il^ilifp 'poputoton. etc..J The «:i7e of the diagram should be neither too big nor too paper. /u. crossing.or 'productton' on Y-ax>s and -years' or 'months "n X-ax.^ scale slould be selected to su. so that peoples attention is automatically drawn towards it. 4 Scak ■ A diagram should be drawn with the help of geometric . They are called one-dimensional because of height of the bar 90 . . -Xc^ore. production. Draw a b«r A:. --. T/oo' " " the series. population.500 '' .500 36. sa.2000. A Jimple^rdlTlrc^ b'^ vertical base.Xt e^dilr one category either in years. export of computer software (1997-2002) Scale : 1 cm = Rs 7. ^fonnation of Tu or shading to make them more simpfc bar diagram the scale is detet^ned'^teiTof ^ Illustration 1. 2002-03 p 144) Solution. months w«ks et T or groups. It is used for vistil ? T "" horizontal o. .000 crore Y 42000 35000 ff 28000-S O r 21000^ 140006.. ' . &o„omfc Survey.es.significance and not the width of the bar Foil " u (-) Simple bar diagram (b) ^^ j ^^^ ^ypes of bar d^ Sub-divided bar diagram (c) Multiple bar diagram id) Percentage bar diagram (e) Broken bar diagram (/) Deviation bar diagram (a) Simple Bar Diagrams • The variable can be presented. All the bars L be brau^LZ attractive. "" of computer softw^r relating to expo.OOf-0.. 000 crores 2001-02 2000-01 1999-00 1998-99 1997-98 0 36. Export of Computer Software (1997-2002) Scale : 1 cm = Rs 7.500 ■ 28.000 crores.940 .150 10. ^ „ V ^vU w T Years on X-axis.150 10.500 —r 7000 14000 21000 2^00 3M00 42000 Rupees (in crores) . 1 2001-02 "tr rrr aw——— .Vertical base. Value (Rupees m -^ores on Y-ax s W 2 • Years on Y-axis.350 4 17. 6.940 YEARS Fig. vertical base showing horizontal b«s as under : Alternative solution .350 17.28. Scale : 1 cm = Rs 7. Value (Rupees m crores) on X-axis. .he guZrr/a Sir given ^^^^t^stics for Economics-X (h) Sub-divided Bar Diaeram • TJ.. e First advance estimates (Khanf only).0. 3 92 ReauirPnt^^t j ' .-05 -06 1999-00 -01 -02 . .!^^^^^^^ ^Component Ba values of the given data is to be dividedTo v^no F-t of all a bar representing total s CwrXn '^e to.03 -04 -05 -06 Fig. 2006.00.. ...!: o"^^^^ . j In general sub-divfdedo.______Dri>>a nhanaes given below : Poodgrains Production (in million tons) Wholesale Price Changes 52-WeeteAverageJnflat^^ '1999-00 -01 -02 -03 -04 X (Provisional) Average up to Jan. dotting or designs can remember that the various componentrshouFd be t ' tndex' IS to be given alongwith the Lgram to ? " ^^^^ bar.crores) m 1997-98 to Rs per year for last four years. Z ^^ proportion to the values given in the dat^ S/ ? be'used to d^sttguish-os-g . are . -TwoXtl^rr..^^^^ i„ the export of computer ^^^^^^^ the software export have .o^ -nomlc survey . 14. P^^s i.. above ™o „e ui^L ^ . 400-1 O a loot e robbery ■ murder 200 . Draw a suitable diagram to represent the following mfo^ation : Year 2001 2002 2003 2004 2005 Trains Murder 108 131 97 102 75 Solution.Illustration 2._ ' ^^ia . Draw a snir. Robbery 82 115 144 70 68 Loot 321 386 352 285 245 Total 511 632 593 457 388 CRIME IN RUNNING PASSENGER TRAINS (2001-2005) Scale : 1 cm = 200 crimes 800 600co m E .Kl ^ differences. Draw a suitable diagram of the following data : Statement of CrimeJnR^g Passenger Trains .s sn.agram to represent the above data. Statistics for Economics-} Illustration 4. Suh-d.2001 pnno -----^-2002 2003 YEARS 2004 jr 2005 Fig.ab. 5 94 .E..n . Ltion. d be given. ' 93 Qigrammatic Presentation ^ifU^w^ofD^riation S.v. Asia 2003-04 2004-05 YEARS Fig.e to the ahove data.aea bar a. YA 100 Q Other Regions ® Africa pfi West Asia o-E. 4 (c)M between t inter-relaC of drawin. Asia West Asia Africa Other Regions Total d.ta.a. In this cai spacing isi in a set. .000 tons for exports as against 1 54 000 ton / . Year Murder Robbery 2001 108 82 321 2002 131 115 386 2003 97 144 352 2004 102 70 285 2005 75 68 245 Loot CRIME IN RUNNING PASSENGER TRAINS (1998-2002) (Scale .Solution. consumption exports during the sam"! fortnigriast sfasfm^ (t) Present the data in a tabular form (Hi) Present these data diagrammatically. 1 cm = 100) 500 4001 2 300H tr o 200 looses 321 352 285 B Murder ■ Robbery Q Loot 245 tons during the same fortnight last vear(ronnf TK « ? T' during the first fortnight of DecemberToo ^ 2 ssloo^f f"" and 41. export and stools in Sugar Mills in India Scale : 1 cm = 50.) MULTIPLE BAR DIAGRAM SUB-D'VIDED BAR DIAGRAM 400350300250iction K 8. (/) Presentation of data in a tabular form.95 Digrammatic Presentation Solution. off-take for internal consumption. Export arU Stock in Sugar Mills in India. Off-take for Internal Consumption.000 tons. 2001 . 2000 (figure in thousand tons) December. •c i Stock Fortnight Sugar Production. December.000 1 200- tories ■ 150- iption 1 lil for ■ 50- 100- . (Hi) Diagrammatic presentation of above data by (a) Sub-divided bar diagram (b) Multiple Bar diagram INDIAN SUGAR MILLS ASSOCIATION REPORT (Fortnight Sugar production. (First fortnight) Production Off-take from Mills Export Stock Source : muiau du^^i 378 154 Nil 224 387 283 41 63 --------- export and stock which we have calculated. 3 100 Factory B (%) 33. pnofit amo loss 40 30 20 108. 2001 (First fortnigh) (First fortnight) Dec. 13 96 Statistics for Economics-XI j Proceeds per Chair Factory A (Rs) Factory B (Rs) Wages Material Other Expenses 160 120 80 200 300 150 Total Selling Price 360 400 Profit or Loss (±) (+) 40(-) 50 650 600 1 he percentages are calculated as under : Ptrcentaj^^ (For Percentage Bar Diagram) Proceeds per Chair i-______ Factory A (%) j Wages 1 Material Other Expenses Total Selling Price 90 100 Profit or I oss (+) (+) 10 OT UJ 4 m ^ a. 3 tr 97 Digrammatic Presentation % COST Y chmr 3.grams B 0- Dec.3 50 25 . 2000 (First fortnigh) Dec. 2000 Dec. 2001 (First fortnight) Fig. c . .100^ . b„ a suitable dragram. OF STUDENTS .N SCIENCE (2001-2005) Scale : 1 cm = 25 students . . n » PROFIT AMD LOSS : 1 cm = 20% factory b factory a (t UJ 60- QZ CO 40" UJ lU Qrj 20' cc o: -20 M other Expenses Q Material Wages wm Profit or Loss Fig. 8 .pries in which some values may Broken Bar Diagram : Sometimes we may S™^ reasonable shape each bar is written on the ™ ° . Year Number of students 2001 2002 2003 2004 ^___- 25 48 375 125 neces^to brsn^ -^ ^ 98 Statistics for Economics-XI 3 NO. ch have both " "" " ■n plus and minus values to plot th.e export..ron the base l. etc.ne and negative vales bell'fbatTe" Year 1998 1999 2000 2001 2002 . 9 net 'i^r: ^e^l^ipt tt^ t.200175f2 150- 2 LLI Q 125? CO 100- u. wh. o d 75- 2 50250-2002 2003 2004 YEARS 2005 Fig. =3 CC 10 5 0 -5 -10 -15 -20 -25 gg Surplus ■ Deficit u .Export 47 125 20 94 120 Import 30 115 39 no 125 (Rs in Lacs) Balance of Trade 17 10 / -19 -16 -5 99 Digrammatic Presentation . Solution. BALANCE OF TRADE (1998-2002) Scale 1 1 cm = 5 lacs Y 25 -20 -15 -I CO O CO lU LU ti. . 10 2001 2002 B. and different the expenditure over different heads l^ke^ pornons^^^^^^ ^^^^^^^^ heads Namely. Similar^ expenditure of ^^^ ^^ components or the rent.s „™ te^e^ h" "" .«ke„ . 100 . PIE DIAGRAMS _^ ^^^^^ ^^ ^^^^^^^ These Pie diagram or circular diagram is ^^^^ ^ comparatively easier to draw.t 1998 1999 2000 YEARS Fig. _ ^ Pie diagrams are very POP^^-^J/JJ" oercentage breakdowns by represent the portioning a circle into various parts ^ various parts will indicate Lvernment expenditure Transport. then pie a g bar diagram. Pie helpful for comparison.o be equal to 360-. clothing. etc. diagrams are very useful in emphasising exhibited. etc. totals as well as comp^em parts ca ^^^^^ be drawn by making the^/™ ^^^^ ^^^ ^IgllmL ^called I p. Education. If the series is diagrams are less effective than difference among the components is very small. Steps for Construction of Pie Diagram percentage of respective totals. Circles can With circles and sectors. angle at the centre is 360 or 2%. I hereio . education.t . food. ». The sector so obtained wi^ the component.S.6" —...„g each of Degree of any Component part = Component value simultaneously for comparison tbrralTrh " proportional ro the square roots » ^ ■ ^^^^^ -"-a.>o„.ply.n..ry to express each part proportionately ™^rees. From this second line a ble nowT""' ^^^^^^^ ^^^ equal to the degree represented by second c^Z ^^ ^^^ ^--^re the portion of the second component S^ih 7 representing component parts can be coZuTd ' -P-senting differen? Be distinguished Illustration 9. the new line drawn a circumference. Now.„alto^. Before draw™. » 3. with this K ^^ o'clock centre with the help ofp^r t] "^a o t iTete to f ^ ^ ^^^ component.3. Constmrr o a-break-up of the cost of — Item Expenditure Labour Bricks Cement Steel Timber Supervision % Ltgrammauc Fresentation _ ^^^^ percentage into So. .se. is common position on the circle. we Labour Bricks Cement Steel Timber 25 % 15 % 20 % 15 % 10 % 15 .hepercentages oahe .„ceiperce„totthetotaWalne..t.mponent parrs w„l be now converted to degrees by . 0 100.1 28.7 2004-0S 100. (Degrees of angle are rounded off) Statistics for Economics-XI j Items Redymade Garments Cotton Textile WoIIen Textile Total 2003-04 % 52.3 35.2 19.7 23.0 Degree of angle .0 102 Solution.7 100.2 19.Supervision Fig.0 / 41. 11 n- of three textile items in percentage Items Readymade Garments Cotton Textiles Wollens Textiles Years 2003-04 Total 52.1 28. Clothing 3.188 69 103 360 2004-0S % 41.0 100.3 35. Clothing .7 23. basis of 360 taken as equal to the total of the values. basfs of 360 r . Represent the following data by a pte diagram. Food 2. Rent 4.0 Degree of angle 150 84 126 360 export of textile items 2003-04 2004-05 Fig. 12 Illustration 11. Family X Family Y 1. Education 5. Food 2. Miscellaneous (Including Saving) 400 250 15r 40 160 640 480 320 100 60 Total 1000 1600 103 the digrammatic Presentation >ms of Expenditure Rs 1. Miscellaneous (Including Saving) Total Square root 400 250 150 40 160 1000 31. Rent 4.6 360 640 480 320 100 60 1600 40 640 1600 480 .3. ___i 1000 ^x360 = 90 1000 ^360 = 54 x360 = 14-4 1000 1000 1000 x360 = 57.6 400 —x360 = 144 Family Y Rs _ . Education 5. tlie W—HmS-Ot^^^^^ " a . Wore the radU of arcle accordmg to avaUabUtty of space 3.ve .r capacity to g. and ™ tje dte^rb^Lrl^^^-Tt'''""'"-^' in diagrams.r basic fnncti^rrsytdTrj 2.e.2 : 4 (31.i.n the. facts are not possible to show .x360 = 144 1600 1600 1600 1600 x360 = 108 x360 = 72 x360 = 22.50 360 Radii of circle are determined m proportion 3.50 x360 = 13. 24 104 limitations of diagrammatic presentation Statistics for Economics-xl fo/lowng points „„„ rememberedmterpretation of diagrams.= 2 cm expenorrure of family x and y Food im] Clothing m Rent B Education B Miscellaneous FAMILY X FAMILY Y Fig. Diagrams can show * presentation.2 are : Family X : Radius -y = 1-6 cm 4 Family Y : Radius .6 : 40). TpLin JT "ftheir utility. 6. 8 9. easier to understand at a glance.st. and (b) pie diagram Digrams are less accurate but more effective than tables in presenting the data • ' rc^mtlstc:: " —« "-wmg.by tables etc. and W Multiple bar diagrams. they can misrepresent facts diagram for visual Presentation of" n"^^ ^^ ^ P-icular and the object of presentation. W Composition of the population of Delhi by reltgion H Agnculture production of five states of India. What are the merits and limitations of dtagrammatic representation of stat. 4. sucrpresentatiotr^^ newspapers. 3. 105 iigrammatic Presentation . Questions : • a-:: — feplam the various rules of drawing a diagram. Therefore.cal Explain the following with illustration ■ M Sub-divided bar diagrams.^ t ^^^ data A well constructed simple and attractive Ltam sho ^are and caution. it shLld be m. 5. magazines and journals ^^^ mformadon is be seen in financial reports in 2. txplam M bar diagram. J .951 2.729 4.113 2003-04 1. 'i-e He. (c) Deviation bar diagram 1 (e) Multiple bar diagram.167 (Rupees in crores) 2004-05 988 323 1. (b) Broken bar diagram (d) Sub-divided bar diagram iLt the following data by simple bar diagram.p for Economics-x\ Sunp. PRODUCTION OF COAL (Million Tons) Production pillion Tons) bar diagrams DEMAND AND AVAILABILITY OF STEEL (Thousand Tons) Exports (Rs in crores) 4.e Bar diagram.(Write short notes on the following 1(a) Percentage bar diagram .042 2. Present the following data by sub-divided bar diagram.789 1.394 4. «e diagram.744 / Represent the foJlowine • l . total import Food Fertilizers Mineral oil Others Total 2001-02 2002-03 474 795 125 298 341 1. 107 Ipigrammatic Presentation ntmaiic 11 B (Rs) 3 1 75 100 175 150 30 25 20 25 ----- . ■:--—--- ' 7. Export (Rs in crores) Import (Rs in crores) 2002 ~~2m 73 80 85 70 72 74 _2004 ZOOS 8..e fCowmg rab. ■ . Food Clothing Rent Education Miscellaneous Farntly A (Rs) Family B (Rs) P'Xpettdiiure 9 TU --——i—^ 1440 ' ^'iree year's result of XTT ri T _ • ^- "..Yi^f ■ ■■ ■ .. bar chart : COST P^OTRPDS AND fROm AND LCTJ^ Cost per table : (a) Wages (b) Other Costs (c) Polishing Total Cost Proceeds per table Profit (+) Loss (-) Chapter 7 cmpbic presentation 3.Price per Unit Quantity Sold Value of Raw Material Other Expenses ________ Show the following data by percentage bar diagram^^ L^^ of a product . sw Construction of Graphs Graphs of Frequency Distribution Line Frequency Graph • Histogram Frequency Polygon Frequency Curve : asiiiiiifa* . . ''Tt^ol senes graph X-axts ^^ ^ ^^^^^^ ^ 110 ■JJ Statistics for Economics-XI OF FREQOEIICr scale wid. fe' in ^phtc pr^tT^e Figi'r"" " ' frequency graph Scale : 0..T'" "" "" portion of the scL may be t use of 'kinke. ------. Fig. 109 Graphic Presentation .. 1 .: S^rrXs .of graphs..^j of origin 'O' which represent^ Quadrants : G-ph pa^ys ^mded " ^^^ ^^ ^ Y are posttrve. ^^ .^^^ -V-Tdata.„j axes. .75 cm = 10 Workers on V-axis S'graphs. axis .e^lu-'rS^^Str..Graphic presentation gives i rr - as a tool of analysis. re—.ea. " (Line Graphs.75 cm = 10 Rs on X-axis 0.«e„„t . D- equal parts called qaadrants . (See Fig. ol t„ue.. d. d.e difference of lO^wS™ T ' " "" wasting too much of space of ^a7h pSr «« ^ S'^Ph require a lot of space so that X^is is T". . Taking the above illustration they are : (a) Use of kinked hne (b) Starting from 59" . X-axis for variables under study (Heights in inches) 2.(b) Histogram (c) Frequency Polygon (d) Frequency Curve or Smoothed Frequency Curve ie) Cumulative Frequency Curve or 'Ogive' (a) Line Frequency Graph fluency array. Draw a vertical line on each value equal to the length of each frequency 4.^. Both the axes must be clearly lebelled and scale of measurement clearly shown.• . represents the frequency of that variable on kaphic Fresentation 111 Heieht in incht » 60" 90 61" 80 62" 120 63" 140 64" 132 65" 70 66" 40 Nc -~d----f) ^ Metlwd {. X-axis can conveniently be determined according to the need of the problem.. We can have three varieties of X-axis. of students) 3. on graph by which the line is drawn. Y-axis for frequencies (No. 13 il2 Statistics for Econo >mtcs~} . Solution. See the graphs given {d) Both axes must be clearly labelled and the scale of measurement should be clearly shown.(c) Starting from 60" (use thick line to read the data properly). HEIGHTS OF STUDENTS Scale 140 120 - ^ 100 - lU Q 3 is 80- LL o 60 - d 40 - H 20 (a) using kinked line 1 cm = Frequency 20 Students 1 cm = r on X-axiis Fig. oe. 4. X-axis for variables under study (Marks) . Both the axes must oe ucdny clearly shown.iF-pu 20 StudL onV-axis ^ I cm _ 1 on X-axis Scale 1 starting FROM 60" ! ■ ^ = Frequency 20 Students on V^axls^ 1 cm = 1" on X-axis yf (b) Histogram ' which each and also called a frequency histo^m : '' " ' ^^-dimensional diagram Cases of Constructing Histogram U) Histogram of Equal Class Intervals {« Histogram when Mid-points are given Histogram of Unequal Class intervals Method 1. 3. reW.y-axis for frequencies are freq Thus is pn («)K n obtai 113 \ Graphic Presentation class with frequency.STARTING FROM 59" (X-axis) . — histogram Scale : 1 cm = 10 Marks on X-^is 1 cm = frequency 4 on Y-axis 1- in for all the classes and the frequencies . Solution. =5~ . ' the measurement should bet^ sS::^^ first mid-pomt.^ interval Histogram = ^en ^ff^^^Tora the following aisnr. of Students (f) 150 160 170 180 190 200•8 10 25 12 7 3 Statistics for Economics-XI Graf mid-poirns different classes from the given 2. /.. class mterval is 10 for all SSe!rr: -Se for each class can be deaded(c^^ frequency) Class (Marks) 0-10 10-20 20-30 30-40 40-50 50-60 Class frequency (f) 4 10 16 22 18 2 10 X 4 = 40 10 xlO = 100 10 xl6 = 160 10 x22 = 220 10 xl8 = 180 10 Total Area = 720 Total frequency^^. 10/2 of we get lower and upper lim. . X-axis for variables under study 5: (Marks). 4 iTZ r fr^q^es (No.^^______^ .. 114 Method Marks (Mid-points) No. get the difference ^vide the difference by 2..ion of total marks ojrr^-"'a'^Sjra Boara H—.. with frequency he clearly shown. of Students)..bu.In the above illustration 2.e. 115 ^Graphic Presentation histogram X-AXIS^starting from 135 marks Scale 1 cm = 10 Marks on X-axis 1 cm = 5 Students on V-axis i UJ .H. (iii) I ni Solution. the class decided is 145 ~to 155 ^ ^ = ^^^ . histogram kinked line method Scale : 1 cm = 10 Marks on X-axis 1 cm = 5 Students on V-axis histogram x-axis-starting from 145 marks Scale : 1 cm = 10 Marks on X-axis 1 cm = 5 Students on V-axis 165 175 185 marks Fig. 3.^" U^g the same -jet ^h^ mid-points as under : No."PP. of Students : g IQ ^25 ^ ^hri ways : ^ See the Figs. below : ^ ^^^ Starting from 135 marks. 2. 4.Thus. 7 Sd Nc histogi Metho 1. erval.erval. the adjusted frequencies are : histogram Scale : 0. frequencies n. '^"I'^Take . . 2.ogram would give a misleading picmie. Do no.s. ^rr^^^^^^ each .us.5 cm = Rs 5 on X-axis 1 cm = 5 Workers on V-axis daily wages in rs Fig. 116 Thus. otherwise . O dZ 135 145 155 165 175 185 195 MARKS Fig.ecan^e of h. Rep«se„. Constru^^ ' r -Jji^s ^ Students W . adius. he adiusred. class in. Hit) Histogram of Unequal Class Intervals 'Tst^lo 4.he his. 9 205 215 S.he frequencies of i„.he Cass — are unequal.he fonowing^^ans of of Workers . 10 japhic Presentation Histogram : When Class Intervals are given by " Method ^ IllustrLn 5.ogram hu.he class which has d>e lowes. .o 3 u. ■ widths will be according to class limits. 5 32 24. „ inrliisive method (where lower and upper Note : Since the class intervals are given ^ . Adjusted Class Limits Marks Students (f) 4. Method J ^^^^^^^^ .5 25 19. Statistics for Economic^y.^d upper limits of Adjustment : Find the difference between lower hmn^ _ and so on.5-14.5 5 9.5 13 29.5 14.5 marks Fig.5-34.5-24.5 29.5.117 5-9 10-14 15-19 20-24 25-29 30-34 4 17 25 32 13 6 Solution.5 19.5 17 14.5-19.5 6 histogram Scale : 1 cm = 5 Marks on X-axis 1 cm = 10 Students on V-axis 9. 24 118 (c) Frequency Polygon («) Without histogram.5-29.9.5 24. o 10- d 2 50.Histogram .de of each rectangle ciearly sro^^^"^' ^^^^^ ^^^elied and the scale of the meas Solution.loISo" --20-30 5 30-40 12 40-50 15 50-60 22 60-70 14 4 —" . measurement should histogram and prequencv polygon JO on X-axis 1 cm = 5 Students on y-a*is 25' CO 20- & UJ s 15- 1co u. Jotn these md-p„i„„ „f ^.!>uitaDJe histogram kepnm„ 2.the of the '"Ho " ™T """"P'- 3. ofstfj^nirw 1 . This dotted area which was under histogram but is not under the frequency polygon.-Frequency Polygon Fig. But the shaded area has been included under the polygon.. ■BJ Hr Mui-pomts 15 5 25 12 35 15 45 22 55 14 65 4 Solution. with histogram). This dotted are is excluded from the area of frequency polygon. we can get the frequency polygon without histogram as Method 1. (ii) Frequency Polygon : Without Histogram und™"^ ^^^ illustration. Take the mid-point of each class interval. we observe that some area which was under the histogram has been excluded and some area which was not under histogram has been included under frequency polygon.e. Therefore. Thus there is always some area included under the frequency polygon instead ot the area excluded from histogram. Scale of X-axis can either be decided on the basis of class interval or midpoints 3. the total area excluded from the histogram ts equal to the area mcluded under frequency polygon. Join the points plotted for the mid-points corresponding to their frequencies by straight lines. No. We will get the same figure as obtained by the first method (i. 2. 24 Graphic Presentation jj^ While drawing the frequency polygon. This was not under histogram. 6-69. of Students (f) 2 58 64 25 ~1MS Students (f) 19. Frequency Distribution of Marks Tally Bars Gi foi Marks 20-29 30-39 40-49 50-59 60-69 m( Total -^xuxc preparing 1 exclusive method.5-A9.5 29. i." " ..5 39.5-29.5-59.« .ude„.s i„ an 2.e.frequency polygon Y Scale : 1 cm = 10 Marks on X-axis 1 cm = 4 Students on V-axis Fig.5 2 5 8 6 4 J--'-WUTtlUN Scale : 1 cm = lo Marks on *-axis 1 cm = 1 Student on V-axis » i* marks Fig. No. 14 ...5 49..5-39. 13 120 Illustration 7 V St'^t'stics for Economics-XI exa JSr -ks secured by 25 s...5 59. « .1 - in . Solution. 150-200. 250-300 and 300-350 (b) Draw a frequency polygon. of households (f) . and what per cent spend more than Rs 200 per month? Solution.121 Graphic Fresentation lllusmtion 8. 50 on X-axis 1 cm = 2 Households on V-axis *■ X 100 150 200 250 300 350 EXPENDITURE IN RUPEES Fig. Total 30 Tally Bars 13 (b) frequency polygon Scale : 1 cm = Rs. ju r-^nt \c) What per cent of the households spend less than Rs 250 per day. - . We have the following data on the daily expendttute on food (in rupees) fot 30 households in^alocaU^: ^^^ .s r/o r/s r/o .. 200250. (a) lit Monthly Expenditure on Food (Rs) 100-150 nil 4 150-200 mil 6 200-250 mim^iii 250-300 M 5 300-350 11 A. 15 400 122 No.s (a) Obtain a frequency distribution using class intervals : 100-150. s . 16 >X 350 400 We observe that : 123 Graphic Presentation statistics. (e) Cumulative Frequency Curve (Ogive) .I _____. It is a uni-modal distribution curve.h v drawn with care frequency polygon to get a smoothed freq^ ett " histogram for the data given in IllustraZ 8 fo^ U" ^^ constructing frequency curve Scale : 1 cm = Rs. 50 on X-axis 1 cm = 2 Households on V-axis 200 250 300 expenditure in rupees Fig. frequency polygon and frequency curve _____1—-----. histogram.6% spend less tha.fc) Ont of ^sn J.ri^ol 124 ■JJ Statistics for Economics-XI ie) y-Shaped Curve (Curve E) : In this case. u Statistics for Economics-X ^^ Hence 76. maximum frequency'is at the ends of rh. xtlr '' ^^ spend more than (d) Frequency Curve or Smoothed Frequency Curve generalVby^'el^^^^^^^^^ frequency curve.ust the same as^I Tthf poL^^^^ "u ^ ^^^^ ^^^"he required to be done carefully to ge^ co rect "eS Smoothing the frequency polygon shows neither more nor less area of the rectanLs of . It is drawn area mcluded . g in above illustration.„. than 10 is 46. and so on.ng marks more In 0 . moi. the number of students obtain. of StHdents 44 7 10 Marks 40-50 50-60 60-70 No. of Students 12 obra. S" ^^^^^^^^ 125 Graphic Presentation of each class ..s 50.„g mar.I- O" the graph paper by rwo (a) 'Less than' method (b) 'More than' method Illustration^^Draw^^ for the following data : Marks 0-10 10-20 20-30 30-40 Ma. more than 20 is 42. of Students (c. Cumulative Frequency Distribution Marks Less than 10 Less than 20 Less than 30 Less than 40 Less than 50 Less than 60 Less than 70 No.f-) 4 8 15 25 37 45 50 . of Students (c. ""Ilet the cumulative frequencies of the given frequencies either by 'less than method' or 'more than method'. Plot the various points and )om them to get a curve (i.e.) 50 46 42 35 25 13 5 We get a rising curve in than method'. Cumulative 'Ogive on Graph Paper (Cumulative Frequency Curve) by 'less than' method Scale : 1 cm = 10 Marks on X-axis 1 cm = 10 Students on V-axis by Scale more than' method 1 cm = 10 Marks on X-axis 1 cm = 10 Students on V-axis >X 10 20 30 40 50 60 70 80 MARKS 10 20 30 40 50 60 70 MARKS .f.calculated cumulative frequencies 4. „ . 2 X-axis — the variables under study 3. ugiv 5. if the above case of 'less than method' and declining curve in case of 'more cuLlative frequencies are plotted on the graph paper.Marks More More More More More More More than 0 than 10 than 20 than 30 than 40 than 50 than 60 Nr. be clearly lebelled and the scale of the measurement should be clearly shown.. Y-axis . „o„ of Weekly Wages Workers (f) 100-109 7 110-119 / 13 15 32 20 8 120-129 130-139 140-149 150-159 Method 1.h. and «ale of the measurement should be . >jet cumulative frequencies Both the axes should be iX l beM T clearly shown.oHo„i„. Must the lower and upper of he classes. 19 126 ■JJ Statistics for Economics-XI FAv by less than' method Scale : i cm = 10 Marks on X-axis 1 cm = 10 Students on V-axis Less than method Fig.Fig. 20 ^ c„„e for . 18 Fig. dis„.he . f 0-5 5-10 10-15 15-20 20-25 25-30 30^35 35^0 1 7 10 20 13 12 10 14 9 ----.. 99.Number of Students Cumulative Frequency (Less than) c.5 119.5 7 13 15 32 20 8 7 20 35 67 87 95 ogive (less than method) ocale ■ 1 cm = Rs.5-129.. cm = 5 Marks on X-axis 1 cm = 20 Students on /-axis or h . 10 on X-axis • 1 cm = 20 Workers on V-axis be .f Cumulative Frequency (More than) c. Adjustment of da.5 139.5-109.Fig.limits and calculation of cumulative frequencies by less than method. Statistics for Economics-XI It I Marks r\ r..5 149....5-159.5 109..5-119.5-149.127 Graphic Presentation Solution.5 129:5-139. 21J and indicate the value o^Ae^i^ ----------128 Solution...1 7 17 37 50 62 72 86 95 88 78 58 45 33 23 9 than' ogive Scale . month. For example. The information arranged over a period of time (e. students (2000-06) . in minus figures). Presentation of this type of information by hne or curve on the graph paper is of great use in economic statistics..Kg. months. week) is never in negative (i. Give titles to X-axis and Y-axis. The pair values will give different dots on the graph paper.. days etc. As the time (year. 23). month or week according to the problem. is taken on X-axis. .. Year. or arithmetic hne graph. X-axis can start either from 1999 or 2000 (See Fig. Time series can be sbown on the graph paper.g. weeks.. Start Y-axis with zero and decide the scales for both the axes. ^ . 4. 3.. values corresponding to time factor are : Years Students 2000 50 2001 150 2002 100 2003 150 2004 200 2005 225 2006 200 These dots obtained of pair values are joined by straight line which is called line graph or histogram (See Fig. 2. . (a) General Rules to Construct a Line Graph 1. 23).. there is no need of using Quadrant II and III.e.. 22 Graphic Presentation 129 ^ graphs of time series .) is termed as a time series.. For example. on every 1 cm for Y-axis one may represent an equal gap of 50 students and 1 cm for X-axis a gap between 2000 arfd 2001. years. These graphs are known as hne grapjhs or histograms. Scale : 1 cm = 50 Students on V-axis 300250CO H Z 111 200Q Z) H OT 150- O d 100- Z 500- . / / N s f / S / / 2000 2001 2002 2003 2004 YEARS Fig. m One Variable Graph Kendriya Vidyalaya ■ Method 1998-99 1999-00 ^^e given below • J™" Ae dots . It is not advisable to ■ by a straight iine and not by a curte " -is r ^^^^ of un. 23 2005 2006 130 5. . od Sc . the.2000-01 2001-02 2002-03 2003-04 2004-05 120 400 567 490 760 834 750 Gra Wh in v (yea timt orig largi < year Y-as smal line . Select Y-ax. for variables under study (students) 3. by STUDENTS-KENDRIYA VIDYALAYA (1998^)5) Scale : 1 cm = 200 Students on V-axis z UJ Q =3 Ico u. G„ble title and scales to X-J Ld C = ^o Its value and . 2. I 1 Select X-ax for the time factor (years). two reqa porti line. 1587 2003 1490 2004 1760 2005 1734 2006 1675 STUDENT&-<30vt. Keeping doln^—^ . C T in crranh r nresentation See Fig. Dortion of the scale may De omiueu wm^n ^ciix ---. 24 J Graphic Presentation ■ What is a False Base Line? 131 't rlvldTuse faUe base U„e according to ne^ of tbe ptoblent.. Present the following information on the graph paper.n e. stude^ ts ^ one t^usaud .o out tequiretnents by using False ^se Line) mustrafon No. 25). £ that is the use of False Base Line in graphic presentation (See Fig. . school (20(hm»6) Scale : 1 cm = 200 Students on r-axis 2000 2001 2002 2003 2004 2005 2006 YEARS Fig.2001-02 2002-03 2003-04 2004-05' YEARS Fig. higher sec. illustration 13. Year__________ Students 2U00 1120 2001 1380 2002 . 25 132 Statistics for Economics-XI .b n-bet of wmmmmM P" : . 6 - 1.8 6.Year (1) Agriculture and allied sectors (2) ■ Industry (3) 1994-95 1995-96 1996-97 1997-98 1998-99 1999-00 7.0 6.9 9. ^^ ^^^ Year 1997-98 1998-99 1999-00 2000-01 2001-02 2002-03 .0 - 0.9 Services (4) 7.2 0.3 8.n gdp at factor cost t>Cdle : 1 cm = 2 per cent growth rate in years — -----Services Agriculture and allied sectors Industry YEARS Fig.0 5. 26 133 IGraphic Presentation 1(d) Graphs of Different Units different units.0 10. we will have two different scales.9 5. estimated sectoral growth rate . When two values are given into two ^^"erent unn .2 Gr (d) Wt as axi data as * rime series graph.9 4.1 11.1 9.8 9.3 7.0 8. Quantity (in. 150 crore Quantity -S-Rupees 1997-98 134 Two figures of graphic presentation 10000 exports (Provisional) (US $ IMillicn) Statistics for Economics-Xl\ are shown below to understand time series graph. Average of Quantity: 12 Approximately Average of Value : 695 Approximately trade of tea in quantity and value (1997-2003) qcale • 1 cm = Quantity 3 thousand Tons or7-axis : 1 cm = value in RS. 28 m Questions : . ^ IMPORTS (Provisional) (US $ Million) 5000 Fig. '000 tons) 9 10 12 11 14 15 Value (Rs. in crores) 300 596 782 900 762 640 Solution. „ rbe «*en an the class intervals h^trtheTa.■ TSL^ Name ehe . and (b) Ogive? Explain their construction with the help of sketches. ^^ Give iUusrrarion. Explain the importance of graphic presentation of data. of Students : 3 10 14 10 . prepared.Wha. 19. berween -Bar dra^ar^'irH™ JO... i4. ^ . Sm Vim Graphic Presentation ^^^ What is a false base hne? Under what conditions would its use be desirable? What is meant by (a) Histogram.cance of Rs 715 f^r the gtven « of da™ " '"-e 13. 17. 7.e„. 18. Describe the procedure of drawing histogram when class intervals are (i) equal..«e. Distinguish Histogram and Historigram clearly with illustrations. 1516.f. Marks : 0-10 10-20 20-30 30^0 No.„„ve and s.apb . ! r Tr' '"O'-ency curve fop rXt"iara presented .gn. Probl^^s : ^ The frequency distribution of marks obtained by students in a class test is given 40-50 3 below: .t^X''""'" pS - f the ess than type. « a 'Cumulat. and (ii) unequal. What is a smoothed frequency curve? Discuss briefly various types of frequency curves. Draw Histogram from the following data : r Marks Obtained Number of Students : 6 : 10-20 20-30 10 In a certain colony a'sample of 40 households was selected. Comment on the shape of the histogram.Dtaw a histogram to represent the frequency distribution of marks. What is histogram? Present the data given in the table below in the form of a Histogram: Mid-points : 115 125 135 145 155 165 175 Frequency : 6 25 48 72 116 60 38 3/ Make a frequency Polygon and Histogram using the given data / Marks Obtained : 10-20 20-30 30-40 40-50 /Number of Students : 5 12 4. The data on daily income for this sample are given as follows : 200 120 350 550 400 140 35085 200 15 30-40 15 22 40-50 10 185 22 50-60 14 50-70 6 195 3 70-80 4 70-100 3 5. 180 170 210 430 . 136 ■JJ Statistics for Economics-XI Frequency .0-. 15-19. (b) Show that the area under the polygon is equal to the area under the histogram. u ij ' Size of classes t'::. .35-.O . 20-24TQ (b) What percent of th^ hr.„ ^ 10 15 "'Z. 0-. I .. 'tr '"^r ".. (Hint.0. 30-3.30 3.-. 210 170 250 300 (a) Construct a Histogram and a frequency polygon..^r cZ''^"" Frequency : 4 ^ ^^ 1^-24 24-30 30-36 u..110 90 185 140 110 170 250 200 600 800 120 400 350 190 180 200 500 700 350 400 450 630 110..ao-z^ Students . 3. Get a frequency distribution table to obtain a continuous series). ^ Workers : 9 12 15 Weekly Wages of B cia .0 .0 .O-. -.f s ''''' 'st: it: S. o„ of a scooter ma— . Year ^ Production f? (in units) 1 Total Cost I (Rs in lakh) 2001 8500 24 2001 2003 2004 2005 9990 117001330015600 .o^.0 « ^ ^ iwi ■ 137 Waphic Presentation I P^ . -a —^^^^^^^^^ tr r " \ Profit (Rs in ^^ 65 80 95 f i graph from the Ps " --.e foUo™. a„a . . 3.:::^ — co.. company.ot. . proauct. 'p:.Companies : 2 3 j ^— a-OOO. 35 3. 12 » 25 31 29 27 35 U.--------------------------Im-hnrtv t Year 'T99"o-91 1991-92 1992-93 1993-94 1994-95 1995-96 1996-97 . !i' the average . of Correla«p„ l-rtion to lBde» lV«™be«» Chapter 8 nusmcs of centum. As satisfy fl represents the marks of aM .29 34 45 49 <53t »TICAL TOOLS AND INTIRPRETATIOlf ii ■ Average p^^ ••—ye. '' ^^ everyday income of hctnr . An average is a fi. tendency Weaning and Importance ----Objects and Functions Of Averages Characteristics of a RepresentatL Averaoe Three Orders Of Measurement ® Arithmetic Average or Mean J-'st of Formulae and Abbreviations MCAMte AND IMPORTANCE the whole group is ealleH 7 briefly average Th^ j / observations. ^^P^^^^^ts marks of students -n a . J generally <^entral tendency or ^easur^T. 139 Measures of Central Tendency According to Croxton and Cowdon : "An average value is a single value within the range of the data that is used to represent all of the values in the series. The average of one group can be compared with averages of other groups.general conclusions. The mean of a sample gives a good idea about the mean ot the population. For example. Thus the purpose of an average is to represent a group of individual values in a simple manner.complex data. a sales manager may need to know the average number of calls made per day by salesman in the field. Since the average is somewhere within the range of the data." £ 1 According to KeUoy and Smith : "An average is sometimes called a measure of central tendency because individual values of the variable usually cluster around it." and functions of averages 1 To represent the salient features of a mass complex data : It determines a single figure' of the whole series. very often. In statistical enquiries. 3 To know about universe from a sample : Averages also help-to obtain a picture of complete group by means of sample data. 2 To facilitate comparison : Averages are useful for comparison. It is helpful in reducing the mass information into a single value for drawing. it is sometimes called a measure of central value. It is a tool to represent the salient features of a mass of. A railway officer will require information regarding the average number of passengers carried by rails on the . easily at a glance or the average monthly sales of Department A are compared with average monthly sales of Department B. It is difficult to generalise anythmg from the ages of crores of Indian People. sample method is used. But if it is said that the average age of an Indmn is 55 years one can draw conclusions about health conditions of the people. so that the mind can get a quick understanding of the general size of the individuals in the group. For example. 4 To help in decision making : Averages are helpful for making decisions in planning ■ in various fields. the average marks of students in section A can be compared with the average marks of students in section B. they are expressed in averages. an average becomes essential. Definiteness can only come.„d by pLons of ordln^rSltn 'e.. estimating and planning and other managerial decision areas. If the calculation of the average Statistics for Economics-XI u„de«.. groups or classes. i gh^tl^ of a repiuesentjirive average > As the average represents statistical information and it is used for comparison. kfNDS OF STATISTICAI... if big and small mills cotton mill industry fnTdfa s^pUt' u ^^ ^^^^ ^ female workers.. 141 Measures of Central Tendency students). uiju- 1 It should be simple to calculate and easy to understand : An average should be calculable with reasonable ease and rapidity only then it can be wide y used.^ Arithmetic Mean] tSeometric Mean or Mean X . 5 To trace mathematical relationship : When it is desired to trace the mathematical relationship between different. Measures of second order. It should not involve heavy arithmetical calculations..^. 1.. """ are not separated the aver^cotton cloth per mill.various passenger runs...AVERAGE ♦►Moving Average . it must satisfy the following conditions : . ! Harmonic Mean (HM) (GM) . Averages are valuable in setting standards. Measures of first order.. Measures of third order... adillt worts ^^^ ^^ OF MEASUREMENT There are three orders of measurement. 2. 3. (B) Discrete Series. Calculation of Arithmetic Mean 3.e 1010+1020+1030 3060 --= Rs 1020 /. median. mode are mo^ '"M^lirt cl of qualitative data which caunot be measured quantitatively for rdatiou to all the values. Meaning 2.e. Miscellaneous Problems ^ Merits and Demerits of Arithmetic Mean (B) Weighted arithmetic average or Weighted Mean (b) Short Cut Method (Assumed Mean Method) ic) Step Deviation Method Let us see the calculations m the following senes. A. Series of Individual Observations of v^updLtS^^^^^^^ we . (A) Series of Individual Observations. average wage taken by the workers is Rs 1020 Direct Method . Symbolically. which has highest demand. etc. most fashionable garment. median should be the choice. Mea 1M .. naturally. 142 ■JJ Statistics for Economics-XI I 1. Mathematical Properties of Arithmetic Mean 4. mean. (C) Continuous Series.Progressive Average Composite Average -^Quadratic Mean Specialised Average (Index Numbers) Of the above mentioned average. + +..Iv . — = Rs 1020 N3 Therefore. Arithmetic mean SX = sum of all the values of observations /. X„ N = Number of observations Alternative equation . X =.3 XX = 3060 Special 1.e. X. I these « ieasures of Central Tendency 1.../. 2. + X.. average of the workers is Rs 1020 where. Obtain IX by adding all the values of variables. Divide the total by number of observations (N).. Symbolically.....x„ X —143 N . + X3 + ..XX • 3060 A.( M Alten where denote Worker Wages (Rs) X A B C 1010 1020 1030 X N = 3 . the symbol X is the 'Greek alphabet called sigma and is used xsi mathematics to denote the sum of values.3060 144 2 The sum . If we replace each item of observation by the calculated mean. n where.v=' - A 1010 1020 B 1020 1020 C 1030 1020 N = 3 ZX = 3060 3060 NX = IX 3 X 1020 . n . Workers Wages (Rs) Mean V. 7:r-.■ -Workerc Statistics. for Economics-d rv.X = —Sx.total number of observations Lx = the sum of n values t = i X (1010 + 1020 + 1030) 3060 = Rs 1020 iSpecial Features of Arithmetic Mean 1.------------1 AB C N=3 Wages (Rs) X . then the total of these replaced values will be equal to the sum of the given observations. This assumed mean.A. i. Rs 1010.X-X 1010 Xj 1020 X^ 1030 X3 ^X = 3060 -10 0 +10 2(X-X) = 0 Symbolically.. observations and finally the product is added to the b] r Worker X ^ — A (d) A B C lOlu 1020 1030 N=3 4 0 10 20 Id =30 Measures of Central Tendency . deviations of X variables from assumed mean . Use of following formula. A = assumed mean d = X . 3.1020) 0 . 145 ..e. X= A + 1010 + — = 1010 + 10 = 1020 N 3 i. To total . ^X-X) =0 Alternatively. ^ ~ ^ ^^^^^ ' ^^^O) + (1030 ..s divided by the number of ^^^^"I-ted. 2. 30. mean the total of tt deZl-? ^^^^^ « calculated. Get the sum of the deviation {Jd).K + Jf^ + + . average wage taken by the workers is Rs 1020. 4..Steps : 1. i. where..3 X 1020 = 3060 . Calculate deviations of items from assumed mean (d).xj -nx = (1010 + 1020 + 1030) .3060 = 0 Short-Cut Method (Assumed Mean Method) get the anthmetic.e. Decide assumed mean {A).e. 3.e.. Get the sum of step deviation (Zd').e.0 0 B 1020 10 1 C 1030 20 2 N=3 ■ X-A Id'=3 Steps 1. Decide assumed mean. Divide these deviations by common factor. All deviations taken by assumed mean are divided by common factor. 3. Use of following formula : -Ld' X=A+ N = 1010 + J X 10 = 1010 + 10 = Rs 1020 i. Rs 1010. Symbolically (Assumed mean = Rs 1010) Worker Wages A 1010 . Calculate the deviations of items from assumed mean (d). i. i. i.e.. average taken by the workers is Rs 1020. N = number of observations ^ Step Deviation Method We can further simphfy the short cut or assumed mean method. 5.A). 146 where v a u for Economics-XI ... i.. 4.e."Ld = Z (X .e. sum of the deviations of X variables taken from assumed mean. 2. C = 10. C ..„us„a. . assumed mean d' = brcom'r flz"" ~ . 3. X = Arithmetic Mean '"lit <iev. St 1.ees Workers Daily Income (in Rs) Solution. 4.fo^m.a„„„. A B C D E F G H "3^ J T J ^6^ l>Jiiy Incuwe [Rst X ZX = 2400 X = ^ il^ N ~ \o ^ rupees.ons.Common factor T . 2.neome of ten wor. . 'Measures of Central Tendency 147 asures of f^entrai. done (a) Direct method Direct Method rr^^^^^^ . Calculate the anthmetic mean of the marks given m illustration l by he short-cut method (Assumed Mean Method).wnere. lenui^nc^y Dlustration 2. ^ = Number of observations . The average daily income of workers is Rs 240. of X vanabies fron. in a factory. "" 10 = 200 + 40 = 240 rupees The average daily wage of workers is Rs 240. „ 1 w niustration 3. /. Calculate the deviations from assumed mean.Worker ABCDEFGHI J Daih income (Rs. suppose A = 200.t^'i Worker . Calculate the anthmetic mean of the marks given m illustration 1 by step deviation method.e. v-A-d 2.. = +400 Steps : 1 Decide assumed mean. Get the total of the deviations calculated from assumed mean (d).^ 3. Use the following formula : X=A+ = 100 + 400 N . A A . . 4.) N = 10 120 150 180 200 250 300 220 350 370 260 X-A JdJ -50 -20 0 +50 +100 +20 +150 +170 +60 Id. +2 +15 +17 +6 Id' . suppose 4 = 200. 120 150 180 200 250 300 220 350 370 260 -50 -20 0 +50 +100 +20 +150 +170 +60 X-A m -8 -5--2 0 +5 +10 H. A 50 . Discrete Series Students .-Marks : Solution.ABCDEFGHI J Marks X X-A <<i) .Decide assumed mean. X ^C ■JJ Statistics for Economics-XI N = 200 X 10 10 = 200 + 40 = 240 rupees The average dady wage of wbrkers is Rs 240 B.40 148 Steps : 1. Multiply the frequencies with variables (fX). 4 . 3. ieasures of Central Tendency Steps : 1. Get the sum of the products (I^fX).B 100 c 50 D 150 E 100 F 50 G 150 H 100 I 50 J 100 tx 200 400 f^x. 2. HO tTC . Divide the total by number of observations {Lf or N).each. /1+/2+/3+••■/« 149 Z/X ^f 900 10 or .00 each and .. 40 X. Number .Frequency ' N = L/j i.e. 20/. Marks : 10 No.of observations fx = Product of variables with their respective frequencies Direct method : Illustration 5. E/X = sum of the products of variables and their frequencies f. of Students : ^ Solution. . Following tables gives the marks obtained by 100 students in a class. of Students ■ 10 Xj 5 A 50 f^X^ 20 Xj 200 f^X^ ■ 30 X3 40/^3 . 1200 /■3X3 800 f^X^ 50 X.S/X N = 90 marks where. Multiply the frequency with the variable X. 15 f^ 1250 E/ = N = 100 IfX = 3500 Steps : 1. 20 10 30 40 40 20 50 25 Marks No. Calculate the arithmetic mean.. The followmg formula is . Get the sum of the product (LfX)..e. 800 . S/" or (N). ISO t Statistics for Economics-XI j X= or N 3500 100 If = 35 ••• Average marks of students is 35. 1200 . 3. i. Alternative equation : where. ^^^ Short-Cut Method (Assumed Mean Method) : We can use this ^method to calculate arithmetic mean in order to simplify arithmetic calculations.2. 1250] " 3500 = 35 Marks ^ 100 x 35 = 3500 = -475 + 475 = 0 Aver Step taken by calculate -XI Measures of Central Tendency . ^ = -^Ifx n '' • n ' 100 J_ 100 100 " + 200 . Divide Z/X by total number of observations. suppose A = 30.. X .100 -20 -10 0 +10 +20 -100 -100 0 +200 +500 = 500 n of :heir Steps : 1. deviations of variables taken from assumed mean •Lfd = Sum of the product of frequencies and their respective deviations Dlustration 6. i.frequency X . 2.e. Calculate the deviations from assumed mean..e.A.used : Ifd X=A+ N Here. Multiple deviations by frequency and get fd. Sliort-Cut Method (Assumed Mean Method) cies 25)] Marks 10 20 30 40 50 5 10 40 20 25 N .A = d. 3. i. A = Assumed Mean N = Number of observations f . Calculate the average marks of students given in Illustration 5 by short cut method. . Decide assumed mean. . 1.4. Divide deviations by common factors ^Zii.. /. Use the following formula : Ifd X=A+ = 30 + N 500 100 5) = 30 + 5 = 35 Average marks of students is 35. ^ common factor = Q 10 ^ .„ I„u.m.especve devSrtrd/^^'™'"^ -"ents given . 2. 3. Statistics for Economics-XI j X = A.A = d. 5. The following formula is used to calculate the arithmetic mean by step deviation method. All deviations taken by assumed mean are divided by common factor.Common factor f = frequency - ' Step deviation '''' ' fr^f -d ehe. seep Step Deviation Method Steps . . Add the product of deviatioiis and frequency. C N ^ ^ A = Assumed Mean N = Number of observations C . 152 til. Step Deviation Method : We can further simplify the short-cut method. Calculate deviations from assumed mean.„aeio„ 5 b. suppose A = 30./x . V Here. Decide assumed mean.. The only difference is that m continuous series mid-pmnts ot trcSsfLrvals are required to be obtained. . Add the product of step deviations and frequency 6.. Marks (X) No. Continuous Series t In continuous series.^e ^^ ^ the case of discrete series.. (i) Direct method.g. («) Short-cut method.4. Multiply step deviations 'by frequency 5. the method of calculations of arithmetic mean . we can use all the ^^-e me^ods o^ca^^^^^^^^^^ of arithmetic mean in the same way as we used m discrete series. Direct Method . Marks 0-4 4 4-8 8 8-12 2 12-16 1 lliuil. Mid-point = Here presents lower limit and presents upper limit. . the mid-point for a class 5-10 can be obtained as : ^^ = 7. These methods are. After obtaimng the mid-points. 50 = loo ^ 10 = 30 + 5 = 35 Average marks of students is 35 153 Measures of Central Tendency C.5. The following equation can be used to get the mid-points.. and {Hi) Step deviation method. of Students (f) Mid-points (m) . Use the following formula. m. 3. 8 48 f. i../■„ X .0-4 4-8 .. Alternative Equation ■X= « where.V. 8-12 12-16 14 X/^ = N = 15 4 A 2 A 1/. Divide Sfm ky total number of observations (N). /. Multiply the frequency with mid-point (fm)..e.. 2 wij 6 m^ 10 W3 14 m. ■JJ Statistics for Economics-XI /1+/2+/3+. Obtain mid-points (m) of the classes. = lower limit and = upper limit 2. Here. Get the sum of products (Lfm). •Lfm = 90 Steps : 1.m.Mean marks are 6. 20 f.' Symbolically. 4. « = total of frequency (E/) ^ frequeLts ™cl-points of classes and their respective V = ^ X [(4 X 2) + (8 X 6) + (2 x 10) + (14 x 1)] J_ 15 1 X [8 + 48 + 20 + 14] x90=90 15 15 . S'ii ll 154 l'!. Obtain mid-points.6) + 2(10 . 6. NX =-Lfm 15 X 6 = 90 IhlT of mid-points from arithmetic mean being multiphed by their frequencies is always equal to ZERO. Multiply deviation by frequency and get fd.= '6 Marks.6) = -16 + 0 + 8 + 8 = 0 Measures of Central Tendency Short-Cut Method (Assumed Mean Method) „ u u ^ ' . ' '""uipuea oy •Efim-X) =0 = 4(2 .A = d. Add the product of deviation and frequency. Decide assumed mean (A = 2). 3.6) + 1(14 . Use the formula : IM N X=A+ . ^AK^d Illustration 9. m .e. Special Features of Arithmetic Mean 1. of Students ■f Mid-point m -----.. Thyotal of frequencies multiplied by Arithmetic Mean is always equal to the sum ^of the product of mid-points of various classes and their respLive frequtd" . Calculate the deviation from assumed mean. Calculate arithmetic mean of Illustration 8 by short-cut method. Marks XNo.m-1 d ■ /a --— 0-4 4-8 8-12 12-16 4 8 2 1 0 32 16 12 1 6 10 14 N = 15 0 +4 +8 +12 60 Steps : 1. 5.6) + 8(6 . 4. i. 155 )01U11UU. 2. =2+ 60 15 =2+4=6 Mean Marks are 6.hmetic mean of m„s„a. I'. Solution..«e . Calculate the deviations from assumed mean M = 4.o„ 8 by rbe deviation method. =2+1x4 = 6 Mean Marks are 6. worSlfXory"^-'- ^^ folWin. -y 15 =2+—X4 15 ^ A^ w . 7: ui't^^'frrmr fr^-ncy. Decide assumed mean (A = 2) 3.Obtain mid-points. Divide deviations by conmion faaor (Common factor = C <■ e 10) 5. frequency. Multiply step deviation b. of Workers . rcl. 2. ' ■'■ m-A tS6 Steps : for Economics-XI 1. distHbu^on of daily wages of Daily Wages (Rs) Below 120 120-140 140-160 160-180 Above 180 No. .10 20 30 15 5 80 confidl^.«« ..25 Mean wage of workers is Rs 146. Income (Ks) No.25.„ .ose the ends Calculation of mean. J?aily Wages (Rs) (X) 100-120 120-140 140-160 160-180 180-200 Measures of Central Tendency Applying formula. Illustration 12.tfr r^tl^S:. of families More than 75 85 140 95 115 105 95 115 70 125 60 135 40 145 25 150 . Calculate the arithmetic mean. Following information pertains to the daily income of 150 famdies.. we get lfm If 11700 157 X= or lfm N 80 = 146.« . 95-105 105-115 f15-125 125-135 135-145 145-155 (f) 10 25 20 25 10 20 15 25 N = 150 Mid-points m-100 (m) 80 -20 90 -10 100 0 110 +10 120 +20 130 +30 140 +40 150 +50 m~lOO 10 id') -2 -1 0 +1 +2 +3 +4 +5 m -20 -25 0 +25 +20 +60 +60 +125 Ifd! = +245 Applying formula. X=A+ Id' N = 100. Jncome jRs) (x) 75-85 : 85-95 . First.:. get the class frequencies from given more than c inxuiative frequencies..1^x10 .Solution. X = A + ^ x C -Lfd = 69 md' +1) = 119 . to a of tl Marks Students X m m-lS d f fd' f(d'* 1) 10 ) d' 0-10 10-20 20-30 30-40 40-50 50-60 4 6 20 10 7 3 5 15 .33 = 116.33.= 100 + 16.^^^ calculatrngTmr ^ Illustration 13. The formX i^af le" ^f{d + 1) = -Lfd' + Zf Equal values on both sides of the above formula is a proof of correct calculations We add one more column to a table of calculations prepared in discrete and contiCus eri ?hTcoLn ifrrr'. tiJ'ii 158 ^ Statistics for Economics-XI Charlier's Accuracy Check CO "P ^y given by Charher while computing arithmetic mean by the short-cut method and the step deviatiL methodTa frequency distnbution m discrete and continuous series).33 ll Arithmetic Mean is Rs 116. Calculate the mean for the following marks obtained in Statistics bv 50 students. Also apply Charlier's accuracy check for verifying calculations ' f-f = 0-10 10-20 20-30 30-40 40-50 "50-60 Students : 7 3 4 6 20 10 Solution.25 35 45 55 -10 0 +10 +20 +30 +40 -1 0 +1 +2 +3 +4 -4 0 +20 +2C +21 +12 0 6 40 30 28 15 N= 50 Arithmetic Mean. . the calculation is correct. Ix = 0 .X) = Le. Mean is a point of balance and sum of the positive deviations is equal ^o the sum bf the negative deviations. Applying Charher's test : md' + 1) = Lfd' + Zf 119 = 69 + 50 119 = 119 Hence. The sum of the deviations of the items from the arithmetic mean is always equal to zero. Marks x~x X 5 -10 10 -5 15 0 20 +5 25 + 10 LX = 75 nx-x) = o -_ ^~N5 = 15 Z(X . of obi H« X ires of Central Tendency - ^^^ ithematical Properties of Arithmetic Mean The arithmetic mean has the following important mathematical properties : 1. H hi 2.N = 15 + 69 50 X 10 = 15 + 13.8 = 28.8 Hence mean marks are 28.8 or 29 approx. _ N. X J 2 = Combined mean of two groups X1 = Mean of first group X2 = Mean of second group Nj = Number of observations in the first group N^ = Number of observations in the second group («) Three related groups : .2 . = Combined mean of three groups Mean of first group Mean of second group Mean of third group No. of observations in first group No.X ) = Total of the deviations from arithmetic mean.X) = 0 2. of observations in second group No. In case of discrete and continuous series Zfx or I.f(X .3 x.-77—~T7 NI+N2 Here._ N1X1 + N2X 2 A 1.Here. The combined mean formula is as under : (f) Ttvo related groups : .XI+N2X2+N3X3 X U. of observations in third group T .= N. We can calculate the combined arithmetic mean from the means and the number of observations of two or more related groups. Ex or E{X . ■c 160 T^u £ . X.. = 40 and Xi = 35 ^ ^ (60x40)+ (40x35) 60 + 40 ~ _ 2400 + 1400 100 38 the 3800 100 = 38 marks. -- .3. l-alculate the combined mean of all the students of Solution. N^ = 40. X2 +N3X3+ . " ^ ^ marks. = 60..2. of Students 40 N. B ■ 35 X2 No. Section : I. Statistics for Economics-XI The formula can be extended for more groups groupraVuIl?''^^ of combined mean of more Combined Mean X 1. Here.Nj(„ ■ Ni+N2+N3+.. = + N. A 40 Xi 60 N.N„ of /o'Sr in^IL™? cllcd^^^^^^^^^ rr ^ seaions A and B. N. If the mean marks of section A are 40 and that of section B are 35. The mean marks of 100 students of combined sections A and B are 38 marks. Find out the number of students in sections A and B. Illustration 16. Solution.N. of Stilts Mean —— —_ A 40 Xi 60 Nj B ? X2 ■ 40. fmd out ^rt s'^lS^r'''^ Section No. Nj = 60. of Students .N1 + N2 2 = 38. mean of the students of section B is 35 marks. N^ = 40 and Xi = 40 (600x40) + (40XX2) 38 = 38 = 60 + 40 2400 + 40X2 100 _ 3800 = 2400 X 40 X2 .ttere ar^n"4?s:udrsr^a^dt ^«- <n s^ion . H Measures of Central Tendency Combined mean (Xi. are 40.2) =38 161 -N2X2 where.40X2 = 3800 . M Mean No. V.2400 40X2 = 1400 X2 = 35 Marks Hence. 162 3 The sum of rh 12 3 4 5 XX = 15 Set I ---(X . B 35X2 ■ ?N. Combined Mean (Xi. 1.2) = 38 _ N1X1 + N2X2 N1+N2 Here Xi. 40 Nj .3) (X) -2 4 for Economics-XI .2' = 38. = 100 Hint. Xi = 40.60) = 40.100 A 40X1 ? N. the students in section A are 60 and in Section B are (100 .35 Nj = 3800 .Nj) (Nix40) + (100-Ni)x35 =-Too 3800 = 40 Nj + 3500 . X2 = 35 and N^ + N. N^ = (100 .35 N^ 2.3500 - 5 Nj = 300 Nj = 60 Hence. E three values of the formula are known. ^^^^^^^ 4.-1 1 0 0 +1 1 +2 4 = 10 X 12 3 4 5 Set II ~ X~2 (X-2)^ (x') (x'^J -1 1 0 0 +1 1 +2 4 +3 9 = 15 H -- N5- ^ V ~ ^' ' f taken from mean.. ' 'f"' ''^v'^tions taken from any value cha^:o. W. Arithmetic mean i. eh.>. calculated by a simple formula... . h . . the third can be calculated" _ J^X . -n-e arithmetic mean of a series of 40 as Rs 265. X = 120 x 2000 = Rs 2. Total wage bill = N. which is explained in the following illustration. X = or IX = NX if any two of the X■. Fmd the correct arithmetic mean. ^ ' ^^^^^--—--—1 X 10 30 20 30 30 30 40 30 50 30 i:x = 150 150 Measures of Central Tendency 163 150 = 30 N NX = SX 5 X 30 = 150 150 = 150. ^ m„s„ado„ 17. e.40.. ^ This property has great utihty in calculatton of wage bills. Bnt while calculating ii an item Rs 115 was misread as Rs ISO.000..g.Tf . Solution. The relation NX = ZX can be easily used for correcting the value of mean. . average wage Rs 120.. of workers 2000. . No. 12. N NX -■"''e. and (it) in terms of paise... is wrong as the us get correct EX by subtracting the incorrect item and adding the correct item Incorrect EX = 10600 Less : Incorrect item 1^0 10450 Add : Correct item Correct ^^X = 10565 Hence. Since Statistics for Economics-XI j Here. _ EX X = N EX = NX Here. . of observations. X = 265. i. N = 40 EX = 40 X 265 = 10600 Calculated EX. corrected arithmetic mean = Rs 264.Since. 10600. Tie arithmetic mean of a given set of data (i) in terms of rupees. 40 Illustration 18. 164 f Solution. . 1+2+3+4+S EX IS value. . sa.1 Exa 1 2 3 4 __^ "" "" observations are Rs 5 less . I. ^ value = 25). ™ean of 5 items (1. 2. • .e. 2 3 4 5 .) is 3.Calculated ZX i c Tnn values. ^ Incorrect « correct H (5 observati^s x Rs f^st T ^ ^dd : Corrected balance of 5 observations (5x5) -25 Correct es —---- . N " . Let us cor^t M u 1. 5. • * ' 4.— = Rs 105 («) Corrected mean expenditure in terms of n ' Rs 105 X 100 = 10500 ^^ = paise 10500. Corrected ZX 525 ^^. we get tbe Alisi ( ll requ ] £ Lo to be < examp class M 2. IX = IS X = 3 X+2 3 4 5 6 25 5 X~2 -1 0 +1 +2 +3 Xx2 Column 1 : X = 3 (+) Column 2 :X . after aadition. ■ u .uUip^on by -b»caon and *a. s.on by *e same constant to their means.wUrr^f^uc mean These problems are re^r rb^^e rS^^e mean. . .b„action and . Miscellaneous Problems .i: a j j j (X) Column = 2) = 1 ^ = 6 Multiplied 2 = (3 X 2) = 6 51 2 4 6 8 10 30 6 We a JVIaifc Studo Cunn before ap Thus Marks Stud^ 165 . 1. first example. 2. ^^^^ Marks : 5-10 1(^5 ^^^^^^ Smdents : 5 =4 25-30 (25-10) = 5 =8 -^ ^^0-16) . W o^^ j^^r^^L^niS^^ to be defined by marking an —and last classes.. the same class mterva IS decided M. In the case of open-end classes Example Marks Below 10 10-15 15-20 20-25 Above 25 No. of Students 58 3 4 5 Lower limit of the first class and upp. Thus. In the case of cumulative frequency distribution Example Less than 10 Less than 15 Less than 20 Less than 25 Less than 30 5-10 5 5-15 13 5-20 16 5-25 20 5-30 25 We are given Marks : ^tive frequencies ar. 5) class would be 5-10 and last class 25-30.:equired to be converted into ascending class frequencies Thus we get. ■tr m 166 Example : Statistics for Economics-X. it is the most 3. 1. We are given.a. It is rigidly defined and not affected by personal .' th representative measure.s the most popularly used because of the following merits ■ . bias Its values is always definite. of Students More than 5 More than 10 More than 15 More than 20 More than 25 25 20 12 9 5 5-30 25 10-30 20 15-30 12 20-30 9 25-30 5 . They are : Marks : 5-1 o (25-20) Students : =5 10-15 (20-12) = 8 15-20 (12-9) = 3 25-30 20-25 (9-5) =4 5 Merits and Demerits of Arithmetic Mean Merits : Arithmetic Mean . It is based on all the observations of the series Therefore it i. It IS simple to understand and easy to calculate. ^neretore. Marks : Students Marks No.ive frequencies. " 2.ue^citTef: "r" above descending cL„. 4 Arithmetic mean can be a value that doe.g. Average calculation is not . tt IS a good base for comparison.. The calculation of arithmetic mean does not require any specific algement of ^iS m SS^~ . 5 Arithmetic mean gives more impot«. which is not an item of the series. It needs mathematica. typiS^Rs 167 Measures of Central Tendency „ ^ Rs on 000+ Rs . Uses of Weighted Mean 3.000 paid to the General Managet. not extst m the senes at all.. (B) Weighted Arithmetic Average or Weighted Mean 1.S.500+ Rs 2."presenmive^ I. .^-re mathematical 7 It is fluctuations of sampling and ensues . is affected by an extreme value of Rs 20. calculations. rnfpe^f Sloor"^' -P'ovee-y clerk Rs jjoo.ability m calculations.on.n other .4. n«h''o7o?:fcuI onTj^^^i^^^^^^^^ '^e average and be used with caution mean should ffc^o 000 a " General Manager's sala. /. 8 and 9 is = 7. . thL are number One item may be more impot^mt hi—rlorl^^^^^^^^ to different ^o^ as jeights. Calculation of Weighted Mean (a) Equal Weights (b) Unequal Weights arithmetic mean gives equal importance tt/aif the ^^^^^ fact.000 _ g oqO per The average salary will be--4 month..nce to the bigger items and less importance to titr^cflXaecided inst by observa.. in a firn. the average of 4.500 + Rs 4. Meaning 2. . « » "ve Cual weigh. of payment of wage per hour by three ways Simple Arithmetic Mean (x) : Workers Man Woman Child 8 6 4 ZX=18 EX X= .verstt.tems. Xw = Weighted Arithmetic Mean W= Weights X = The variables Steps. 3.es or boards. WeStled mean is used for comparison of the results of two or more un.wir«=ights are figures to indicate d>e relative mtportauce of . (/) Multiply weights by X and obtain WX Hi) Divide the total (ZWX) by total weights (XW) Solution. . . 4. 1. 168 Statistics for Economics XI Calculation of Weighted Mean The formula for calculating weighted arithmetic mean is as under : IWX Xw = where.. It is used in the construction of Index Numbers. to different categories of employees in a fa«o^ 2.u^ It is used' to calculate standardised birth rate and death rate. Xw = X Rs 6 = Rs 6.=Rs6 LW 150 Weighted Mean is Rs 6. weighted arithmetic mean will be equal to the simple arithmetic mean. women and child workers are 10. 20 and 50 respectively then our ---------Vorkers lyfJtr 'x W . (a) Weighted Mean (Equal Weights) : (Xit.--_U-^^^---^ .N 8+6+4 18 : ■: 3 .— -. . 4X3 50 Wj 50 50 W3 I.w.) terSn' . lb) Weighted Mean (Unequal Weights) : (X«^) Suppose men. 6 X.ana •n Type SWlfittlBil^Bl® Wages (Rs) ■ X Workers W Man Woman Child 8X.. when all : items are given equal weights. + XM+2W .V .W= 150 400 300 X^W^ 200 X3 W3 IWX = 900 -—-—-'ires of Central Tendency 169 Xw X. = Rs 6 per hour.n Thus. Suppose men. Type Man Woman Child X Workers W 50 20 10 SW = 80 WX 400 120 40 . assumed weights can be assigned to the items on the basis of their relative importance. women and child workers are 50. then our answer would be different. But. u However.W = 80 120 ZWX = 400 . __ Xw < X Rs 5 < Rs 6 . normally they are not equal. the weighted arithmetic mean will be less than the simple arithmetic mean when items of small vflues are given greater weights and items of big values are given less weights. 20 and 10 respectively.Man 8 10 80 Woman 6 20 Child 4 50 200 l. in the absence of given weights.SWX 400 Xw = = Rs 5 LW 80 Thus. fiiBsBslPlSSaHi^B^^^^B Food Articles kg W Flour 5. ^ Statistics for Economics-Xl Thus the weighted arithmetic mean will be greater than the simple arithmetic mean when items of small values are given less weights and items of big values are given mor^ weights.50 Ghee 58.28 Price in Rs per kg Qty.ZWX = 560 __ ZWX Xw = 560 80 =7 Weighted Mean is Rs 7. Articles of Quantity Consumed Food (per kg) 3 Price in Rs Flour 11.60 58.50 5.0 .35 Solution.5 Oil 20.28 8.8 Ghee 5.16 2. Calculate Weighted Mean by weighting each price by the quantity consumed.4 Sugar . Xw > X Rs 7 > Rs 6 niustration 20.2 .4 5.60 Sugar 8.2 Potato.8 11. Consumed in . ' Oass IX X XI XII Total School .89 Weighted Mean Price is Rs 22.55 I.W 17.89 WX iMM 66.5 .040 •2.35 20.A Appeared 30 50 200 120 400 Passed 25 45 150 75 295 School B .400 7. state which or them is better.436 = 22.700 327.16 Oil .Potato2.Xw = ZWX 403. From the results of the two schools A and B given below.55.0 Total 17.436 . lUustration 21.296 0.000 IWX = 403. School A Class Appeared w Passed Pass % X IX X XI XII 30 50 200 120 15000 7500 25 45 150 75 LW .400 _ SWX _ School B : Xw School B is better. 29504 400 = 73.5 100 120 100 80 XW = 400 School A : WX 8000 9504 7000 EWX = 29504 LWX 29499 _ 73 75 Xw .400 WX 8.33 90 75 62. converting into percentages. An exannnaaon was held to decide the award of a -hola* .2 70-62.Appeared 100 120 100 80 400 Passed 80 95 70 50 295 171 Measures of Central Tendency ntf Use Weighted Anthmetk Mean after obtaining homogeneous figures.76 m„s«a»„ 22.50 2499 4500 LWX = 29499 School B Class Appeared W Passed Pass % X IX X XI XII 5000 80 95 70 50 80 79. WX.Subject Statistics Accountancy Economics Business Studies Weight 4 32 i Marks of A 63 65 58 70 Marks of B JAarks of C 60 64 56 80 65 70 63 52 DUSIIICSS jiuun^o _-________ Of Ihe candidate gening the highest marks . 4 63 252 60 240 65 260 Accountancy 3 65 195 64 192 70 210 Economics 58 116 56 112 63 126 Business Studies 1 70 70 80 80 52 Total IW = 10 EWXj = 633 SWX^ = 624 EWX3 = 648 Statistics EW 10 2:WX2 LW 2 10 624 Marks of C WX. 52 .s to be awarded the scholarsWp. Subfect W Weight Marks of A Marks ofB X. who should get it? 172 Statistics for Economics-XI Solution. . deviations of X variable from x= The variables.e. (1) 2./v O^ I Weighted Mean of B. zx- Sum of all the items of the variable X. Mean Xw = ^^ .(X X) = 0 ljc = 0 If(X .e.-Lfm ^ N X=A+N Mathematical Properties of Arithmetic Mean Here. Xw^ = ^^^ = 62. X-X=x Mathematically. U= Sum of the deviations of X variabic i taken from an assumed mean. Properties of ■ Arithmetic Mean (3) E(X . i. Discrete Series (Grouped data) ^N X-A + ^^f xC N 3.EWX3 648 ^t Vt .N1X1+N2X2 + N3X3 N1 + N2 + N3 (4) NX = IX ii- Measures of Central Tendency 173 Abbreviations X = Arithmetic Mean.N1X1+N2X2 Similarly . hence he is entitled for scholarship. C = f= Frequency.ZX N Short-cut Method Sfi^ Deviation Mjethod X = A+'^f'xC N 2. Ix^ is minimum ZWX Weighted. OF FORMULAE AND ABBREVIJmONS [Arithmetic Mean. Individual Observations (Ungrouped data) Direct Method .X ).4 Marks.. X — A .e. d = X-A. | N= Number of observations or (Z/). an assumed mean. i. Properties and Weighted Mean] Type of Series 1. The weighted Mean of C is the highest.X )Ms the least..0 Ifx = 0 . d' = Common factor. i. Continuous Series (Grouped data) . Xw^ = ^^^ = Marks. step deviations of C E/X = Sum of the product of variable (X) X-variable from assumed mean . Weighted Mean of C. X and weight.e.e. Xw = : Weighted Arithmetic Mean. i. N. 174 StattsUcs for Economics-XI . : Number of observations in the first zwx = : Sum of the product of variable 1 group. X-X = X. m= and divided by common factor.. W= N. - : Arithmetic Mean of second group. A= Assumed mean Lfd = Sum of the product of and their respective deviations. What are the functions of an average? Discuss the characteristics of good average. Which of the average possesses most of these characteristics? 3. lLfd' =Sum of the product of deviations and their respec ive step deviations. Name the commonly used measure of central tendency. (X. Xi- : Arithmetic Mean of the first group. EXERCISES Questions : 1. square of the deviations of X variable from mean. What is meant by 'Central Tendency'? Discuss the essentials of a measure of central tendency. What is a statistical average? Mention different types of averages. Yd' = Sum of step deviations.. = : Number of observations in the : Weights.and the frequencies (/). 4. lfm = Sum of the products of mid-points frequencies and the frequencies. = Combined mean of two groups. i.Xf = X'. 2. Mid-values. second group. deviations of X variable from the mean. X. 66. 670.^ -sure ' ^of the va^es of the vanable from in ^ " unweighted mean? lU. 52. Problems : 1. 152. What are the uses of weighted mean? II. 62. 56. 25. 151 2. .67 and D = 47. 148.67. 39.. 147. Calculate arithmetic averages of the following information : (a) Marks obtained by 10 students : 30. c = 147. 144. 490. 24 (b) Income of 7 families (In Rs) : Also show 590.. B = 56.fr^q-ncy -. Also explain properties of mean. ic) Height of 8 students (In cm) : 140. 435. 890. 12. C= 62.5. ^ = 600.^. 145. .33] Frequency : 6 ^ fI^J^/"^ 11 Inning 20 50 10 40 26 60 8 46 Calculate mean of the following series • 5 Mutch m .] Match II I Inning 6 irmx 65 15 I Inning 7 8 9 12 15 28 20 14 expenses of following 10 firms . 47.. 575 = o 550.. Define the mean. and (b) Weighted Mean.u______________________• " : Name of batsman Matt 1 ■ Inning ch I II Inning ABCD 60 40 100 20 42 40 80 140 52 . 150.. Write notes on (a) Central Tendency.12 cm. 6. ^ II Inning 10 36 18 84 100 70 100 ••^ = 42. Why xs arithmetic mean is the most commonly used measure of central tendency^ Llutionr "" """ ^^ ^ . 000.94] 10-20 20-30 .000] ieasures of Central Tendency Calculate mean of the following frequency distribution 62 64 67 70 73 82 103 176 212 180 Values Frequency 60 54 77 115 n Calculate arithmetic mean of the followmg data Profit (in Rs) : 0-10 No of shops : 12 18 2/ 30-40 20 175 81 85 89 78 _ 50 21 [X: R> 70.10 5 [X= 7. Expenses = Rs 14.06] Firms Sales (Rs in '000) Expenses (Rs in '000) 1 50 11 2 50 13 3 55 14 4 60 16 5 65 16 7 65 15 8 60 14 9 60 13 10 50 13 [X= Sales = Rs 58. 25 years and UK = 29.45] Compare the average age of mal^injhe_ two countries :_____________________ population of U.K. Age Group 0-5 5-10 10-15 15-20 20-25 25-30 30^0 40-50 50-60 60-65 (in lakhs) 214 258 222 157 145 161 267 184 120 100 18 19 20 18 16 14 27 25 19 17 8.40-50 50-60 16 [ X = Rs 30.404 years] Calculate simple and weighted ar^hmetic averages of the folbwing items : Items Weights Items Weights 68 1 124 9 85 46 128 14 101 31 143 2 102 1 146 4 108 11 151 6 110 7 153 5 . [Average Age India = 25. Later on it was discovere — Ll ' ^^ —^^ -. of Students Less than 10 Less than 20 Less than 30 Lesi than 40 Less than 50 -:100 _ 5 15 55 75 A = OU iViaiivai Also get 5:f(X-X) = 0 176 11. Statistics for EconomicsThere are two branches of an establishment employmg 100 and 80 nerso.32. ^ ^^ 3.07 and Weighted Mean = 108.2 = Rs 252. 14. Fmd out the mean of marks obtained by both th g^ups of students taken together.112 23 172 2 113 17 [Simple Mean = 121. rrS monthly salrfes by thrtwo b^rh are Rs 27^ and 225 respectively. H obtained in the same examination by another group of 200 students were 52.81 to be"49 Tth'by a group of 100 students were found If Too . 12.71) Marks No.correspondingTot . 13. The mean marks of 1 >0 students were found to be 40. 15. fmd out the arithmetic mean of the salaries the employees of the establishment as a whole. [Combined Mean = Xi. 32]! 19.2 = 59.' -ru [X2 = 57. [Xj.3 = 63.25 Marks]! R^18r4"T 1000 workers of a factory was found to be taken af29ranri67" 'T?"' ^^ workers were wron^ taken as 297 and 165 mstead of 197 and 185. Calculate Combined Mean 70 80 90 100 j 70_ 78 83 85 [X = 48. [Correct X = 39. .-ru •.25 kg] Calculate mean ot the following data : Marks Beloiv : 10 20 30 No. Find the average wage of a worker from the following data • : Above 300 310 320 330 340 350 360 3701 No. of Students : 5 9 17 40 29 50 45 60 60 16.5 boys m group of the same class is 58 kg Find the mean weight of 60 b^! .2.41 Marks] Section Mean Marks No. of Students A 75 50 B 60 60 C 55 50 17 .7 Marksl The mean weight ot 25 boys in group A of a class is 61 kg and the mean weiS of .125 Marks] average ot 31 marks. ' [Corrected Mean : Rs 180. of ivorkers : 650 500 425 375 300 275_ 250 100 .■ [Xi. What were the average marks of the other students. . Find the correct mean. .. A candidate obtains tbe followmg percentage of marks : Sanskrit Mathemat^ 84..50 5. Calculate weighted mean by weighting each price by the quantity consumed: Food items Flour Ghee Sugar Potato Oil Quantity Consumed 500 kg 200 kg 30 kg 15 kg 40 kg Price in Rupees {per kg) 1.23]j Measures of Central Tendency 177 effip* .29 °C) 21. Geography 47.50 0.[X = Rs 339. Politics 57.43 Marks] 22... ^ o. Economics 56.50 ..8. English 78. Mathematics and Sanskrit. ^ is agreed to give double weights to marks m Enghsh. of day -40 to -30 10 -30 to -20 28 -20 to -10 30 -10 to 0 42 0 to 10 65 10 to 20 180 20 to 30 10 [X = 4.. What is he weighted and simple arithmetic mean?= 68.00 4.. X = 64. History 54..25 20. M. Calculation of Median 3. Merits and Demerits of Median Definition .35] 23. rmmmv«jm (a) Median.5 2 7 2 % pass S2 76 73 76 [Weighted Mean Mumbai : 72.A. A distribution consists of three components with total frequencies of 200.0.Com. of students M.A.les). of students Np. B. T-l 83 73 74 65 66 4 5 2 3 3 65 60 1 3 6 7 3 7 81 76 74 58 70 73 2 3.[Xw = Rs 6.Com. Mathematical Properties of Median 4. of Students are in hundreds Courses of study Mumbai % pass No. M. Definition 2. Mumbai is better] 24. (b) Partition Values (Quart. Comment on the performance of the students of three universities given below using weighted mean : No. Kolkata : 7U. 10 and 15 respectively Find out the mean o^ combined : distribution. median 1.55. = Chapter 9 TOOTioMHTnaet ui. of students Kolkata % pass Cher tnai No. B.5 4. B.Sc. 250 and 300 having means of 25.Sc. and (c) Mode.6 and Chennai : 72. "Median of a series is the value of the ttem actual or estimated tvhen a sertes ts arranged in order of magnitude which divides the distribution into the tivo parts. 'uf "sr 147 151 140 Anurag Deven 149 M Suresh 142 At Mayoor 147 AtuI 144 144 "" heights of 7 students in a class.--.^ev^r shit t s^r z"-. Bowley. "If the number of the group are ranked m order according to the measurement under consideration then the measurement of the number most nearly one half ts the median. . nL! O™'''''' ." ^ ■ According to Secrist.---e sxss lics-XI Positional Average and Partition Values 201 According to AX.. we arrange our data in ascending order as follows : 140 142 144 145 147 149 151 mm Deven Mayoor Satish Himankar AtuI Suresti f ■ . ' So. This arrangement facditates locating the central position so that the series may be divided into two parts one less than the central value and the other more than the central value.Satish 145 145 Himankar The first and most important rule for obtaining the median is that the data should be arranged in an ascending (increasing) or descending (decreasing) order. (b) Discrete series.) Anurag 151 Deven140 Suresh 149 Mayoor 142 Atul 147 Satish 144 Himanka-- 145 Name of Students Height (cm. Name of Students Height (cm. ^ ^ impute the Solution. That is. Median is the central positional average of given data.Anurag If we arrange the above data in descending order we get : Name of smdents : Anurag Suresh Ami Himankar Satish Mayoor Deven Height (cm) : 151 149 147 145 144 142 140 From this ordering also we observe that 145 cm or value of the 4th item is the median. median has a position more or less at the centre of the values and it divides the series roughly into equal parts. Calculation of median (a) Individual observations. (c) Continuous series.) . 180 Statistics for Economics-XI {a} Individual Observations meiarhetht. Locate the median by finding the size of 3. we get fN+^^ Me = Size of = Size of fN + V th item Pos But fN h the h 7+1 item .Deven140 Mayoor 142 Satish 144 Himankar Atul 145 147 Suresh 149 Anurag 151 Steps : 1. Applying the formula. Arrange the data in ascending order. \th item 2. The above data must be arranged either in ascending or descending order to get the value of median. -XI the f ositional Average and Partition ValuesI g j But when the number of item in a series is even 2. the central item.= Size of 4'"' item Median is the Himankar's height. i. N+V the item will be in fraction. 6. Solution.e. which will be the »tn item m the list. the central item.. 4.e. 10 etc. and calculate the median height.) . 7. i. 8. /. 145 cm 8th tTm ^^ cm. Arranging the data in ascending order including the height of Rajesh. nil 1 is when it is an odd number.. 9. 5.e.. When the number of items in an individual series is 3. we get et Name of the Students Deven140 Mayoor 142 Satish 144 Himankar AtuI 145 147 Suresh 149 Anurag 151 Rajesh 152 Me = Size of = Size of Heii^ht (cm. 11 etc. that 'N+iK ■ th Item will be a whole number. Serial No.5'*' item Medkn is estimated by finding the arithmetic mean of two middle values. Marks Serial No.. 2. adding the height of Himankar and AtuI and dividing by two. Marks Serial No.+ Item Item = Size of 4.„ an ascending order in the Serml No. Marks 1 2 3 4 5 6 11 11 12 ' 15 15 17 7 8 9 10 11 12 32 13 14 15 16 17 18 32 33 35 35 38 41 Median = Size of the 18 20 211 22 J 23 .e.fN + lX^ . '& Size of 4. i.5"^ item = item + item 2 145 + 147 292 Median height = 146 cm. Marks Serial No. Marks Serial No. 1 17 7 41 13 11 2 32 8 32 14 15 3 35 9 11 15 35 4 33 10 18 16 23 5 15 11 20 17 38 6 21 12 22 18 12 Marks JI 182 Statistics for Economics-XI """sed . Z^lue of the 9"* item + Value of the 10^'' item = 11^. of Students 10 1 20 8 30 16 40 26 50 20 60 16 70 7 80 4 Marks 10 20 30 40 50 60 70 80 No. of Students 1 8 16 26 20 16 7 4 N =99 Ctwtulatiue frequencies c.f 1=1 10 = 2 26 = 2 52 = 2 72 = 2 88 = 2 95 = 2 99 = 2 16 16 16 16 16 16 .5.21. Hence Median = 21.5 (b) Discrete Series Illustration 4.5"' item = . Marks No.item = Size of the 18 + V th = 9. Calculate median of the followmg distribution : Solution.5'^ item The value of 9. Arrange the data in ascending or descending order. Apply the formula.26 26 26 26 26 20 20 20 20 16 16 16 up to (c) Cc nil the m( 7 7+4 Positional Average and Partition Values 183 Steps : 1. Me = Size of fN + n th Item. 2. Compute cumulative frequencies. Median is located at the size of the items in whose cumulative frequency. 3. Median = Size of = Size of (N + l . the value of (N + U th item falls. 4. Find out the value of median from the following data : Daily wages (in Rs) : 100 50 70 110 80 Number of Workers : 15 20 15 18 12 Solution. . Illustration 5.th 2 (99 + ^^ Item = 50th item Median Marks = 40 Marks.f. 12 47 100 15 62 110 18 80 Cumulative Workers Frequencies Median is the value of fN + l rso+i^i or th or 40.) 50 20 20 70 15 35 80 . Find the median size of land holdings. (c) Continuous Series Illustration 6. Wages in Number of Ascending Order (Rs) (f) (c. Thus the median value would be Rs 80.5'*' item. All items from 35 onwards up to 47 have a value of 80. The size of land holdings of 380 families in a village is given below. of Families (in acres) Less than 100 100-200 89 200-300 148 300-400 64 400 and above 40 39 J' 184 IvV i Solution. Do not use Less than cumulative . of families (f) frequencies 0-100 40 40 100-200 89 129 200-300 148 111 300-400 64 341 400-500 • 39 380 Steps : 1. Statistics for Economics-XI Size of Land Holdings (in acres) No. Compute less than cumulative frequencies. u th item.Size of Land Holdings No. f = Frequency of the median group.x i where. Applying the formula. Median = /j + . f = 190. /• = 100 185 Me = 200 + 1^1^x100 = 200 = 200 + .f = Cumulative frequency of the class preceding the median class. Median item is located by finding out size of the item in continuous series. Locate the median group in cumulative frequency column where the size of fN^'' the item falls. we get --cf Me = /j + : Xi Ositional Average and Partition Values where. = 129 f = 148. c.f. c. Calculation of Median Me = size of = size of 2 380 item item = 190^'^ item Median lies in the group 200-300.f. /. Apply the following formula to calculate the median from located group : —— c. 3. = 200. = Lower limit of median group. 4. I = The class interval of the median group.2. 22 acres of land holdings.. Calculate median from the following data : Age (in years) 55-60 50-55 45-50 40-45 Number of Age Persons (m years) (f) '7 35-40 13 30-35 15 25-30 20 20-25 Total Number of Persons (f) 3U 33 28 14 160 «I Note : If the given question is in deseending otdet of values then Wore giving the question. Median size of land holding = 241. . . .148 61x100 148 241. the dafa is Required to arrange ■„ ascending order to calculate less than cumulative frequencies.22 acres of land holdingr^d 50% of famihes are having more than or equal to 241.) Illustration 7. (ie 50% of the families are having less than or equal to 241.22 acres.216 148 •. ) 14 42 75 105 125 140 153 160 186 Statistics for Economics-XI In the above example median is the value of lies in 35^0 class interval.•• Median Age = 35. Calculate the median from the following data : . This question has been solved below after arranging the series m ascending order. N. of persons (f) 14 28.f. Me = + X i 80-75 f^l th or .. or or 80''^ item which = 35 = 35 30 5x5 30 X5 = 35 + 0.83 = 35.83 . 33 30 20 15 13 7 Cumulative frequency-(c. "___ Age in years (Ascending order) 20-25 25-30 30-35 35-40 40-45 45-50 50-55 55-60 No.Solution.83 years Illustration 8.1) ri6o> th I2. ) 4 10-20 12 16 20-30 24 40 30^0 36 76 40-50 20 96 50-60 16 112 60-70 8 120 70-80 5 125 ha' on Middle item is ri25 xth or 62. If the data are given in the form of cumulative series they have to be converted into simple series in order to find out the frequency of the median class which IS needed m calculation of median.Value Frequency (f) Value Frequency (f) Less than 10 4 Less than 50 Less than 20 16 Less than 60 112 Less than 30 40 Less than 70 120 Less than 40 76 Less than 80 125 Solution. Value Frequence (f) 0-10 4 Cumulative frequency {c. Once it is done that rest of the procedure is the same as in any other continuous series.f. which lies in 30-40 group. lics-XI Positional Average and Partition Values 187 .5* item. ich ^ -c. be lich me Size Frequency (f) 10-20 42 42 20-30 25 67 30^0 58 125 40-50 40 165 Cumulative frequency (c.X i ^^ 62. Me = /.5* item which lies in 30^0 group. . th . + .) ri65Y' Middle item is —^ or 82. in !. e>umulative frequency taoie is oi more man type.f.5-40 = 30 + —trr.u».f.5x10 36 = 30 + 6. Calculate the median from the following data Size Frequency (f) More than 50 0 More than 40 40 More than 30 98 More than 20 123 More than 10 165 Solution.25 Median = 36.X 10 = 30 + 36 22.25 Illustration 9..u eases mc ucua have to be converted into a simple continuous series and median is calculated of ascending order series. eI N Me = /j + .67 188 Statistics for Economics-XI Illustration 10.5 i.. ' .5x10 58 = 30 + 2.. 10 = 30 + 58 15. Compute median from the following data • MMues : 115 125 135 145 155 165 175 185 195 Frequency : 6 25 48 72 116 60 8 22 3 he'^^rdls^^Z^: Th 7fT ^^^ of ^he class-intervals of a contmuous trequency distribution The difference between two mid-values is 10 hence 10/2 . and so on up to -——---- Uass-intervals 110-120 120-130 130-140 140-150 150-160 160-170 170-180 180-190 190-200 Total ..X i ^ 30 . iM^iZ >. The classes are thus 110-120 170 l^n ^ ^ 190-200.67 Median = 32. upper limit ot a class. if the arithmetic mean of the data given is 28 Find rh.79 Illustration 11.Frequency 6 25 48 72 116 60 38 22 3 390 _ 6 31 79 151 267 327 365 387 390 The middle item is (390^ th or 195"' item.79 Median = 153. which lies in the 150-160 group. and (b) the median of the series. Me = + - ^ 195-151 = 150 + -r^^-X 10 116 = 150 + 44 X 10 116 = 150 + 3. I ^ ■ • frequency. ^ ^^^ Profit per Retail shop (in Rs) : 0-10 10-20 20-30 Number of Retail shop 30-40 27 40-50 17 50-60 6 j2 jg . 2100-= 35 X . Let the missing frequency of group 30-40 he X.fm = 2100 + 35X Applying formula. (b) Calculation of median : .Positional Average and Partition Values 189 Solution. we get Ifm X= or 28 = Ifm If " N 2100+ 35X 80 + X 28 X (80 + X) = 2100 + 35 X 2240 + 28 X = 2100 + 35 X 2240 . {a) Calculation of missing frequency. the missing frequency is 20. Profit per Retail shop X 0-10 12 5 60 10-20 18 15 270 20-30 27 25 675 30-40 X 35 35X 40-50 17 45 765 50-60 6 55 330 N = 80 + X Number of retail shops f Mid-point m ' —-----1 fm Y.28 X 7X = 140 140 X= 7 = 20 Therefore. . 190 Statistics for Economics-XI Me = /j + -ly. Illustration 12. 0-10 12 12 10-20 18 30 20-30 27 57 30-40 20 77 40-50 17 . Let the missing trequency of the group 20-40 be X and the missing frequency of 60-80 group be Y.X -in ^0-30 20x10 = 20 + —-X 10 = 20 + 27 " ■ 27 = 20 + ^ = 20 + 7.41. which lies in the 20-30 ^group. Now Z/" (total frequency) = 100 i. However.Profit (Rs) (X) Fret. 100 = 14 + X + 27 + y + 15 .e. In the frequency distribution of 100 famiUes given below.407 Median = 27. Find the missing frequencies. 94 50-60 6 100 N= mcHC\ (f) (c-f) 100 The middle item is 100^ or 50th item. of families : 14 ? TI 15 Solution. the number of families corresponding to expenditure groups 20-40 and 60-80 are missing from the table. the median is known to be 50. Expenditure : 0-20 20-40 40-60 60-80 80-100 No. 27 .5 = 50 .f.13. Me = /. which means it lies in the classNow..5 Since the frequency in this problem cannot be in fraction so X.14 . f Xt 50 = 40 + 50-[14 + X] 17 X 20 50 .15 or X + Y = 44 Expenditure No. f^ wouId be taken as 23.e.14 .5 = 22. of Families (f) 0-20 14 Cumulative frequency (c. ^lOOV*' Middle item of the series is also interval 40-60.40 = 27^20 f ositional Average and Partition Values191 10 X 1. (Given median = 50) N or 50* item.f.X X = 50 .35 = 50 . i.[14 + X] 13. + -c.14 .or X + Y = 100 .) 14 20-40 X 14 + X 40-60 27 41 + X 60-80 Y 41 + X + Y 80-100 15 100 Median is given in this problem as 50. X + Y = 44 or /■j + = 44 . The sum of the deviations of the items about the median./. For example : X : 10 11 12 Deviations from Median : 2 10 Deviations from any poirit. Median is an average of position and therefore is influenced by the position of items in arrangement and not by the size of items. It is easy to calculate and understand. mean cannot be graphically determined. It can be determined graphically. 2. (say 10) : 0 1 2 The sum of the deviations taken from median (12). . 4. It is proper average for qualitative data where items are not converted or measured but are scored. = 44 . 2. ignoring ± signs. 3. less than the sum of the deviations taken from an\ 13 1 14 2 = t> -f — »J ipomt (1® Merits and Demerits of Median ^^''^i^ilJAN t Merits 1. It is well defined as an ideal average should be and it indicates the yalue of the middle item in the distribution.fj or 44 . Mathematical Properties of Median 1.23 or 21 Thus the missing frequencies in the question are 23 and 21. will be less than any other point. For getting partition values the most important rule is that the values must be arranged m ascending order only.5. we can arrange the data either m ascending or in descending order but here there is no choice-only ascending order is possible for calculating partition values (Quartiles). 6. Calculation of Partition Values Definition When we are required to divide a series into more than two parts. we have a piece of cloth 100 metres long an^d we have to cut it into 4 equal pieces. partition values (quartiles) 1. In the case of open-end distribution it is specially useful since only the position is to be known. It is not accurate when the data is not large. Demerits 1. It is affected by fluctuations of sampling. 2. 4. Suppose. Definition 2. 3. we will have to cut it at three places. Characteristics of Partition Values 3. In the case of finding out the median. It is useful in a distribution of unequal classes. It cannot be given further algebraic treatment. For example. It is not based on all the observations of the series. the dividing places are known as partition values. we have the following data of heights of 7 students in a class • Name of students : Anurag Deven Suresh Mayoor Atul Satish Himankar Height (cm) : 151 140 149 142 147 144 145 . Interpolation by a formula is required to calculate median in continuous series This reqmres the assumption that all the frequencies of the class interval are uniformly spread which is not always true. For median data need to be arranged in ascending or descending order. 192 Statistics for Economics-XI 5. It is not affected by extreme values. 6. Quartiles are those values which divide the series into four equal parts. iwuim ------------- . for getting correct results. They help us in understanding how various "ems are spread around the median. quartiles are not averages like mean and median. Therefore. Positional Average and Partition Values Now we arrange the data (in Illustration 1) in ascending order : 193 Name of Students (cm) Deven140 Mayoor 142 = Q == First quartile or lower quartile Satish 144 Himankar AtuI 145 = Me = = Second quartile or middle quartile 147 Suresh 149 = Qj = Third quartile or upper quartile Anurag 151 AS we Know tne meuiaii is uic nci^iii. that is in understanding the composition of a series. the first quartile is the average of first half of the series and third quartile is the average of the second half of the series. the special use of partition values IS to study the dispersion of items in relation to the median.Therefore. (b) Discrete Series. quartiles are averages of parts of series For example. Characteristic of Partition Values The difference between averages and partition values is as follows : While an average is representative of whole series. Calculation of Partition Values (a) Individual Series. the data must be arranged in ascending order in all the cases. Thus. (c) Continuous Series. e. Q. The middle or second quartile (Q^) is the central positional value of the data. is always less than Q^ and Q3 (Q^ < Q^ and Q3) and median falls between Qj and Q3. middle quartile and upper quartile. and third or upper quartile (Q3) is the central position value of upper half of the data. 11 400 26 300 12 170 27 580 13 440 28 370 14 480 29 380 15 620 30 350 S. Wages (in Rs) . = 142. lower and upper quartile. = 149. (a) Individual Series Illustration 13. median. The first or lower quartile (Qj) is the central positional value of the lower half. No. = 145 and Q. second and third quartiles.. In the above data. Wages (in Rs) 1 330 16 240 2 320 17 330 3 550 18 420 4 470 19 380 5 210 20 450 6 500 21 260 7 270 22 330 8 120 23 440 9 680 24 480 10 490 25 520 . No. It must be remembered that Q. By definition quartiles will divide a series into four equal parts and so number or quartiles will be three.cm. (Q. These are also called first. suppose we have to calculate quartiles. i. S. They are known as lower quartile. Now. From the following information of wages of 30 workers in a factory calculate median. Wages (inRs) . No. fN+iY^ rN+v""' and N+1 Nth items Median Me = size of N+l 2 th Item r 30 + 1* = size of —J. Locate the item by finding out.item = 15. * Wages (in Rs) 1 120 16 400 2 170 17 420 3 4 5 210 18 440 240 19 440 S. Arrange the data in ascending order.194 Solution. 1. No.5* item ^ size of 15th item + size of 16th item 380 + 400 Statistics for Economics-XI S. 2. 75 (320 . Lower Quartile is Rs 315. Upper Quartile Qj = size of = size of .size of 7* item) = 300 + .300) = 300 + 15 = 315 .75th = size of item + |(size of .-.260 20 450 6 270 21 470 7oi 300 1 ^ 22 01 320 23 480 9 330 24 490 10 11 12 13 14 15 330 480 25 330 350 370 380 380 500 26 27 28 29 30 520 550 580 620 680 F I Uj {b)l I Calci = 390 Median is Rs 390. Positional Average and Partition Values Lower Quartile 195 Qj = size of .size of rN+n th item (30 + 1 ah item = 7. of Shoes (f) 4 8 12 15 20 35 50 40 20 15 24 12 5 3 .5 9 9.size of 23^" item) = 480 + ^(490 . uu Illustration 14. first quartile and third quartile. Size of Shoes 4.480) = 480 + .5 7 7. (&) Discrete Series r .50 Upper Quartile is Rs 482. Following are the different sizes and number of shoes m a shoe shop.5 5 5. Calculate median.5 8 8.25(10) = 480 = 480 + 2.5 6 6.25* item = size of 23^" item + -^(size of 24* item .5 11 No.5 10 10.VN+r"' item V3O + 1Y'' item = size of 23.50. 5 20 59 7 35 94 7. Locate the item by finding out : fN + U th fN + V . Calculate less than cumulative frequencies. Statistics for Economics-XI I Steps Sue of shues " ----- 4.5 20 204 9 15 219 9.5 5 260 11 3 263 dat: 1. 3.5 50 144 8 40 184 8.5 4 4 5 8 12 5.196 Solution.5 24 243 10 12 255 10. 2. Arrangement of the data in ascending order is necessary.5 12 24 6 15 39 6. th and Af + 1 th Wk " "" " feo-cy. N + l^ th 4 263 + 1^*'' Qj = size of = size of = size of 66* item = size of 66* pair of shoes Medial Ap: item item lics-XI Positional Average and Partition Values First Quartile = 7 size of shoes. Third Quartile 197 Q = size of .size of = size of fN + lY'' . Item r263+r th item = 132th item First Quartile = size of 132* pair of shoes = 7.5 size of shoes. the tule of Median Me . we get . of persons (in Rupees) 800 16 1000 24 1200 26 1400 30 1600 20 1800 5 Income ■ ^H (in Rs)1 800 16 16 1000 24 40 1200 26 66 1400 30 96 1600 20 116 1800 5 121 Median : Applying formula. First Quartile and Third Quartile from the following data: Solution. Income No.Vn+T''* Item = size of r 263 + 1^ th item = size of 198* item Third Quartile =8.5 size of shoes. niustration 15. Calculate Median. = size of = size of (N+l^ th Item .5* person ••• Qi = Rs 1.Me = size of = size of fN + 1 Nth item ri2i+i^ th item = 61* item = income 61* person 198 Median = Rs 1200 First Quartile Statistics for Economics-XI Qj = size of 4 Item = size of 121+n Item = 30.5* item = income 35.000 Third Quartile Q. 5* item = income 91.000 Q3 = Rs 1.400 (c) Continuous Series Marks Students Solution.400 Me = Rs 1. = 91. = Rs 1.f 14 30 48 71 89 97 100 H lics-XI Positional Average and Partition Values 199 .200 = Rs 1. of students (f) 14 16 18 23 18 8 3 c. Me 30-35 14 35^0 16 40-45 18 45-50 23 50-55 18 55-60 8 60-65 3 Marks 30-35 35-40 40-45 45-50 50-55 55-60 60-65 Mo.5* person Q.T21 + n th 4 item Thus. .X i Nr — -C. 2. and fN' 4. 4. Calculate less than cumulative frequencies. Apply the suitable formula to get the value : Me = /j + . (N^ v4. th Items .Steps : 1. Locate the median group. first quartile and third quartile group by cumulative frequency column where the size of respective fall. Median. 2.f. 3. . first quartile and third quartile items are located by finding out th n/»T\th u. th N4 th . and N Item m continuous series. fN) 4] -c. median lies in class 45-50 N .f. f ^ 23. Q^ lies in class 35-40 ^ Statistics for Economics-Xl 14 j item = = 25* item . c. 200 First Quartile Qj = size of Hence. XI Median Median = size of-^ item = = 50* item Hence. / = 5 50-48 Me = 45 + = 45 + 23 2x5 23 X5 = 45.4% marks. median is 45.43 Hence. Me = /j + .X i th 100 where. = 45. ^ = 50.f = 48. 4% marks. ~= 25. / = 5 . where. Calculate the Median and Q^ using the following data : Mid-points marks : 5 ■ No. / = 16. of 15 25 .f. c.f = 14.f = 71. ' — = 75. ^ 35.f.11 lo Hence.—-c. first quartile is 38.11% marks. ^ „ 75-71 = 50 + = 51. Third Quartile Qj = size of Hence. i = 5 = 35 + 1^= 38. Q lies in class 50-55 (N^ th rioo^i Item = ^ UJ I4J = 75* item -c. f = 18. third quartile is 51. c. XI where. niustration 17. = 50. f X/ v4.43 16 Hence. we get Median = size of th U> r5o^ item = I2 J = 25* item Hence. Median Applying formula. = 20. Given mid-points are required to be converted into class intervals.X 10 = 20 . Me = /j + ^^— X i where I.2 1 Calculation of Median and Q3. 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 3 10 17 76 4 . Median lies in class 20-30 Applying suitable formula to get the median value "l-c-f.^ = 25. f = 17. i = 10 2S-1 3 Me = 20 + . c. = 13.f.students : 3 10 17 35 7 45 6 55 465 2 75 1 Positional Average and Partition Values 201 Solution. Q^ lies in class 40-50 Applying suitable formula.17 12x10 Hence. i = 10 .f. Third Quartile 17 = 27. = 37.05 marks. f= 6.5* item Qj = size of Hence.f = 37. c.05 n v4y item = ^^ = 37. we get /XT / Nl Qs = K + v4. = 40. Median is 27. where.5. f X/ 3 13 30 37 43 ■47 49 50 202 Statistics for Economics-XI . -c. Illustration 18. we get ( N^^ Median = size of v2. 250 . median hes in class 50-60 Applying suitable formula to get median : Me = /. of Students : 15 35 60 84 96 127 198 250 Solution. + 2 f Xt .83 marks.83 Hence. Calculate the Median and Quartiles for the following : Marks (below) : 10 20 30 40 50 60 70 80 No. Item = -— = 125* item Hence. of^tttdentfi 0-10 15 15 10-20 20 35 20-30 25 60 30-40 24 84 40-50 12 96 50^60 31 60-70 71 ■ 198 70-80 52 250 127 Total 250 Median. Applying formula. Before calculating Median and Quartiles.10 . 40 . third quartile is 40. first we convert the given cumulative frequencies into class frequencies : [ W. = 40.03 = 40.^^. 5* item Xt 62. •i Tl in a Positional Average and Partition Values First Quartile rN 2) Nth item = 250^ Q^ = size of Hence. N = 125.35 marks. '.. lies in class 30-40. c. Q. i = 10 125-96 2 Me = 50 = 50 + 31 29x10 31 .5-60 = 30 + X 10 .f = 96. Applying suitable formula.Hence.35 . we get E^cf.where / = 50. median is 59. f= 31.X 10 = 59. = 62. Q. = Median = 59. Q.item = Hence. = size of -J.04.52 marks. lies in class 60-70. Thns.35 and Q. The following series relates to the da. Q. is 31. = 68. Third Quartile fN th r250 4J Q.52 marks.04 marks. nlustarion 19.. (b) minimum income earned by the top 25% workers. 204 Statistics for Economics-XI . 30 . ^ 31.ly income of workers employed in a firm. Q. = 187. Applying suitable formula.5* item Q3 = ^ f XI 203 r" Hence. we get 4. Q. and (c) maximum income earned by lowest 25% workers. is 68. Compute (a) highest income of lowest 50% workers. = 31.04 24 Hence. ) qutdln"' ^^ ^^^ ^^^ to the given Positional Average and Partition Values SoiutLoa. 205 DaUy Income (m (X) 9. 1. where Area of lowest 25% of workers ■ At this point a worker is earning maximum daily income of lowest 25% workers (i. The data is arranged in ascending order..5 19.5 29.5-39.5-29.. The data is arranged in ascending order. where .5-14. (Median) Nth . The data is arranged in ascending order.25-29 30-34 35-39 20 10 5 Daily Income (in Rs) : 10-14 15-19 20-24 Number of workers : 5 10 15 Before solving it let us understand the question. we are required to convert the classes into class boundaries.lasses 2.5 34. As the data are of inclusive class intervals.4.5 24. where Area of 50% of workers in highest income group At this point a worker in the centre earning highest daily income of the lowest 50% of workers (. lower quartile value = O.5 5 10 15 20 10 5 5 15 30 50 60 65 (a) Computation of highest daily income in lowest 50% of workers. Median value) 3.100% data --------1_ .5-34. ^.e.5 14. H-4 At this point a worker is eaming minimum daily income of top 25% workers .5-24.e.5-19.Area of top 25% of workers 4. (b) Computation of minimum daily income earned by top 25% workers (Q^) th 3x65 ^ l\ Q.125 . we get . N Me = /j + -c.5 .T. = Value of — item = Hence. Q3 lies in class 24.5th item which lies in 24.f.5 class interval.13. = 48.529.Median is the value of th — (65\ item or Uv - I2j or 32.29. Applying suitable formula.5 + = 24.-.5 + 0.5 + f 32.625 = 25. Xt = 24.5.75* value Qs = ^ . Applying suitable formula to get median value.5-30 20 2. Highest data income of lowest 50% workers is Rs 25.5x5 x5 20 = 24. f Statistics for Economics-XI (c) Computation of maximum daily income earned by lowest 25% workers (Qj) Qi = Value of UJ item = 4r = 16. It.-c.92.5 + 4. = ^ f XI = 19.75-30 _ = 24.5 Applying suitable formula.24.25* value Hence.5 + 16.25-15 15 1.19.5 + -TT-x^ = 24. we get Q.687 = 29.f. 20 13 17 10 14 9 .5 .416 = 19. Graphical Determination of Median and Quartiles Illustration 20.5 + = 19.5 + 0.916 Maximum daily income earned by lowest 25% workers is Rs 19.25x5 15 x5 = 19. f Xt 48. Q^ lies in class 19.5 + 20 18.187 Minimum daily income earned by top 25% workers is Rs 29.75x5 20 = 24. Determine median and quartiles graphically from the following data : Marks : 0-5 5-10 10-15 15-20 20-25 25-30 30-35 35^0 Students : 7 10 . f ■ Mjr/fes less than More than cumulative 0-5 7 5 7 0 100 5-10 10 10 17 5 93 10-15 20 15 i7 10 83 15-20 13 20 50 15 63 20-25 17 25 67 20 50 25-30 10 30 77 25 33 30-35 14 35 91 30 23 35^0 9 40 100 35 9 Less than cumulative Marks more than N = 100 First Method (only for median). median value is determined. lics-XI Positional Average and Partition Values 3. 2. 207 Second Method (For Median and Quartiles). From the intersecting point of two ogives. Calculate ascending cumulative frequency (less than). Secc Akrrifes . 2. The point where perpendicular touches X-axis. 5. 3 4. . 4. Draw two ogives—one by 'less than' and other by 'more than' methods. Calculate ascending cumulative frequencies (less than) and descending cumulative frequencies (more than). Steps 1. Steps 1. Determine the value by the following formulae : . draw a perpendicular on X-axis.Solution. . i. Locate 50. N4 th Item. 100 4 3^100^ = 25* item = 75* item V^/ 3. 4.e. i. 25. i. 208 Statistics for Economics-XI VehficaUon Median Group 15-20 Me = /j + . From these points where they meet the ogive draw another perpendicular touching X-axis... 75 values on Y-axis and from them draw perpendiculars or cumulative frequency curve (ogive).Me = size of Qj = size of Q = size of th UJ item. ^^ = 50* item u Item. Qj. The points where perpendicular touches X-axis.e. 5.e. Me and Q^ are located. 30 -cf xt ic 75-67 ^ 8x5 = 25.NU -cf f Xt 50-37 ^ 13x5 Median = 20 Marks Lower Quartile Group 10-15 -cf XI 25-17 = 10 + —-X 5 . = 29 Marks Less than method' cumulative frequency curve is the reminder of the rule that at the .10 20 8x5 20 Qi = 12 Marks.29 Q.— . = 12 Upper Quartile Group 25 . Thus 15 marks are not ^yp. the data is arranged in ascending order. For a better understanding of mode let us look at the following information about frequency of students in relation to marks obtained. There are more frequencies (18. 3. Therefore. 7 shoe. Although 15 have the highest frequency. a more careful examination of the information shows that the highest concentration of the frequency is around 40 marks. Definition According to Coxton and Cowden. Thus. that is value which is most typical. m the neighbourhood of 40 marks. It may be regarded as the most typical of a series of values.c^/ of the series of valLs. to define accurately. Marks : 5 10 15 20 25 30 35 40 45 50 : 2 3 25 2 1 18 20 24 14 10 According to the explanation of mode given above. mode is that value of observations which occurs the greatest number of times or with the greatest frequency. 14. Definition 2. it means in a given data maximum number of people wear size No. Howeven median can be ocated on graph even by more than 'ogive' or calculated by arranging the data m descending order. 7. 20. 1). 10) as compared to the neighbourhood of 15 marks (2. 2. If garment manufacturers say that short collars are now in fashion the statement implies that maximum number of people now-a-days wear short collar shirts If we say the mode is size No. mode is that value of observations around which items are most densely . "the mode of distribution is the value at the point around which the items tend to be most heavily concentrated.hrst step of calculation of quartiles. the modal marks will be 15 because maximum number of students (25) have obtained 15 marks each." x/r j • The word mode comes from French la mode which means the fashion Mode in statistical language is that value which occurs most often in a senes. 40 marks is the mode and not 15. Merits and Demerits of Mode 1. ^hb^ Positional Average and Partition Values 209 1. For the reasons given above. Determination of Mode 3. That is. Determination of Mode (a) Series of Individual Observations and Discrete Series (b) Continuous Series (c) Graphic Location of Mode id) Mode from Mean and Median.or heavily concentrated. in case of three values occurring most frequently then the series is called tri-modal. Bowley "The value occurring most frequently in a senes (or group) of Hems and around which the other item^ar^ distributed most densely. If two or more observations occur the same number of times (and more frequently than any other observation) then there is more than one mode and the distribution is multi-modal.„. The mode is defined as the most frequently occurring value. where there is one mode. (a) Series of Individual Observations and Discrete Series In a senes of individual observations. The mode as a measure of central tendency has little sigmficance for a bi. the mode can be located in two ways • " -- a cl^r""" ^^st ^^ ^^^ Marks : 4 6 5 ' in 98 Solution. then there is no mode in that distribution. as against uni-modal.oae .or "Mode is that value of the graded quantity at wh.- . 10 4 7 6 5 Modal value 8 7 -rks obtamed by 15 students i . " -A.L. 2. If two values occur most frequently then the series is bi-modal.ch the instances are most numerous." 210 Statistics for Economics-XI . If each observation occurs the same number of times. A grouping table normally consists of six columns Frequencies are added in twos and threes and total are written between the values.. values are first arranged in ascending order and the frequencies against each item are properly written. The mode can be determined just by inspection in discrete series. .e. if the items are concentrated at more than one value. It necessary. attempt is made to find out the item of concentration with the help of grouping method. 21 times. By inspection. In such situations it is desirable to prepare a grouping table and an analysis table for ascertaining the modal class. In grouping method.of Persons ■ Solution. Column 1. we can determine that the modal wage is Rs 225 because this value occurred the maximum number of times. Illustration 22. (a) (i) Array : 4 4 5 5 6 6 : Mode = 7 Marks («) Discrete Series. In discrete series the mode can be located by two ways : (i) By Inspection. Find out mode from the following data : Wages (in Rs)___ 125 3- 175 8 225 21 275 6 325 4 375 2 ' No. they can be added in fours and fives also.. the size around which the items are most heavily concentrated will be decided as mode.7 7 8 8 9 9 10. {ii) By Grouping. (i) By Inspection. In discrete and continuous series. Converting the above data into discrete series. i. we get Mode = 7 Marks lics-XI Positional Average and Partition Values 233 (b) Discrete Series. (ii) By Grouping. The maximum frequency is observed by putting a mark or a circle. Column 6. Column 5. 212 Statistics for Economics-XI Illustration 23. Leaving the first frequency. Frequencies are grouped in twos. we shall be in a position to determine the modal class.Column 2. If the same procedure is adopted in continuous series. other frequencies are grouped in threes. Grouping Table Wages (m Rs) • <l) 125 3 No. other frequencies are grouped in threes. Column 3. of persons (3) m ■' (6) 11 175 8 225 21 32 29 ■ 27 275 6 325 4 ii 35 31 10 12 Analy. Column 4. other frequencies are grouped in twos.sis Table 6 375 2 . Leaving the first two frequencies. Leaving the first frequency. put a mark or circle on every total. An analysis table is prepared after completing grouping table in order to find out the item which is repeated the highest number of times. We shall now see how mode is determined by grouping method in a discrete series. After observing maximum total in each of these cases. Frequencies are grouped in threes. Find out mode of a data given in Illustration 20 by grouping. 125 "■ 225.: 3 8 10 12 16 14 10 8 17 5 4 i beW ^^^ be done as shown lics-XI Positional Average and Partition Values 213 Grouping Table 2 3 3 8 4 10 5 12 6 16 7 14 8 10 9 8 .2 3 4 5 6 7 8 9 10 11 12 13 Frequency.Column No. 32S 1 2 3 4 1 5 1 1 1 1 1 1 1 1 1 1 1 1 1 6 3 1 6 Total 1 3 ^^ Smce the value 225 has come largest times..- 275. Compute the mode from the following : Size of the item . hence the modal visage IS lUustration 24. 6 times. 10 17 11 5 12 4 13 1 11 22 30 18 22 J: 18 28 21 24 25 _42 35 10 30 40 30 38 26 The analysis can be done separately also as shown below : ! 12 . 3 4 5 6 Total to ■1 m 32 I It j The value of 6 has come the largest times (5). 12 531 13 ^ Statistics for Economics-Xl 214 (b) Continuous Series applying the following formula? " determined by Mo = I + or Mo = / +Xt ifl~fo) + {fl~f2) X /■ where. hence mode is 6. Mo = Mode . Mo = Mode /j = lower hmit of the modal class modal class and the frequency of the class before the modal class . g ^ 10 12 20 12 215 Positional Average and Partition Values Solution. \f _ f^l Jhe difference between the frequency of the niustration 25. ^ ^ ^ ^ ^^ Frequency . precedmg class (ignoring signs) '" A.5 . .. Since the central sizes are given. Grouping Table Qass Imervai 0. = frequency of the modal class /o = frequency of the class preceding the modal class = frequency of the class succeeding the modal class i = class interval of the modal class The above formula can also be expressed in the following way : or Mo = Mo = /j + X/ A1+A2 "/"i-Zol + l/i-Zil Xt where. 3 .5-1./j = lower hmit of modal class /". Fmd out the mode from the following frequency distribution • Central snes : 1 . we must convert them into class intervals. = (Read delta 2). 5-10.5 3.5-6.5 2.5 5.5 8.5-5.3.1.5-2.5-7.5-8.5 7.5-9.5^.5 6.5 4.5 9.5-.5 (V 6 10 12 20 12 5 32 14 22 32 16 32 17 24 44 10 28 37 . Find the mode of the distribution from the following data : Below 15 20 10 .. we should apply the following formula.5 = 5...5) /j = frequency of the modal class (20) 216 Statistics for Economics-XI fo = frequency of the class preceding the modal class (12) fi = frequency of the class succeeding the modal class (12) i = class interval of modal grdkp (1) 20-12 Mo = 4.5.5 + 0.5 + = 4. .5 + 2x20-12-12 8 X1 40-12-12 8 X1 16 Mode = 4... ..5-5.. Illustration 26. To determine the value of Mode. fi-fo Xt where.Analysis Table 42 20 123456 1 111 11111 Total 1 3 6 3 111 1 1 By Inspection Mode lies in the group 4.5 + = 4. /j = lower limit of the modal class (4. mode of the given distribution first convert the given data into class intervals. For calculation." 25 26 " 30 38 " 35 47 40 52 " 45 55 Solution. Grouping Table 10-15 15-20 20-25 25-30 30-35 35-40 40-45 3 10 7 23 16 - 28 12 - 9 21 14 5 - 8 3 - Positional Average and Partition Values 217 Analysis Table Column No. lO^IS 15-20 20-2S 25-30 30-3S 35-40 123456 1 111 11111 1111 Total 1 3 6 1 4 2 11 1 . Mo = /./j = 16. The class intervals are not equal.46 Mode when Class Intervals are Unequal The formula to calculate the mode from the modal class discussed above. before calculating the value of the mode. They are made equal by combining two or more classes. I = 5 16-7 Mo = 20 + 2x16-7-12 X5 = 20 + ^ X 5 = 20 + 3. Applying the formula. Compute the mode from the following data : Class Frequency Class Frequency 0-3 3-6 6-10 10-12 12-15 15-18 4 8 10 14 16 20 18-20 20-24 24-25 25-28 28-30 30-36 24 14 16 11 10 6 218 Statistics for Economics-XI Solution. When the class intervals are not equal. we must take them equal and the given frequencies should be adjusted presuming that they are equally distributed throughout the class. + ^XI /j = 20. we get fi-fo where. Illustration 27. is apphcablt in a series where there are equal class intervals. = 7.= 12.The mode lies in the class 20-25. Grouping Table .46 Mode = 23./. /•„ = 36. 6-12 18-24 24-30 30-36 1 2 3 4 5. = 36 - - 74 18-24 24 + 14 98 = 38 - _ - 75 24-30 16 + 11 + 10 = 37 - Ill _ 43 30-36 =6 - - Analysis Table Column No. f^ = 38.Class Frequency (1) 0-6 . 6 1 111 11111 Total 1 6 3 3 111 1 1 The mode lies in the class 18-24 Applying the formula. a. = 37. . i = 6 38-36 Mo = 18 + 2x38-36-37 X6 81 . we get X/ where /j = 18. (V (4) 4 + 8 = 12 (V (6) -] 36 6-12 10 + 14 = 24 - -1 72 60 • 12-18 - 16 + 20 . The highest rectangle will be the modal class. 4. 3. / = 10 . f^ = 14. Illustration 28. Prepare a histogram of the given data. where = 20. 2. = 12. Followmg are the steps of locating mode on graph. f^ = 10. Determine the value of mode of the following distribution graphically and verify the results. Draw two lines diagonally inside the modal class rectangle to the upper corner of the adjacent bar. From the point of intersection of these lines.= 18 + J X 6 = 18 + 4 = 22. draw a perpendicular of X-axis which gives the modal value. 219 Positional Average and Partition Values (c) Graphic Location of Mode The value of mode can be determined graphically in a frequency distribution. Mode is 22. of Students Solution. f. . . 1. + . 10-20 20-30 30-40 40-50 50-60 12 14 10 8 6 Marks No. ' Ifi-fo-fi X i.0-10 5 GRAPHIC LOCATION OF MODE Scale: 2 cm = 10 Marks on X-axis 1 cm = 2 Students on V-axis 30 40 MARKS Verification : Mode lies in the class 20-30 Mo = /. This relationship does not exist in moderately —' will pull^a^lt Asymmetrical Distribution. (Negatively Skewed Curve) Negative X < Me < Mo Symmetrical Distribution. gives the modal value.L h I 1 ^^ber of cases above the mean value and below the mean value are equal.33 6 Mode = 23. 220 Statistics for Economics-XI is ^ ^"'Tt't luchZt? u" ^ ^ perpendicular. ^^^ the per^ndicular touches the X-axis.33 Marks. vakr. (Positively Skewed Curve) Positive X = Me = Mo . Mode cannot be determined graphicSSy if two id) Mode from Mean and Median (vJ'uvrj'^l't T'r intl W^ of distribution curves in Chapter 7' a symmetrical distribution mean. (Bell-Shaped Curve) Peak Asymmetrical Distribution. median and mode are orZZ Ll ^fl ^^ ' distributicm of frequencies on either side of the maximum.Mo = 20 + 14-12 2x14-12-10 X 10 = 20 + — X 10 = 20 + 3. negatively skewed). positively skewed) and ha greater concentration m lower values mean and median will be more than the Lde (X and Me > Mo). In other words. {b) If mode in a tolerably asymmetrical distribution is 12 and median is 16.(2 X 58) = 67 .X < Me < Mo. X > Me > Mo.Median) = Mean . Dlustration 29.. mode is lowest. If we know any of the two values out of the three..Mo < Me < X of tie (^'^^^"caly distribution. if the distribution tails off towards higher value lit r conint r" ""'T ^ ^^'-s. if the distribution tails off towards lower value of the data and has greater concentration in higher values. then mode (X and Me < Mo). valufbf Ae^dl'Vyi"^^^^^ distribution. median and mode reveals that in a moderatelv assymetrical (skewed) distribution the median lies between the mode and the arS^Sc mean.2X In most of the cases if the distribution is moderately asymmetrical. approximately 2/3rd distance from the tpode and l/3rd from the lafTS relationship is expressed as follows which is given by Karl Pearson Positional Average and Partition Values 221 Mode = Mean . we can calculate the third value from the above relationship.3(Mean . this IS called b.3 Mean + 3 Median = 3 Median . mean and n^edian are less.2 Mean Mo = 3 Med . Mode = 3 Median . what would be the most probable mean? Solution. {a) In an asymmetrical distribution mean is 58 and the median is 61 Calculate mode. (i. Inhere may be two values in a series which occur with equal frequency. In case of bi-modal distribution or mode is ill-defined. its value may be determined by the above formula which is based upon the relationship of mean median and mode.2 Mean = (3 X 61) .e.-modal series. the value of mode calculated from mean and median would not differ significantly from the value calculated by other methods.e. The relationship between mean. i. mode is highest. In other words. 5 66. -12 -9 -6 -3 0 +3 +6 +9 +12 -4 -3 -2 -1 0 +1 +2 +3 +4 -12 -24 -28 -30 0 +28 +32 +30 +20 3 11 25 55 91 119 135 145 150 N = 150 . of terms f yield X Midpoints m m . musttation 30.i - lfd' = 16 Median : N = 63. Statistics for Economics-XI Production -- No. Production (in kg) : 50-53 53-56 56-59 59-62 62-65 65-68 68-71 71-74 74-77 No. Calculate the mean.5] fd' c.2 Mean 2 Mean = 48-12 = 36 Mean = ^ 2 = 18 Mean =18.2 Mean - 12 = (3 X 16) .5 d I 3 ) d' fm-63.Mode = 67.5 69.5 57.5 75. of farms : 3 8 14 30 36 28 16 10 5 222 Solution.5 63.5 . median and mode production yield.63.5 72.5 54.82 Mean = 63.5 + 0. The following table gives production yield in kg per hectare of wheat ot 150 farms m a village.5 60. Mode = 3 Median .82 kg per hectare ' \7\th Median = the size of f — Item 150 .2 Mean 12 = 48 .f 50-53 53-56 56-59 59-62 62-65 65-68 68-71 71-74 74-77 3 8 14 30 36 28 16 10 5 51.32 = 63. N /.th Item = the size of V2 = the size of 75* item Median hes in group 62-65.67 kg per hectare Grouping table Analysis Table 223 Rupees X (V No (2) (■V (4) .666 = 63.f =55. Me = /. + X f where. we use the following formula : N T -C.f. To interpolate median. c.67 Median = Rs 63. = 62. f= 36. — = 75. of receiver iS) . i = Positional Average and Partition Values 75-55 Mode = 62 36 ■x3 = 62 + 1. nspection the o. f^ = 28. /■„ = 30. 59-62 62-65 65-68 68-71 123456 111 -111111 1111 Total 3 6 4 1 1 224 Statistics for Economics-XI ^Jy .ode hes i„ the group applying the Mowing formula. if 44 *« 62-65 36 80 66 65-68 28 - - 64 94 68-71 16 44 71-74 10 26 74-77 5 1 54 - i!0 31 15 - Column No.50-53 3 - 53-56 8 56-59 14 11 25 - 59-62 30 22 . f^ = 36. we Mo = L + _fi~fo_ 1 _/■_ Xi Here. = 62. 2fi-fo-f2 I. / = 3 36-30 Mo = 62 + = 62 + 2x36-30-28 x3 14 . Median is even more simple to calculate and is almost as stable as mean. Mode. Mode has its own uses and advantages as we have seen. - -. Mode is not A i.can be determined in open-end distribution.29 kg per hectare. not sm^table when relative importance of itemst fctaT tnt —• " "Ot c:. Mean is simple to calculate. vduts L nottt™. but as compared to mean and median.may differ . mode is less suitable. 3. Mode cannot be decided in bi-modal and multi-modal distributions 2.om one "Lg"' " Ae size of the class interval decided lics-XI Positional Average and Partition Values 225 Comparison of Mode with Mean and Median We find that as compared to mean and median. Merits and Demerits of Mode Merits compared to mean LSian Inf^o f A?""' " most typical and cogent ues orr ^rm^^ 2. it can be given algebraic treatment and is not affected by fluctuations of samphng.a'ble of algebraic 4."'' " if the extreme 3.285 x3 Mode = Rs 63. it is not so precise and accurate. "" -hmetic mean can not be ascertained 5. Mode is not based on every item of the series. Mode . its value is definite. M^e is helphU in describing the quaUtative character of the product Demerits 1. '' of shoes etc. although it is influenced by fluctuations and cannot be given algebraic treatment. Mode is the most popular item of a series and is also easy to calculate and simple to understand.= 62 + 1. But it is not suitable for most elementary studies because it is not based on all the observations of the series and is unrepresentative. OF FORMI Individual Series and Discrete S . Median Me = Size of fN+l^ V2 th Item 2.—— X I Qj = Size of l4j Item . rN+n Item 4J th Item Me = Size of th Item X. I N/2-c.f.1. Me = /j + --j. Lower Qj = Size of Quartile 3. Upper Qj = Size of Quartile fN + 1} th I4. Grouping method for discrete series. Deftne mode. and (c) Anthmetic mean 7.be -- 5. Explain how mode can be read on graph paper> • nr^a—ft and tendency -. Discuss ks merits and dements.an .Q3 = Size of f fN . 2. What is meant by the following ? (a) Mode (b) Median. After grouping. Write short notes on : 4 De&rr"'?' of a dismbution. 3 wTT """" -1-s. decide the Modal Group and use the formula to find modal value in continuous series Mos: Xf 226 exercises Statistics for Economics-XI Questions : Define median. ■ " can be read W .4. Item Q3 = + f ' Mode : 1. •5. a„ as measures of and mode of a frequency " rtrsLr ^^ -. 70. 42. Size Number of households : 4.--<b) Average inrelligence of srudents in a 'class. [Me = 210] 60. 48. 95. . and (c) Average production per shift in a factory .80 [Me = 52. 55. . 50./Find out median (a) Serial No. Calculate the arithmetic mean and the median.■ "raTr^ndelf ^ -d. . Median W Arithmetic Mean] Problems : Calculate median of the following data : 145 257 130 260 200 300 210 345 198 360 234 390 159 160 178 Fmd out median of the following information : Marks : 10. 20.5 marks] Positional Average and Partition ^^^ues 227 'e have the following frequency distribution of the size of 51 households.. .5 4 3 1 [Me = 61.6] 5 22 25 ... Q. Me = 5] 6 20 7 12 25 9 30 .. of Persons 2 3 6 /? 61 ^62 63.4 9 21 11 75 Total 51 1 2 3 4 5 2 4 10 8 15 5-- 10 - 15 20 25 2 •4 6 8 10 [X = 5.. = 61 and Q3 = 63] 6. (b) = 20] out median. furst quartile and third quartile of the following series : Height (in inches) : 58..' Compute the median... = Below 20 20^0 40-60 60-80 Above 80 No. [Me = 65.. 59 60 No.. of Students : 0 16 _^. 64 ' 6^ 66 15 10 . /The percentage of marks obtained by 68 students in an examination are given below ..[Me (a) = 12.. Get Mode on Histogram./ [X = 146.30% 20% 63. . Calculate the meanof the following distribution of daily wages of workers in a factory: Daily Wages (in Rs) : 100-120 No. of Students 30-35 14 .67] 8. Me = 146. of Students More than 70% 60% 18 50% 40 40% 45 " . Mean cannot be obtain!? o^ HiLl^^rT:' Marks No. Calculate the median. calculate the median for the distribution of wages given above.) 10% 65 8 50l [Me = 53.7.4 Marks] 228 j„ . (ft) Draw a histogram and indicate mean and mode therein. Statistics for Economics-XI ■ 26 8 ^ 2 50 j 16-19 20-29 30-39 40-49 50-59 60-64 1 ^^ 46 49 ■ 32 28 14 — —iiicuian or tne above data. Marks No.75. of Workers : 10 140-160 160-180 180-200 Total 30 15 5 80 41so./The following table gives the marks obtained by 65 students in statistics in a certain examination. 1] Size of items Frequency Size of items Frequency 1 3 8 10 3898 4 10 10 17 5 12 11 5 6 16 12 4 13. Calculate mode for the following data 7 14 13 1 [Mo = 6] 1 26 2 113 3 120 4 95 5 60 6 42 7 21 8 14 9 5 10 4 [Mo = 3] .4. Q. = 38. 40-45 18 Compute mode from the following series : 45-50 50-55 55-60 60-65 23 18 8 3 [Me = 45.43.35-40 16. Q^ = 51. Median and Mode from the following data : 59 1 61 2 . of Candidates : 263 113 49 9 2 is/uk of electric lamps is given in the following table. of Candidates : 6 Marks : 50-59 29 87 181 247 60-69 70-79 80-89 90-99 No. Below 400 4 400-800 12 800-1200 40 1200-1600 41 1600-2000 27 2000-2400 13 2400-2800 9 Above 2800 4 [Mo :e Mean.Positional Average and Partition ^^^ues 229 ^^ind out the Mode from any of the following two distributions : X : 30-40 40-50 50-60 60-70 70-80 80-90 90-100 f ■■ 6 10 16 14 10 5. 2 And Marks : 0-9 10-19 20-29 30-39 40-49 No. Calculate the median and the mode. 5.75.48] i Followiiig is the distribution of marks of 50 students in a class : [Marks (Kore than) : o\ 10 20 30 40 50 50 \ Calculate theNMedian Marks. Statistics for Economics-XI 32 20 43 11 61 31 47 . . [Me = 27.15 52 56 64 20 35 21 50 22 10 43 42 49 62 75 77 persons is ' 97 given below : 35 30 30 95 . 25. If^60% of students pass this examination.63 9 65 48 67 131 69 102 71 40 73 17 Total 350 [X= 67.9. find out the I minimum marl^btained by a pass tandidate.5%] 46 40 20 10 230 j„ . Mo = 67. Me = 67. LA = jy. 19.he age gronp of 2S to 57 . of workers : 23 25-29 10 55-59 10 30-34 15 60-64 5 5 .ears Mo = 36. of Students Less than 10 Less than 20 Less than 30 Less than 40 Less than 50 Less than 60 Less than 70 Less than 80 Less than 90 5 15 98 242 367 405 425 438 439 20.22] Age in years 20-24 iVo.60 27 53 31 9 45 22 36 13 46 73 81 40 40 55 67 54 23 42 25 51 modal age.j:). Me = SX 44 For the data given below find graphically the folWing • [a) The two quartiles. Determine the value of mode for the foil ^ Mode = 3. ' (b) The central 50% limit of the age ' W The nnntber of workers falhttg .n . of workers : Age in years : 50-54 No. Median = 2 M^n = ^^ : 21. c1. [Modal age = 42 years] Ma No. The following table gives the distribution of the wages of 65 employees in a factory. Wages (in Rs) : 50 60 70 80 90 100 110 120 (Equal to ormore than) " ^^0 Number of employees 65 57 47 31 17 7 ' 2 q 23.35-39 25 65-69 2 40-44 65 45-49 40 Draw a -less than' ogtve front the following data and hence find out the value of Class 20-25 25-30 30-35 35-40 40-45 45-50 50-55 55-60 Frequency 6 9 13 23 19 15 9 6 231 Positional Average and Partition ^^^ues 253 22. of students 0-10 0 . Draw the histogram and estimate the value of mode from the following data : Marks No. 10-20 2 20-30 3 30-40 7 40-50 13 50-60 11 60-70 9 70-80 2 80-90 1 24. Weekly wages : No. Represent the following data by means of a histogram and find out mode.:. of workers : 10-15 7 15-20 19 20-25 27 40-45 25-30 30-35 35^0 . 7i?'r ti!^™ ^sir ^^ -Chapter 10 measures of dispersion Introduction Objectives of Measuring Disperelon Metliods of Measuring Dispereion (A) Dispersion from Spread of Values (B) Dispersion from Average (C) Graphic Metiiod—Lorenz Curee Absolute and Relative Measures of (A) Absolute Measures (B) Relative Measures Graphic Method . 15 12 12 8 W ind^idual incomes. the 'less than type' Ogive and the . a single fig^e. Brooks and W. Dick. Hef therLre decidl7thTh Tu""^" ^^ his mer on foot. "Dispersion is a measure of the extent to which the individual items vary. TTiere ifnTed to nt ^^^^of a majority of the people Measures of Dispersion 233 Definitions According to D.L. But it so Wpen^af hf ma^^^^^ Ae height of youngest child^of the ^^ ^s^v 12? m W had happened to the statistician famity ^ ^^ ^^ "" ^^^t must r-^ -7 ^^ there may be great may be below poverty line.eafr^r:?:-^^^^^ -.R. look at the following data about salaries paid to employees of three different departments of an organisation. L.F.C. "Dispersion is a measure of the variation of the items. A ■ ." According to Prof." According to A." Now. These the form of an average. Dept.Comparison of Measures of Dispersion List of Formulae . These av^Les Sf uf frequency distribution in magnitude of the distriLion but ZrS tfir ^^put the general level of Measures of central tendency a^e som^i^r^^^^^ happens when the extent of variationf ^iZl f ^his relation to the other values is lajr^n any ^^^^^^^^ ? ^^ - not only to know the average ^ow about the measures of He knew that the average depth of the r^ter w^ Too 1 uTl' ^ family was 130 cm.L. Bowley. Connor. "Dispersion or spread is the degree of the scatter or variation of variables about a central value. .e.Deviation t.500 + 500 + 1000 0 . i. C Deviation fWf MS 5000 5000 5000 5000 5000 00000 4500 5500 6000 5000 4000 .. B mmmmi Dept.500 -500 Total : Mean X : 25000 Rs 5000 25000 Rs 5000 25000 Rs 5000 We find from the above table that the average salary paid to employees in each department is the same. but the constitution of series is quite different. . Rs 5000. Rs 5000. In this case lowest value is Rs 4000 and the highest value is Rs 6000 and the difference between the highest and the lowest value is Rs 2000. and the highest deviations from the mean are -1000 and +1000. In department 'B' though the mean is Rs 5000.1000 10000 2000 4000 4500 4500 5000 - 3000 - 1000 . i. In department 'A' the salary paid to each employee is the same.e. hence mean is fully representative of the values of the items in the series. If the dispersion of variabdity m the values of various items in a series is large the average may be unrepresentative of the series. (b) To Serve as Basis for Control of Variability : The study of variation is done also for the purpose of analysing why large variations happen or occur and this may help to control the variation itself. some deviations are large and others are small. . wherein different series of three departments the mean was a common value and the variations differed. 234 j„ . This point has already been made clear in the above dlustration. If on the other the variability is small. The lowest value is Rs 2000 and the highest value is Rs 10. in industrial production to control the quality of the product and the causes of variations in product are obtained by inspection and quality control programmes. the average would be a representative value. Not a single item in the series is represented by its mean. the heart and pulse beat are recorded and an attempt is made by the doctors to control these through provision of medicines.000. In department 'C though the mean is the same. The difference between the highest and the lowest value is Rs 8000. For such a study we have. in some major human health problems the blood pressure. we must clearly define the objectives. we are required to make an overall summary of these differences (scatteredness) in all values about the central value. (c) To Make a Comparative Study of Two or More Series : Measures of variability are also useful in comparing two or more series with regard to disparities or . For example. Therefore. In social sciences where we have to study problems relating to inequality in income and wealth. measures of dispersion are of great help. Statistics for Economics-XI |bjectives of measuring uispmm Before we go on to describe the specific methods of studying variabihty. It is clear that we must not only know the composition of a series but also observe how the composition of a series differs from another. This summary is called the measures of dispersion or measures of variation. ia) To Test the Reliability of an Average : Measures of dispersion enable us to know whether an average is really representative of the series. From the above illustration we observe that some deviations are positive and some are negative. a statistical tool called measures of dispersion or measures of variation. which deviate from mean by -3000 and +5000 respectively.The mean in this case. does not adequately represent the values of the items in the series of department 'B'. Similarly. Similarly. but there is wide gap between the values of items. share values. performance of individuals and studies relating to demand. While the mean deviation and standard deviation are from an average defined in terms of deviations from a central value. They are calculated from the values of the variable at a particular position of the distribution. Following are the important methods of measuring dispersion : M^ODS OF MEASURING OlSPERSiON i from Spread ilues Range Interquartile Range and Quartile Deviation from IMean Deviation or Average Deviation Standard Deviation Method-Curve Measures of Dispersion 235 First two measures. While a low degree of variability would indicate high uniformity or consistency or stability. They are not based on deviations from any particular value. termed as positional measures. Same points are for characteristics of a good Measure of dispersion. kurtosis correlation. regression etc. Comparative studies of varmbihty are very useful in many fields like profit of companies. id) To Serve as a Basis for Further Statistical Analysis : Measure of variability which IS measure of second order is very useful in the use of higher measures such as skewness. Note: Characteristics of a representative average are explained on Page 139 and 140 ot this Book. Range and Quartile Deviations are from spread of values. supply and prices. Measures of dispersion in terms of spread and position are as under : (A) Dispersion from Spread of Values (a) Range .. viz.differences. A greater degree of dispersion or variability would mean lack of uniformity or consistency or homogeneity of the data. Lorenz curve is graphic method of studying dispersion/variability. etc. Symbolically. Meaning 2. Thus if we want to compare the variabihty of two or more distributions with the same units of measurement. L = Largest item S = Smallest item Relative Measure To compare the variability of two or more distributions given in different units of measurement. Uses of Range 1. Meaning Range is the simplest measure of dispersion. In case of a frequency distribution. It is determined by two extreme values of observations. we may use absolute measure.(b) Interquartile Range and Quartile Deviation (a) Range 1. Calculation of Range 3. This relative measure is called coefficient of Range. Merits and Demerits of Range 4. It is obtained by applying the following forniula : Coefficient of Range = L-S L + S 236 j„ . It is common practice to use coefficient of range even for the comparison of variability of the distributions given in the same units of measurement. S = 5 . In case of the grouped frequency distribution range is defined as difference between the upper Hmit of the highest class and the lower limit of the smallest class. Range as defined is an absolute measure of dispersion and expressed in the units of measurement of the given data. we cannot use absolute measure but we need a relative measure which is independent of the units of measurement. the frequencies of the various classes are immaterial since range depends only on the two extreme observations. Range is the difference betu/een the largest and the smallest value in the distribution. Statistics for Economics-XI Range = L-S Here. range is located by the following formula : Range = L . L = 30.S where. Range = 30 .? Absolute Measure of Range " ^^^ ^ ' au ■ .11 ^35 Range = L..5 20 = 0. of Persons : 10 15 17 Calculate range and the coefficient of range.35. Coefficient of Range = kzS L + S _ 30-5 30 + 5 = 0.714 . Coefficient of Range = IlzI L + S _ 60-0 ^ 60 + 0 = 1 Age (in year) : l^^o 21-25 26-30 31-35 No.5 = 20 years .18 = 15 years.0 = 60 Range is 60 Marks.5-15.Range = 35.S Here. Range = 33 .39 . L = 60.5 = 25 Range is 25 Marks. Mid-value of highest class =33 . ai ci Measures of Dispersion Relative Measure of Range 237 Coefficient of range = L-5 L + S 35. ^ last class will become 30 5 .. r Range = L-S Alternatively (from mid-values) T. S = 0 Range = 60 .5-15. and 3«» Quartiles) Q. U.v.40.and 37.. Marks of 12.S'^-c.20 6 10 20 . of Students Less than 10 4 Less than 20 10 Less than 30 30 Less than 40 40 Less than 50- 47 Less than 60 50 Solution.5* student in class 30 . of Students 0-10 4 A 10 . Marks of 12. ----------------.50 7 47 50 .f.30 20 30 30-40 10 40 40 . and Q. We arrange the data in continuous series. The following are the marks obtained by 50 students in Statistics.5 51 Illustration 3.e. Marks No.60 3 50 lb get marks ot miame ou /o stuucms.30 and marks of 37.. 1.5* student (i.Alternatively Coefficient of range ^ 33-18 _ 15 _ 0 29 33 + 18 ~ 51 ~ 35. wc ^it . Np.5 + 15..5* student hes m class 20 . Calculate the range of marks obtained by middle 50% of the students. .5* student = l^ + ^ Xt 20 . 25 Marks = L-S = 37.S = 72 .S 72 . When shortest man (61 inches is omitted) the range will be = L . 65. 70.61 = 11 inches.27% 11 3.25 marks Solution.64 = 8 Change in the range = 11 .5 . 69.8 = 3 Percentage change in the range ^ x 100 = 27.25 = 16. rj• " . 67. 72 Range height = L .5-30 XI 10 X 10 = 30 + X 10 = 37.5* students = /j + = 30 + 20 37. f 37. Statistics for Economics-XI = 37.25 Marks Marks of 37. It gives broad picture of the data quickly. We arrange the data in ascending order 61. Merits and Demerits of Range Merits 1. 68.25 Marks Thus. range of marks obtained by middle 50% students is 16.5'*' Smdent-c. 64.21.5 Marks 10 Largest Value Smallest Value Range Marks 238 j„ . 67. Range is simple to calculate and easy to understand th^ rr.5^arks = 21. 68. 66./". ban a very accurate picture of variability one may compute'rangl ' 2.= 20 + ^ X 10 = 21. It is rigidly defined. lb) Measure of fluctuations : It is a very useful measure to study fluctuation. 4. is usually provided by the probable limits in the .: of series Variations in the prices of share. the range would shoot up from 20 to 80 centimetres. the answer to the problems hke daily sales in a departmental store'. It depends on unit of measurement of the variable It has the Demerits le es :st he ies )m 239 Measures of Dispersion 160 to 180 centimetres. Thus. For example. 4. It does not tell anything about distribution of items in the series relative to a measure of central tendency. a single variation in the value of an extreme item affects the value of the range. 'monthly wages of workers in a factory'^ or the expected return of fruits from an orchard'. 3 It is influenced very much by fluctuations of sample.3. Range is subject to P uctuations ■ of values from sample to sample. Uses of Range Despite various limitations. if a dwarf (shortest) student whose height is 100 cemimetres is admitted in our data. other commodities arJ money rates pnd rate ot exchange can easily be studied with the help of ran^e. However in small samples. Range has a great significance in quality control measures. the range is very unsatisfactory measure of dispersion and should be used with great care and caution.'sed measure of variabihty in our day-to-day life. " (c) Use in day-to-day life : Range is by far the most widely . it is uscxul in certain circumstances. It cannot be calculated in case of open-end distributions because extreme values ot the distribution are not known. Thus. the range is useful in the following areas: (a) Quality control : Range is used to study the variation in the quality of the items produced of a manufacturing concern. 5. 4. It is also a measure of dispersion.R = Interquartile Range = Q _ q ..form of range. In fact 50% of the values of a variable are between the quartile (i. (b) Interquartile Range and Quartile Deviation 1. 240 Statistics for Economics-XI Semi-interquartile Range or Quartile Deviation ■ A. = Q3-Q1 2 Symbolically. it would give us what is called the 'Interquartile Range'.Q. Q. Merits and Demerits of Quartile Deviation 1.) and as such the interquartile range gives a fair measure of variability. tU I.^similarly if the diffLnce in the two values of quartiles is calculated. It is an advantage over range m as much as. and Q. Calculation of Quartile Deviation 3. Meaning 2.D. Q. Q3 = Third Quartile.e. Id) Use in meteorological department: Range is also used in a very convenient measure by meteorological department for weather forecast since the general public is interested to know the limits within which the temperature is likely to vary on a particular day. QazQi Coefficient of Quartile Deviation = _2__ Q3 + Q1 2 . it is not affected by the values of the extreme items. Meaning -i i ■£ u Just as in case of range the difference of extreme items is obtained. = First Quartile Semi-Interquartde Range or Quartile Deviation or Q. 2. It IS rigidly defined. of Q. It does not depend on all the values of the data v^rLblf deviation are the same as those of the 2. 198 234 159 160 178 257 260 300 345 360 390 fromi Measures of Dispersion SJiT-i-ti. Calculation of Quartile Deviation (a) Series of Individual observations (b) Discrete Series (c) Continuous Series ! (a) Series of Individual Observations the -ijtr ^^^^ - 145 130 200 210 .D. 241 ' Y S NoC ' ■ Income (Rs) 1 130 9 234 2 145 10 257 3 159 11 260 4 160 12 300 5 178 13 345 6 198 14 360 7 200 15 390 8 210 Income (Rs) ^he quartile . = fc^ 1. It IS simple to calculate and easy to understand.Coeff. f 3. Qj Here. Locate the value by finding out Qj = size of item and Qj = size of fN + 1 th item.D. Thus. Apply the formulae to get interquartile range. = Hzl^ = Rs 70 Coefficient of Q.Steps 1. quartile deviation and coefficient of quartile deviation.304 242 j„ . we get Qj = Size of = Rs 160 Q. = 160 = 300 . = 300 and Q. Statistics for Economics-XI . 3.160 = Rs 140 Quartile Deviation = Q3-Q1 Q. = 2 Q3-Q1 Q3 + Q1 300-160 140 300 + 160 460 = 0. = Size of item = 4* item ri5+n th item = 12* item = Rs 300 Interquartile Range = Qj . Q. 2. Arrange the data in ascending order to get the value of lower and upper quartiles.D. 6. •■• .Solution.30 = 10 Marks Thus.6 ••• Qs + Qa = 50 Now equation (1) and (2) are solved..= 30 Tu tt = 40 .(2) = Q3 ...(1) QizQL Qs+Qt = 0. Q3 .Qi = 30 80 Q3 = Y = Marks Putting the value of Q^ in equation (i) Qs-Q.Q = — 0.= 30 Coefficient of Quartile deviation .D. 30 Q3+Q1 = 0. = 30 40 . We are given Quartile deviation (Q. Upper Quartile = 40 Marks and Lower Quartile = 10 Marks (b) Discrete Series series : coethcient of quartile deviation from the following Heights 'I '' 63 64 65 66 ^ 6 15 10 5 4 3 1 f Coe (in inches) .6 ..) = SlZ^ = 15 Marks r ■ . Locate First Quartile and Third Quartile by (n+IY" and — .5* item = 61 inches Q. 58 2 Range = L .S = 66-58 = 8 inches ai (c) G HI coeffii Ai Nc Sol I'l Measures of Dispersion Quartile Deviation Steps : 1. Values are located at the size of item in whose cumulative frequency the value of item falls. Qj = size of fN+l^ Item = (49 + 1^ th Item = 12. Calculate cumulative frequencies. of Persons : Solution. = size of . 5. Apply the formula to find quartile deviation. 2. 3. Arrangement of items in ascending order is necessary.No. V^y V^y 4. Q.D. = 0.016 (c) Continuous Series Illustration 8. and Range = 8 inches Q.= 63 inches Q. .f. = Q3-Q1 _ 63-61 3x50 . Also calculate coefficient of quartile deviation of the following data. = = = 0. Coefficient of Quartile Deviation Coefficient of Q. Item = -= 37.5* item = 1 inch 243 Height No. of in inches persons (f) 58 2 2 59 3 5 60 6 11 61 15 26 62 10 36 63 5 41 64 4 45 65 3 48 66 1 49 c.OI6.D. Calculate range and quartile deviation and compare them.D. Q3+Q1 63 + 61 124 Thus. = 1 inch Coeff.D. X I Thus. Locate. Apply the following formula : . we get First quartile + J--X / = Size of . Range Range = L . the first quartile and third quartile group m cumulative frequency column where size of respective — and UJ N UJ item falls.+ -J. 3. of members : 3 61 132 154 140 51 3 Solution. 2.4 item in continuous series.Age (years) : 20-30 30-40 40-50 50-60 60-70 70-80 80-90 No. First quartile and third quartile items are (NY' 3/».S = 90-20 = 70 years 244 Quartile Deviation Steps : 1. 4. Calculate cumulative frequencies.T\th j„ . Statistics for Economics-XI located by finding out (-] and V^y . c. of t.An 136-64 an 72x10 " " ~13r = years Hence first quartile is 45.= size of U (544^ 4 item th item = 136* item Hence. Q^ lies in the group 40-50 'i + J. / = lo n .f (years) members (f) 20-30 3 3 30-40 61 64 40-50 132 196 50-60 154 350 60-70 140 490 70-80 51 541 80-90 3 544 KI Measures of Dispersion Third Quartile 245 .45 years. Age No. ^ = 136. h . = 64.f.40.X / where. f= 132. f. XI = 408.) = ^ 64. N -c.4.Qj = Size of = Size of th Vn .D.D.45 2 = 9. i = 10 xlO = 60 + 140 58x10 140 = 64.345 years Coefficient of Quartile Deviation Coefficient of Q.f = 350. f . h = 60. f = 140. = ^^^ Q3 + Q1 .4.14-45. 3x544 Item = 408* item Hence Q3 lies in the group 60-70 where.14 years Quartile Deviation (Q. c. ^ j^g 13. Merits and Demerits of Quartile Deviation Merits : 2 'r'""" " " -derstand._ 64.■] .~ _ -. It ignores half the times-lst 25% and the last 25% 2. 3. . T^S measure .s useful when it is desired to know variah.500 -100% Persons O3 = Rs 45. = 9.69 ~ 109.45 64.345 years Coeff.s also useful where extreme values are likely to affeet the results 4. 4 2Tntb iT" " by fluctuations of sampling.D.000-18.Q3-Q1 45.000 27. = Q3-Q1 Q3+Q1 " 45.000 45.D.428).17 246 j„ . of Q. Demerits : 1.000-18.45 _ 18. 4^ As an ah olute measure it is not sufficient for comparison.17 Thus Range = 70 years Q.000 vv.000 Relative value of dispersion Coefficient of O.14 + 45. ^ ^^^ ^ ^ ^^e data (Rupees) percentage or coefficient of the absolute measure 3.000 + 18. = 0. It .lity in L Itral part o. Statistics for Economics-XI Absolute value of dispersion Quartile Deviation (OD ). h IS a so not possible to give it further algebraic treatment.14-45.59 = 0.D.000 63:000 = 0-428 and^rlti^etatr^tn?:^^^^^ of dispersion (0. '"^"'^ " ^^ for a rough study of 247 Measures of Dispersion (B) Dispersion from Average The range.600 -3. It is.100 -900 £(X-X) = 0 .—= Rs 5. Similarly. the interquartile range and the quartile deviation suffer from common defect. Here.000 4. In gerieral.000 4. therefore always better to have such a measure of dispersion which is based on all the observations of a series and is calculated in relation to a central value.500 EX = 27. As we know the sum of the deviations calculated Lm arithmetic mean is always zero.000) is more from arithmetic mean (Rs 5.000) is quite less than the arithmetic mean.000 Deviations from Mean D E Total (Rs) 2. some deviations are positive and some are negative.400) and the salary of B (Rs 2. such measure of dispersion throws light on the formation of the series and the scatteredness of items around a central value.variX. Let .IX 27000 Mean (X) = — .000 6. Range and Quartile deviations are not calculated in relation to any average. This method of studying dispersion by location of limits is also called the 'Method of Limits . They are calculated by only two values of a series-wither extreme values m case of range or the two values of the quartiles as in case of quartile deviation.us examine from the following illustration about the salaries paid to employees of a departmental store : Monthly Salaries of Employees Employee A Salary (in Rs) m B C 10. some are large and some are small. If we consider an average of these deviations calculated from arithmetic mean. we can get an idea of a measure of dispersion.500 4. This method ot calculating dispersion is called the 'method of averaging deviations'.400-1.400 We observe that the salary of A (Rs 10. positive deviations and . If the variations ot items are calculated from an average.400+1. Calculation of Mean Deviation 3.Meaning ^^'"■'^""ring plus (.D.D. 249 ElDl = (Read sigma D modulus). Calculation o£ Mean Deviation {a) Series of Individual observations [b] Discrete series (c) Continuous series (a) Series of Individual Observations . the measures of dispersion in terms of deviations from central value (average) are as under : (A) Mean deviation or Average deviation Where absolute deviations are obtained from average (ignoring plus and minus signs). adding these deviations directly does not help us. XorMe 2.) and nUnJsAsS^^^l^^^ rith^^ean or Measures of Dispersion where. Therefore. sum of the deviations taken from mean or median ignoring ± signs N or M = Number of observations f = frequency X = Mean Me = Median Relative Measure of Mean DeviaHon Coefficient of M.j . (B) Standard deviation Where deviations obtained from arithmetic mean are squared.negative deviations cancel out each other. Alternatively. = ^ M. 248 Statistics for Economics-XI 2. Demerits and Uses of Mean Deviation 1. Thus. Ments. we may consider either the 'absolute deviations' or 'squanng deviations . 1 22 2 29 3 12 4 23 5 18 6 15 7 12 8 34 9 18 10 12 Calculation of Mean Deviation 12 12 12 15 18 18 22 23 29 34 N = 10 666 300 4 5 11 16 LIDI = 57 . 10. Calculate mean devation and its coefficient from median and mean fro^TL following yeld of rice per acre for 10 districts of a state as under: Districts Rice Yields (in tons) Solution.BtasMrio. Calculate the median of the series fN + lY' j„ . 4. Formula : EIDI . Statistics for Economics-XI Me = size of Item.22 2. Take deviations of items from mean Jg^ormg ± signs and denote the column by 3.5 12 7. '"umg 2. Calculate the total item of finding arithmetic mean.5 34 14. 2. Cdcufate the sum of these deviations.5 15 4.5 18 1.5 23 3. Divide the total obtained by number of items.5 EX = 195 EIDI = 250 1. 3.5 18 1.5 12 7.5 29 9. Cdculate the sum of these deviations. Arrange the data in ascending order.5 12 7. 1. as Take deviations of item from median Jgxonng ± smgs and denote the column 4. D. we get Mean = Median fN + Vth item 4J rio+1^ th <2 Now. we get Median = Size of = Size of = 5. = ^ XT Mean Now. Divide the total obtained by number of items.5 (0) = 18 tons Absolute Measure : JV ~ 10 .5. M. Formula : ZIDI M.D.5* item = Value of 5* item + 0.'on® Relative Measure : Coefficient of M.Value of 5* item) = 18 + 0.5 (18 .D. 57 .5 (Value of item .D.18) = 18 + 0. = N Si"""'' »' Coefficient of M. = N M.D. Illustration 11. Solution.D. {Hi) Mean Deviation about Mean and coefficient.5 = 0. the yield of wheat per acre for 10 districts of a state is as under: District : 1 2 3 4 5 6 7 8 9 10 Yield of wheat : 12 10 15 19 21 16 18 9 25 10 (in tons) . (i) Calculation of Range (Absolute Measure) Range = L .D. from median than that from mean because the sum of the deviations taken from median ignoring ± signs is less than sum of deviations taken from mean. = m. (iv) Meati Deviation about Median and coefficient.N 195 Median ~ = ~ 10 = 19.5 tons Absolute Measure : N=— ~ 10 = 6 tons Relative Measure : Coefficient of M. Calculate : . In order to calculate the quartile and median we arrange the yield of wheat in the ascending order of magnitude. («) Quartile Deviation and its coefficient. (i) Range and coefficient of range. Mean ~ 19.D.S L = 2S.S = 9 Range = 25-9 = 16 tons .307 251 Measures of Dispersion Note It is better to calculate M. (ii) Calculation of Quartile Deviation Calculation of Coefficient of Range (Relative Measure) L-S Coefficient Range = 25-9_16^0.75th item = Value of 2nd item + | (values of 3rd item .Value of 8th item) . Qj = size of rN+n th item = size of srio+i"! th item = size of 8.75 (10 10) = 10 + 0 = 10 tons.47 25 + 9 34 Qj = size of N+l item = size of rio+1 Mh item 9 10 10 12 15 16 18 19 21 25 = size of 2.25th item = Value of 8th item + ^ (Value of 9th item .value of 2nd item) = 10 + 0. = Mean Deviation = 4.D = QiJ:^ Q3+Q1 ^ 19. = Sl^ _ 19.D.75 tons of Mean deviation Statistics for Economics-XI Relative Measure : Coeff.5-10 .5-10 _ 9.= 19 + 0. = .19) = 19 + 0. we get M.25 (2) = 19 + 0.322 Absolute Measure : Mean Deviation from Mean Arithmetic Mean. Q.5 l^XTlO ~ 2^ = 0.=4.2 tons ^^ = 4.5 = 19.5 tons 252 Absolute Measure : ■•■ Quartile deviation Q.D. of Quartile deviation Coeff.2 Mean Deviation from Median Me = Size of = Size of 4 10^ y'' item item Relative Measure : Coeff. of M.25 (21 .D. = N ~ IF tons Applying formula. 5 Measures of Dispersion .5th item = Value of 5th item 2 (Value of 6th item Value of 5th item) = 15 + 0.75 tons M.15) = 15 + 0.5 = 15.D. = 4.D.6 ~ 0.75 tons. of M.D. of M.3 Coeff. 253.wh.D.3 tons (from median) and " - that . deviation.3 tons Relative Measure : M.D.5 (16 . = 0.D. = 0.D.269 = Size of 5.= ^ 10" ••• Mean Deviation = 4. = 4.s why we ate g tfj™ h^ XeTah T "'data. of M. Q""™'' " f" "f series.Mean 15. Range = 16 tons .D.7 tons Coeff.277 15. = Median = 0. = 5. = 4.316 M. Q. 4.chasaLaRe of Ae d^er„rh l all the observations of se™s ttn«25 the value 4. giving us ^^ ^^ ~g of absolute distribution and thus cVcu^ ^^^ -regularities in the true measure of dispersion accurate and Relative Calcularioas of Illustration 10 and 11 Refer to Illustration 10 and 11 p R^eY^U ■ Wl^at Veld M.D.277 .3 rons Coeff.5 tons M. Statistics for Economics~XI 012 3 4 5 6 7 89 10 11 12 15 16 21 10 16 84212202 IDt fm 15 2 30 31 1 16 52 0 0 62 1 10 78 2 32 . ifteretore.eld has greater variati" 6)1" cal-lations we decide rife has lesser variation is more rehable Therefore the vSh 7" ^^^ ^^^P ^hich the yield of rice. 254 Solution. the yield ot wheat is more reliable than {b) Discrete Series Also"tS: ^^^^ lismbution.y. f\D\ N M. Multiply frequencies with deviation and get f\D\. Find out the median. as ^ke deviations of items from median ignoring ± signs and denote the column 6. 3. Locate the item by finding out th item. = Median . Calculate cumulative frequencies 2.86 3 24 90 4 16 92 5 10 93 6 6 95 7 14 97 8 16 97 9 0 99 10 20 Z/'IDI .D. 5. 4. After getting the total of f\D\ column apply the following formula : I. 7. Value is located at the size of the item in whose cumulative frequency the value of « Item falls.194 Total N = 99 Steps : 1. 98 (c) Continuous Series ' Illustration 13. Calculation of Mean Deviation from Mean .96 = Approx 2 Accidents = 0.D.96 Median = 1. = 99 1. of Students : 5 8 15 16 6 Solution.Me = size of = Size of (N + lf 2J item item = 50^^ item Median = 2 Accidents Measures of Dispersion Mean Deviation : 255 M. Coefficient of M.D.D. Calculate Mean following data : Deviation from mean and its coefficient of the Marks : 0-10 10-20 20-30 30-40 40-50 No. = Z/IDI 194 N Coefficient of Mean deviation : M. = If\D\ N . 2. of Students Mid-points m. After getting the total of f\D\ column apply the following formula : If\D\ M. Mean Deviation : A = 25. 0-10 5 -2 -10 22 110 10-20 8 15 -1 -8 12 96 20-30 15 25 0 0 2• 30 30-40 16 35 +1 +16 8 128 40-50 6 45 +2 +12 18 108 5 N = 50 -Lfd'^ 10 m-25 10 d' fd' m. N = 50.D.17 ^IDI • •Lf\p\ = 472 Steps :.-Lfd' =10. 1. = Now. we get Arithmetic Mean : N Zfd' X = A + ^x C N 256 where. 4. C = 10 V ir 10 ^ ^ "" 50 = 25 + 2 = 27 Marks Statistics for Economics-XI M. 3. Calculate Arithmetic Mean by step deviation method. Take the deviations of mid-points from mean ignoring ± sings and denote them by ID!.Marks X No.D. Multiply these deviations by respective frequencies and find out /IDI. where.44 Marks Coefficient of Mean Deviation = Here. we will calculate and Q^ Qj = size of = size of .44 and X = 27 9.44 >27 = 0. = 9. M.f |DI = 472. (/) Calculation of Range Absohite Measwe : Range = L.0 = 50 Marks Relative Measure : Coefficient of Range = L-5 L + S 50-0 50 + 0 =1 (ii) Calculation of Quartile Deviation First. S = 0 = 50 .D.S ^here.349 0-10 5 10-20 10 20-30 20 30-40 5 40-50 10 Marks : No. I. N = 50. X M. M D = ~ 50 Mean Deviation = 9.D. L = 50. of Students : Solution. V4y (50 item th item = 12.5 Marks Q = size of N 14 Item = size of '50 .5* item Q lies in the class 30 .20.5th item Measures of Dispersion 279 Qj lies in the class 10 .4. c. — = 12.-I. f= 10 and i = 10 = 10 + ^^xio = 10 + ^^^^ = 17. th item : 37.* N■4 -cf ■XI Here. formula : 257 Q. 1. N /j = 10. = 30. Applying the following.40 Here. .f = 5.5. = ^^^^^ 32. c. Q. Mid-poitos. = Q^-Qi Q3+Q1 32. /■ = 5 and i = 10 Q.5 50 ■ 258 (iii) Calculation of Mean Deviation from Median Statistics for Economicsr-XI Marks Stud^ts f c.f = 35.Vn) v4y = 37.5-35^^^ = 30 + 5 2.5 15 32.D.5x10 10 = 32. m 0-10 5 5 20 100 10-20 10 15 15 10 100 20-30 20 35 25 0 0 30-40 5 40 35 10 50 40-50 10 50 45 20 200 5 N^ 50 Zf IDI = 450 f«-25 IDI f\m . = 30+37.f.5-17.5 Marks Relative Measure : Coefficient of Q.5 Marks Absolute Measure : Quartile Deviation.5-17.5.5 = 7.5 + 17.D. 2. /■ = 20 and / = 10 Me =.Steps : 1. Multiply deviations by respective frequencies and find out Z/'IDI. IDI. c.20 + ^^x 10 20 + 20 10x10 20 = 25 Marks. J = 25. Calculate median of the given data.D.f = 15. After obtaining the total of f\D\ column. = 20. apply the following formula : n\D\ M. Me = + ^. Measures of Dispersion Absolute Measure : Mean Deviation 259 . Take the deviations of mid-points from median ignoring ± signs and denote 3. 4. /. Median : Median = Size of = Size of Median lies in the class 20-30 Apply the following formula : N l2j 12J item item = 25th item Here. = N Now we get. 2.D. = Now.6 6 4-6 4 5 0 0 0. we get Arithmetic Mean : X=A+ .D.8 3.2 6. Zf \D\ = 450 and N = 50 450 M.8 0 6-8 2 7 +2 4 1.D. 4. Multiply these deviations by respective frequencies and find out f IDI.D.8 4 N = 10 lfd = 2 - I/-IDI = 14.8 I/^WI = 14 Steps 1. Take deviations of mid-points from mean ignoring ± signs and denote them by IDI.36 Illustration 15. Calculate Arithmetic Mean by assumed mean method. 3.M.6 4 8-10 1 9 +4 4 3.8 3. 2. Calculate the mean obtained by 10 students. After getting the total of /IDI column apply the following formula: S^IDl N M. = If\D\ N Here. Marks : 2-4 4-6 deviation from mean for the following marks 6-8 8-10 Student : 3 42 1 2-4 3 3 -2 ■ '-6.2 0. = ^ = ^ = 0. = Relative Measure 50 = 9 Marks Coefficient of M. M.2 260 Mean Deviation: j„ .N=10 • class (the class m which mean hes). N = 10 14.. ' ob«.D. ZnZ. = l/l^^llP^z^KI/B-^ N ^f\d\=14.2) (4) 14 + 0. i. 4 + 3=7 If A = Sum of all class frequencies after the mean' class.A = 5. Statistics for Economics-XI Here.8 14. Now. 2 + 1=3 MD = 10 = 14 + (0.8 = =1. = 2 and N = 10 2 =5+ 10 = 5.M N where.. where.8.8 10 10 . where.e.ni„g Xf W. = If\D\ N X/IDI = 14.48 Marks Alternatively: (Short-cut Method) apply the follo„i°. we get M.D. A = 5. 6 Calculated value. It is based on all the items of the series. value . The strongest objection against mean direction is that while ■ calculating its value we take the absolute value of the deviations about an averap and ignore the ± signs of the deviation. It provides a better measure for comparison about the formation of different distributions. Therefore this method is nonalgebraic.1 6. Merits and Demerits and Uses of Mean Deviation Ments : -d i. 2. The step of ignormg the signs of the deviation is mathematically unsound and illogical.< 4. Thus mean deviation is a better measure of dispersion than range and quartile deviation. . It is not affected very much by value ot extreme items..M. Demerits : uui 1 Ignoring the signs.se and u spite statist deviat cycles.D. The averaging of absolute deviations for an average takes out the irregularities in the distribution and thus mean deviation provides an accurate and true measure of dispersion. 4. It is a calculated value based on the deviations about an average.^ 3.1 261 Measures of Dispersion 3 Based on all items.ec. for this reason it is not in further statistical calculations.48 Marks Note : Take care that assumed mean is close to the true mean. Less affected by extreme values.s P.J 5. = 1. studiei (B) SI 1. Mean deviation is not based on limits like range and quartile deviation. 5 Absolute measure. hence it is affected by ■ every value of the distribution. 3. Mean deviation calculated from various averages will not be the same. it has found favour with economists and business statisticians because of its simplicity. accuracy and also on account of the fact that standard deviation gives greater importance to deviations of extreme values. Uses. median and mode).2 Not well defined. Relation between Measures of Dispersion 6. For fj^^^sting ^u^s cycles. Calculation of Standard Deviation 3. Meaning 2. Despite so many demerits. 3 Harder calculations. properties laid rr " mathematically illogical as in its . this measure has been found useful than others. Meaning It is^e mnf^' Standard deviation was introduced by Karl Pearsons in the year 1893 It i^he most commonly used measure of dispersion. Cannot be calculated. Mean deviation involves harder calculation than the range ' and quartile deviation. (B) Standard Deviation 1. Merits and Demerits of Standard Deviation 262 ■ ' Statistics for Economics-XI -1. Its calculation by an arbitrary origin makes the calculation tedious. 11■•u 4. Other Measures from Standard Deviation 4. It satisfies most of the properties iLd down for an ideal measure of dispersion. Mathematical Properties of Standard Deviation 5. Mean deviation is not a well-defined measure since it is calculated ■ from different averages (mean. In spite of its mathematical drawbacks. It is also good for small sample studies where elaborate statistical analysis is not required. Mean deviation cannot be computed for distribution with open-end classes. mean deviation is not a totally useless measure. 36. 48.nnT!?"'.calculation signs are ignored and absolute deviations are taken. 70. This drawback is removed m die calculation of standard deviation. Calculation of Standard Deviation (A) Series of Individual Observations (B) Discrete Series (C) Continuous Series (A) Series of Individual Observations Standard deviation may be calculated by any of following methods • («) Actual mean method (b) Direct method (c) Assumed mean method (a) Actual Mean Method Illustration 16. deviation. 60 Measures of Dispersion Solution. first the arithmetic average is calculated and the deviations of various items from the arithmetic average are square! ^e ql^d deviations are totalled and the sum is divided by the number of iteL tL sqnaTrrot Symbolically. 45. 42. Calculate Standard Deviation of the following data • 25 50. Calculation of Standard Deviation Steps : . One of the easiest ways of dXg a way devil"'me^r^^^^ ^^ ^^^^^ ^^^ ^^^ Standard deviation is also known as root mean square deviation because it is the square root of the means of squared deviations from le arithmetic La„ . 34. 30. where <y X-X = * - 2. 2. Obtain deviations of the values from the mean. i. = (X. Now we get.e. 5.X). 3.. 4. calculate (X . Calculate the actual mean of the observations. Denote these deviations by X. Square the deviations and obtaiti the total Ix^.X) X= i:X 440 = 44 N 10 Ix^ = 1710. N = 10 jrm = iir ^ \ 10 Values X X-X * 25 -19 361 50 +6 36 45 +1 1 30 -14 196 70 -26 676 42 -2 4 36 -8 64 48 +4 16 .1. Here. Apply the following formula : 263 Here. Divide Ix^ by number of observations and find out the square root. 3.34 -10 100 60 +16 256 i:X=r440 1x^4=1710 = jl7i = 13.076 (b) Direct Method Illustration 17. Vaiues X 25 625 50 2500 45 2025 30 900 70 4900 42 1764 36 1296 . Obtain the sum of square of values. 2. Solution. Calculation of Standard Deviation Steps 1.N. Apply the following formula : = i-w ■i {Xf .Calculate the standard deviation of data given in Illustration 16 by direct method. 48 2304 34 1156 60 3600 . Calculate the actual mean of observations. 2:X = 440 IX^ = 21070 264 Now we get. this method is used to simphfy the calculations. 2. Square the deviations and denote the total LiP 3. N = 10 and X = 44 a= '21070 -(44)2 N "" " V 10 = V2107-1936 = n/171 = 13. Apply the following formula : a= d = X-A N Here. Denote these deviations by d and make the total of deviations. Statistics for Eopnomics-Xl X= EX 440 = 44 N 10 Here. Calculate the deviations of the observations from an assumed mean (X . ZX' = 21070. Calculation of Standard Deviation Steps 1.A). 25 -20 400 . When the mean is in fraction.076 (c) Assumed Mean Mediod Solution. = 6840 N = 10 . find standard deviation of x and y variables : Ix = 235. Here. Ey = 250 Ix^ = 6750.10 a= 1720 f-lO^ 10 10 = VI72-(-1)2 = Vm = 13. N = 10.076 Measures of Dispersion 265 Illustration 19. a= /N N 1720.50 +5 25 45 0 0 30 -15 225 70- +25 625 42 -3 9 36 -9 81 48 +3 9 34 -11 121 60 +15 225 N = 10 Id Id' = -10 = 1720 Now we get. From the following information. ZJ = . /X 4 6 24 -3.94 3.25 = 22.6436 •LfX = 706 266 jj^p^ _ ^tMistics for Economics-XI 43.079 ^684-625 = V59 = 7.8836 17.2180 Ifx'.= 237.8540 7 28 196 -0.3636 56.06 0.6904 10* 5■ 50 . Calculation of Standard Deviation Size Frequency f .06 4.1236 16.75 = 11.7636 52. +2.1816 5 12 60 -2.6720 9 14 126 +1.^675-(23.5)2 = ^/675-552.2436 50. = V684-(25)-= (B) Discrete Series Standard deviation can be calculated by any of the following methods : (a) Actual mean method (b) Direct method (c) Assumed mean method (d) Step deviation method (a) Actual Mean Method Illustration 20.06 9. ::VX ox = yN v m) 6750 10 ay = = \ N InJ lio J 6840 p50Y 10 I 10 J .Solution.X fx^ 8.6400 .0036 0.94 N = 100 X.9232 6 15 90 -1.94 0.06 1. Calculate Standard deviation of the following data : :4 5 6 Frequency : 6 7 8 9 10 12 15 28 20 14 5 Solution.68 .1008 8 20 160 +0. N X= ^fX = 706 and N = 100 706 X= Applying formula. we get 100 = 7. N Zfx'^ = 237. Apply the following formula : a= N IfX Here.64 and N = 100 [237J4 r_ = 1.ake 3.(x'ltrr„ (X -X) and denote these deviations by .06 c= Here. Calculation of Standard Deviation (c) ass X f X-2 .on „ by Solution.r A.by Ae respecve frequences and n.541 {b) Direct Method direar^od" of 'he data given in I„us„at. Z/X^ = 5222.e..e. Calculate mean of the series. Calculate the square of values (X^) 4..45678 64 81 100 -9 10 6 12 15 28 20 14 . Z/X. Here. 3.. Obtain the sum after mukiplying f and X (frequency and size). ZfX" 5. X • 2. i. Apply the following formula : 267 a= Z/X^ N -(X)' \ Ifx^ fz/x^ N J N m 7. i.e. Multiply frequency (/) to X^ and get the total.06 N 100 Now we get. i. 5 24 60 90 196 160 126 50 16 25 36 49 —__ ^ _ 96 300 540 1372 1280 1134 fDO i N = 100 IfX = 706 ^X^^"5222 f Measures of Dispersion Steps : 1. . N = 100 Substituting the values a= IfX" N -(Xf = 5221 . ario„s of s.A) W fXfdf N . Calculate the standard deviation of the data given in Illustration 20 by assumed mean method. Solution.-.3764 = 1-541 .100 -(7. Multiply these deviations by the respective frequencies and calculate the total Zfd where.541 (c) Assumed Mean Method Illustration 22. Now. Standard Deviation = 1.8436 = ^2. Calculation of Standard Deviation Size Frequency Y-7 4 6 -3 -18 54 5 12 -2 -24 48 6 15 -1 -15 15 7 28 0 0 0 8 20 +1 +20 20 9 14 +2 +28 56 10 5 +3 +15 45 N = 100 ■ ^ ' ' —•——^— E/a = 6 Ifd^ = 238 268 Statistics for Economics-XI Steps : r .22-49.06)2 = V52.ze from an assnmed mean and denote these delations 2. Take the dev. we get d= {X. 0^ = V2.3764 jlOO = a/^38-0. Calculate Arithmet. 100 Here. Yfd = 6.c Mean and Standard Deviation for the following Value Frequency Solution. _ Substituting the values.m = 238. Ifd N 0= 238 100 = A/2. 140 1 145 4 150 15 155 30 160 36 165 24 170 175 2 Values ^i^l^l^io^ A^^ Deviation 140 145 150 155 160 165 170 175 .0036 a = 1.541 and — « -- id) Step Deviation Method ^ Jllustradon 23.38-(0. d'. Take the deviations of values from an assumed mean and denote these deviations by (d). i. Zfd'. 3.. multiply these squared deviations by respective frequencies (in other words fd' x d' = fd'-) and obtain the total Zfd'-. Divide these deviations by common factor and obtain step deviations. . Calculate the squares of the step deviations (J"). 2.e.Frequency 1 4 15 30 36 24 N = 120 X ~ iss d -10 -5 0 +5 +10 + 15 +20 d' -3 -2 -1 0 +1 +2 +3 +4 fd' -3 -8 -15 0 +36 +48 +24 +8 m' = 90 fd 9 16 15 0 36 96 72 __32 = 276~ 269 Measures of Dispersion Steps : 1. Multiply step deviations by the respective frequencies and calculate the total 4. 75 X 5 a= Xfd'^ f-Lfd' N N C Here.75 X 5 = 155 + 0. Tfd' = 90. Now. N ^ 120 and C = 5 . a= d' = VN IN C C and C = Common factor xC N A = Assumed mean = 155. Ifd' 9^ = 5.75 = 158.5. we get Arithmetic Mean Here.= 276. N = 120 90 X = 155 + Standard Deviation 120 = 155 + 3. Tfd'. Apply the following formula : where. 75)2 X 5 = /2.ion : No.X) 24.7375 X 5 = 1.he Mean and S«ndard deviation from . Statistics for Economics-XI calXr«t jalT"^ * formula .he following dis.rihu.3-(0.0= (27 il20' f 90 ^ 120 X5 = V2.s used . of Students .59 (C) Continuous series For calculating standard deviation in continuous series any of the following methods may be applied : {a) Actual mean method (b) Direct method (c) Assumed mean method (d) Step deviation method 270 {a) Actual Mean Method j„ . Find .o VN where.318 X 5 = 6. Calcularion of Mean and Standard Deviation Marks X 4-8 8-12 12-16 No. of Students if) 11 .5625 x 5 = V1. x = {X . 4 V « 2 1 Solution.3-0. ^ .x).N= IS Midpoints (m) 1 6 10 14 fm 8 48 20 14 Ifm = 90 (m ~ X) X -A 0 +4 +8 16 0 16 64 fx^ 64 0 32 64 Ifx'. . Calculate the actual mean of the series.N ~15 = 6 marks S/w = 90. x -d-pomts from the mean. N = 15 Measures of Dispersion Standard Deviation 271 .= 160 Steps : 1. Denote these ~* the respective frequencies and ohtam Se ^Sr dS ^^ ^^^ ^^-re root to calculate the Apply the following formula : a= N Mean Here. Calculation of Standard deviation Marks No. N = 15 0 = J^ = M666 = 3. Zfx^ = 160.e. Apply the following formula : 0= . Solution. Obtain the total after multiplying f and m.-.= 700 Steps : 1.. Multiply frequency to m^ and get the sum. 4.265 marks. Calculate the standard deviation of the data given in Illustration 24 by direct method. i. Zfm^. Calculate actual mean of the Series X. 5. Standard Deviation = 3.265 . i.Mid-points X (f) (m) fm w' 0-4 4 1 8 4 16 4-8 8 6 48 36 288 8-12 2 10 20 100 200 12-16 1 14 14 196 196 N = 15 ■Lfm = 90 fm^ E/m. Calculate square of mid-points.e. Zfrn. This method is rarely applied in practice becausc in case the actual mean is in fraction.e. m^. 3..a= If-' N Here. 2. Note.. (b) Direct Method Illustration 25. i. the calculations becomes complicated and take lot of time. of Students --------. = 700 2-^2 Substituting the values Statistics for Economics-XI 15 9o^ 15 = ^ yfiOM ^ 3. x^ cc fd^ No. Zfrn'.5 32.5 47. Multiply these devattons by the respective frequencies and calculate the total.5 d fd 45-50 40-45 35-40 30-35 25-30 22 29 31 47 51 70 52. of Labourers Solution. m\ .5 +10 +5 0 -5 -10 -15 220 145 0 -235 -510 -1050 2200 725 0 1175 5100 15750 N = 250 Svwt tr'"" - Ifd = -1430 Ifd^ = 24950 and denote these 2. Zfrn = 90.265 Standard Deviation = 3. of labourers Jf>Mid-points (m) m -.5 27. N = 15.42.625 marks.5 37. Calculation of Standard Deviation Age in years .5 42. .i Ifm^ (Ifm N N -{Xf = V N • ■ We get. (c) Assumed Mean Method d J!"" ^^^ I^-ation of the followmg frequency 50-55 45 50 40^5 35-40 30-35 25-30 29 31 47 51 70 Age in Years No. Find out the Standard Deviation of the frequency distribution given in Illustration 26 by step deviation method. = 24950. Apply the following formula : i a= where.19 Standard Deviation = 8. d= (X-A) .4. Illustration 27. a= 24950 250 r-1430 = ^99.19 years (d) Step Deviation Method This method is mostly used in practice. Zfd = -1430.718 = V67.8-(-5.72)2 250 = V99.N Measures of Dispersion Now we get. 273 a= l-Lfd^ f-Lfd^ . N = 250 Substituting tbe values.082 = 8. N VN ) Here. Solution.8-32. Calculation of Standard Deviation . 5 -10 -2 -102 -204 25-30 70 27. Jlfd'.5 -5 -1 -A7 47 30-35 51 32. Calculate the squares of the step deviations {d'^Y.Age in years X id' No.4. multiply these squared deviations by respective frequencies (in other words fd' x d' = fd'-) and obtain the total I^fd'-. Divide these deviations by common factor and obtain step deviations.5 +10 +2 +44 88 45-50 29 47. 274 5.4. of labourers f fd" 50-55 22 52.5 0 0 0 0 35-40 47 37. and a= d' = XC IC C = Common factor Now ..5 +5 +1 +29 29 40-45 31 42. 2.286 m. d'. 3. Apply the following formula : Statistics for Economics-XI where.25 5 d' lA'^ = 998 Steps 1.5 -15 -3 -210 630 N = 250 Midpoints m m . Multiply step deviations by the respective frequencies and calculate the total 4. Take the deviations of mid-points from an assumed mean and denote these deviations by d.25 4 Ifd'= . i.e. Illustration 28. ^fj^ ^ _ N = 250 and C = 5 Substituting the values. ^fjn = 993. a= Ifd r2 N Zfd' {N \2 XC where.5 5 23 65 92 100 Solution.5-65.5 Less than 65.5-62.we get.65 = 27 100 .5 65.23 = 42 92 .5 Less than 68.19 years.5 Less than 71.992-(-1. Find the Standard Deviation of the height of 100 students.5 5 23-5 =18 65 . Convert cumulative frequency into class interval. Less than 62.5-71.5 62.309) X 5 = VI^ x 5 = 1.5 71.19 .5-74.5 Less than 74. a= 998 250 286^ I 250 5 = 73.92 = 8 61 64 67 70 73 -2 -1 0 +1 +2 -10 -18 0 +27 +16 20 18 0 27 32 N = 100 Ifd' = +15 m'^ = 97 . Standard Deviation = 8.-.638 X 5 = 8.5 68.144)2 X 5 = V3-992-1. Calculation of Standard Deviation Height (in inches) X Frequency Midpoints im) ni-67 3 (d-) fd' fd-" 59.5-68. Zfd' = 15. = 2. = 97. o= 97 100 100.92 inches. N = 100.Measures of Dispersion Applying the formula. X 3 = VO.9733 x 3 = 2.0225 x 3 = ^/0.9475 X 3 = 0. we get 275 a= < Ifd'^ flfd'^ N N XC where.97-0. Calculate Mean Standard Deviation and mean deviation about mean Marks Students More than 20 50 More than 40 47 More than 80 41 More than 100 21 More than 120 9 . C = 3 Substituting the values.92 Standard Deviation. Illustration 29. Statistics for Economics-XI - 24 ^ " ^^ ^ X 10 = 90 + 4.^. A 90. we get X-A.Solution.8 = 94.47 = 3 30 -6 -18 108 40-80 47 .8 Marks Standard Deviation Applying the formula. IN wj-yo 10 (dl fd' -Lfd'^ = 354 fd^ .8 Mean = 94. we get a= N where.21 = 20 90 0 0 0 100-120 21 . (Convert cumulative frequency into class interval) Calculation of Mean and Standard Deviation Marks X Frequency (f) Midpoints (m) 20-40 50 .41 = 6 60 -3 -18 54 80-100 41 .9 = 12 110 +2 24 48 120-140 9-0 = 9 +4 36 144 1• 130 N = 50 Ifd' = 24 Mean : Applying formula. -Lfd' = 24. C = 10 276 j„ .C where. N = 50. 968 ••• Mean Deviation = 19. = where.4 M-D.N = 50.8 34.8 15.4 and N = 50 \A r. ~ 94. 998.= = 19.XC = 354.968 . N If IDI = 998.8 4.2 35. we get 30 60 90 110 130 ■■m.17 Standard Deviation = 26.D. Zfd' = 24.8 ID! 64.2 M. C = 10 a= SO v50y xlO = V^xlO = 2.617 X 10 = 26.17 Marks Deviation from Mean Marks X 20-40 40-80 80-100 100-120 120-140 Frequency (ft 3 6 20 12 9 Mid-points m N = 50 Mean Deviation Applying the formula. 4 50 998./"IDI 194.4 3. X = 94.D.8)(8) 960 + 38.4 208. < mea (b Measures of Dispersion 111 Let us try the same question by assumed mean method (Assumed Mean = 90) t.8.8 2/IDI = 998.. IM = 960.1/B = 3 + 6 + 20 = 29 and S/A = 12 + 9 = 21 960 + (94. = I/IJI+(X-A)(I/B-IM) N where. we get M.8 96 182.90/lai 20-A0 3 30 60 180 40-80 6 60 30 180 80-100 20 90 0 0 100-120 12 110 20 240 120-140 9 130 40 360 \d\ = 960 Applying formula. A = 90..968 . Marks ■ ■ ■■ f h^ \d\ m m .4 50 50 = 19.4 316.8-90)(29-21) 50 960 + (4. This relative measurement is called by dividing standard deviation by arithmetic mean of the data Symbolically. = ^ X 100 Here.968 3. consistency and variability in two different series. C.V. The only difference between the two measurements is that the variance is the average . The series having greater coefficient of variation. Symbolically. Some of the important measures are as under : (a) Coefficient of Standard Deviation : A relative measures of standard deviation is calculated to compare the variability in two or more than two series which is called 'coefficient of standard deviation'. it is said to be more uniform. it has higher degree of variability). Statistics for Economics-XI consistent or less stable (in other words. It shows the relationship between the standard deviation and the arithmetic mean expressed in terms of percentage. the series having lesser coefficient of variation. more homogeneous. Coefficients of S. (in other words. = ^ X Here. less homogeneous. a = Standard deviation and X = Arithmetic mean. (c) Variance : Variance is the square of standard deviation. C. less m 278 j„ . it has less degree of variability). This measure is used to compare uniformity. In the same way. more consistent or more stable.D.Mean Deviation = 19. a = Standard deviation and X = Arithmetic mean (b) Coefficient of Variation : This relative measurement is developed by Karl Pearson and is most popularly used to measure relative variation of two or more than two series. Standard deviation and variance are measures of variability and they are closely related.V. it is called to be less uniform. Other Measures from Standard Deviation Various measures are calculated from standard deviation. = Coefficient of Variation. squared deviation from mean and standard deviation is the square root of variance. Variance = a^ and Standard Deviation = VVariance Calculation of Variance In Series of Individual Observation : Variance (a^) = N ^ nx-xf N Here. and C = Common factor Individual Observations Illustration 30.X In Frequency Distribution : Variance (a^) = flfd'^ 2' N [NJ Xa Here. The choice is between X and Y on the basis of their five previous scores which are : ■X : 25 85 40 80 120 y■ 50 70 65 45 80 [a) Calculate coefficient of standard deviation. x = X . A batsman is to be selected for a cricket team. [b) Which batsman should be selected if we want (/) a higher run scorer (ii) a more reliable batsman in the team. d' = . Symbolically. variance and coefficient of variation. . N = 5 350 X = = 70 Average score = 70 Runs Standard Deviation a= N Ix' = 5750 and N = 5 a= 5750 = V1150 = 33.D.Measures of Dispersion Solution.91 and X = 70 33. = ^ Coeff. of S. = a = 33.484 Variance E(X-X)2 Ijc^ N .91 70 = 0.D. (a) Batsman X Arithmetic Mean JX X= N EX = 350.91 Runs Coefficient of Standard Deviation Coefficient of S. 88 Runs Coefficient of S.D. = ^ 279 Batsman Scores (X-X)^Batsman X-X Scores X * 25 -45 2025 50 -12 ■ 144 85 +15 225 70 +8 64 40 -30 900 65 +3 9 80 +10 100 45 -17 289 120 +50 2500 80 +18 324 EX = 350 (y-Yf X.Y Y Ex^ = 5750 EY = 310 = 830 .N I(X -XV = = 5750 Batsman Y Y= N EY = 310. N = 5 310 Y= = 62 Average score = 62 Runs a= N Ey2 z: 830 and N = 5 oy = 12. 91 C.44% Here. y .y ^ ^ .Coeff.V = X 100 = 48. = a = 12.D.91 and X = 70 33. = 12.ax c.v.88 and Y = 62 „„ 12.= —= 166 Runs C.88 C-V. .V. than Ae't^ii tt'o^y ST^r. X . .88 and Y = 62 12.207 _ E(Y-Y)2 ^^ ^ NN E (Y.^ = ^ X 100 (J.^ =• Y 100 o = 33. 830 ay.77% ib) H) Batsman X should be selected as a (70 ™„S.X = x 5750 Statistics for Economics-XI ax^ = =1150 Runs Coefficient of Variation „ . = X 100 = 20.Yf = ly = 830 280 Here.88 62 = 0. of S. .1 '""IT of Discrete Series """ * = 48. ^^ Jl-ustration M. Size 4 5 6 7 89 10 Frequency 3 7 22 60 85 32 8 Size X 4 5 6 7 89 10 of Variation Frequency — 3 . Solution.44%. Calculate variance and coefficient of variation from the foHowin. Ifd^ ri/iif Variance (o^) = — .J Here.7 22 60 85 32 8 ~NT217~ X~7 d -3 -1 -1 0 +1 +2 +3 fd -9 -14 -22 0 +85 +64 +24 m = 128 Cot estir fd' 17 28 22 0 85 128 71 a an Measures of Dispersion Variance .59 = 7.0.668 .347 = 1. _ o = VVariance = >/02 = 1. l.668 .32 Coefficient of Variation C.589)2 = 1.fd = 128 and N = 217 362 YmY Variance (a^) = ^ (^217. ^ = 362. = 1.15 ^ 281 + 0.V. = I X 100 Let us calculate a and X.(0.59 . .. V = ^ X 100 = 0. To check the quality of two bulbs and their life in burmng hours was Life (in hrs. we get Here.= ^ X 100 Let us calculate first a and X.N 217 Applying the formula.) Ho. of bulbs Brand A 0-50 15 Brands 2 50-100 20 8 100-150 18 60 150-200 25 25 200-250 22 5 Total 100 100 (i) Which brand gives higher life? (it) Which brand is more dependable? 282 Solution. nomics-Xl Brand A Coefficient of Variation (Brand A^ C-V.1515 X 100 = 15. Standard Deviation .V = I X 100 o = 1.59 C. C. Statistics for Eco.15% Continuous Series itmuous series niustration 32.15 and X = 7. 93-(0.Z/-^-19.376 X 50 = 68.8 hrs.MLjlfd" Xc Here. Arithmetic Mean X=A+ N XC .8939x50 = 1.93-0.19)2 Vl. x50 Measures of Dispersion 283 = Vl.N=100andC=50 CT = /m 1100 '19_ 100 X 50 Co^ffidem ofJS^ation~(BrLd BJ = X 100 Let us calculate first a and X. Standard Deviation a= l^jEfd'^ Xc a= 100 .Z/^.= 193.0361x50 = Vl.100. 5 hrs. Applying the formula. Ifd' = 23. = ^ X 100 . now we get where. a = 68.61-(0.5 = 136.Here.8 and X = 134.23) X 50 = 125 + 11. X = 125 + X 50 Arithmetic Mean Ifd' X=A+ XC N Here.5 hrs. now we get C.32 hrs. A = 125.23)2 .A.V0.15%. Applying the formula. where.5 = 134.5 68 8 C. A = 125.V = X 100 = 51.5 X = 125 + X 50 100 = 125 + (0.V.61-0.h!= 100 and C = 50 19 = Vo.5571x50 0. 100 and C = 50 23 100 = 125 + (0.0529x50 >/0.746 X 50 = 37. I^d' = l9. 134.19) X 50 = 125 + 9. . V. the wages of an employee are wrongly noted as Rs 120 instead of Rs 100.5 X 100 = 27.32 and X = 136. Illustration 33. = |.V.5 hrs).15%).A. (/■) Since the average life of bulbs of brand B (136.5 hrs) is greater than that of brand A (134. of bulbs of brand B (27. = 136. The number of employees.5 37. (ii) Since C. wages per employee and the variance of wages per employee for two factories are given below : No. X 100 Here. therefore the bulbs of brand B give a higher life.34%) is less than that of brand A (51. therefore the bulbs of B are more dependable.V. of Employees Average wage per employee per day (Rs) Variance of wages per employee per day (Rs) (a) In which factory is there greater variation in the distribution of wages per employee? (b) Suppose in factory B.34%. a = 37. = ^ X 100 .C.V. What would be the corrected variance for factory B? Factory A 50 100 120 85 9 16 Factory B 284 Solution. («) Calculation of Coefficient of Variation : Factory A Statistics for Economics-XI C.32 C. x = 120 and a = . = ^^00 + 722500 = 724100 .= il? 100 = 2. = J X 100 Here X = 8S and a = ^ C-V. = ^ X 100 = 4.120 .V.^ = 100 and X= 85 100 x 85 = 8500 It IS not correct ZX Corrected ZX= 8500 .7% W &rrecti„8 Mean and Variation : — zx For Factory B . 100 = Rs 8480 Corrected X = ^ = ^ ^ N Variance = a^ a^ = Here.5% Factory B C. X = 85 and JV = 100 ZX^ 16 = 100 (85)2 T. N 100 .(xp = 16. 04 = Rs 5.<84. and XX^ = 1090 Coefficient of variation (C. The sum of 10 values is 100 and the sum of their squares is 1090.(120)2 .V.(Corrected X)^ 719700 ~ ~Tdr .It IS not correct ZX^ /^^lUU Corrected ZX^ = 724100 .) = ^ x 100 y\. Solution. (100)^ = 724100 . Apply the following formula to get Mean (X) ZX ^=N 10 X = 100 100 10 X= NX = IX = 10 Apply the following formula to obtain standard deviation (a) by direct method.7191.96 Measures of Dispersion 285 Illustration 34. We are given.14400 + 100000 = 719700 Corrected Variance = ^o^reaed ZX^ . Find the coefficient of variation. (X)2 02 = . N = 10.8)2 bet = 7197 . IX = 100. = ^ X 100 Given : X = 800 a = 100 100 800 X 100 = 12.100 = 9 = V9 =3 Therefore. 286 j„ . = 30 Illustration 35. X = 10 and C.X 100 = 30 Thus. The means and standard deviations of two brands of bulbs are given below: Brand I Brand II Mean 800 hours 770 hours Standard deviation 100 hours 60 hours Calculate a measure of relative dispersion for two brands and interpret the result.V.V.V. C.5% Given Brand II C.N 1090 - (X)^ - (10)2 10 = 109 . Statistics for Economics-XI Brand I C. = — X 100 = .V. = ^ X 100 . Mathematical Properties of Standard Deviation Standard deviation has the following important mathematical properties ■ . median and mode are identical.e. But the sum of deviations calculated from Median (ignoring ± signs) is always less than the sum of deviations calculated from mean (ignoring ± signs). . a large proportion of distributions are concentrated around mean. which is used to calculate mean deviation. Mean ± 3 ct covers 99. which is used to calculate standard deviation. i.45% of the total items.11 5 =3 Here. Mean ± 2 a covers 95. Symbolically. Mean ± 1 a covers 68.X. Following are a relationship {i. the bnlbs of brand H are more consistem as compared to brand L 4. deviations taken from Mean..79 % Hence. " ^ ~ ^^ deviations taken from any value Measures of Dispersion 287 less than the sum of the squares of deviations calculated from any other value. Symbolically.e.27% of the total items.X = 770 a = 60 60 770 X 100 = 7. w/ ..73 % of the total items. range of spread of items) can be determined on the basis of mean and standard deviation. x-X. ElX-Mel < ZIX-XI (b) Standard Deviation and Normal Curve : In a normal or symmetrical distribution apart from mean. This can be observed in the following frequency curve : PERCENTAGE OF ITEMS INCLUDED UNDER NORMAL CURVE Illustration 36. Calculate the percentage of cases lying within X + 1 a, X ±1(5, X. ± 3 a from the following data : Size :12345678 9 10 Frequency : 8 12 10 28 16 12 10 2 02 288 Solution. Statistics for Economics-xA Calculation of Mean and Standard Deviation : Mean : Here, N A = 5, I.fd = - 68 and N = 100 -68 X=5+ Standard Deviation 100 = 4.32 CT = Here, ^ 434, Zfd = -68 and N = 100 0= /424 100 -68> —00 I____ liwj = V4.24-(0.68)2 Calculation (a) Cases lying ib) Cases lying (c) X = A ■ ^^^ There is no - y/4.24-0A624 = VIT^ = 1.943 of percentage of cases : X ±10 = 4.32 ± 1.94 = 6.26 and 2.38 between 3 and 6 are (10 . 28 . 16 . 12) = 66 out of 100, 66"/o X ± 2a = 4.32 ± 2 x 1.94 = 4.32 ± 3.88 = 8.20 and 0 44 between 1 and 8 are (100 2) 98 out of 100, i.e., 98% X ±3o = 4.32 ± 3 X 1.94 = 4.32 ± 5.82 = -1.5 and 10.14 negative value. All the cases lie between 0 to 10, /.e., 100%. Measures of Dispersion 289 (c) Combined Standard Deviation : Just as combined arithmetic mean can be calculated, if means and number of items in different groups are given, similarly combined standard deviation can be calculated, if standard deviation means and number of items in different groups are given. Combined standard deviation is obtained as follows : {a) Two related groups : iNiaf + N2O2 + Njd} + Nidj =^ N1+N2 Here, CTjj = Combined standard deviation of two groups CTj = Standard deviation of first group a^ = Standard deviation of second group Xj 2 = Combined arithmetic Mean of two groups (b) Three related groups : '1.2,3 V NjO^ + N^al + Njof + N^d^ + Nzcff + N^dj N1+N2+N3 Here, d^= (X-X 1,2,3), d^ = (X2 -Xi,2.3) and d^ = {X, -X,^^,^) The above formula can be extended to calculate the combined standard deviation of even more groups. Illustration 37. In sample A, N = 150, X = 120 and S.D. = 20; in sample B, N = 75, X = 126 and S.D. = 22. Calculate Combined Mean and Combined Standard Deviation. Solution. Combined Mean : V N1X1+N2X2 - N1 + N2 150x120 + 75x126 18000 + 9450 27450 150 + 75 Combined Standard Deviation : 225 225 = 122 Niol + N2al + N^d^ + N2 . N1+N2 d^ = (X,-Xi,2) = 120 - 122 = -2 d^ = (X2-Xi,2) = 126 - 122 = 4 -i 150 (20)2 ^ 75(22)2 ^ _ (_2)2 ^ 75(4^2 150 + 75 290 j60odoT363ddT6^^ * j„ . 150 + 75 Statistics for Economics-XI [981^ -~ V~22r " = 20.88 Thus, Combined Mean = :22 and Combined Standard Devation = 20.88 follow^ A™'ISirs.''' """ Distributions N A 20 ained by combming ,he ^ 120 C 60 Solution. Combined Arithmetic Mean : 60 50 40 5.D. 8 20 12 H00 + 6000^h2^ _ 9600 200 - ^ = 48 Combined Standard Deviation : 20 + 120 + 60 ^1.2,3 = Here, and I nY+N^J^ = ~ ^1,2,3) = 60 - 48 = 12 = (^2 -X12 3) = 50 - 48 = 2 = (X3 -X12 3) = 40 - 48 = -8 = ' 20 + 120^^60 * 200 [65120 ,_ ~ = ^^25.6 = 18.04 Thus, Combined Mean = 48 Combined Standard Deviation = 18.04 291 measures of Dispersion \d) Change of origin and change of scale : Any constant added or ^"^ttacted (change of origin), then standard deviation of original data and of changed data after addition or Isuhtraction will not change but the mean of new data will change." Any constant multiplied or divided (change of scale), then mean, standard deviation and,variance will change of the new changed data. Illustration 39. Average daily wage of 50 workers of a factory was Rs 200 with standard deviation of Rs 40. Each worker is given a rise of Rs 20. (,) What is the new average daily wage and standard deviation? (ii) Have the wages become more or less uniform? (Hi) If each worker is given a hike of 10% in wages, how are the Mean and Standard Deviation values affected? Solution. We are given N = 50, X = 200, a = 40 (i) Change of Origin Old Series Since, X= IX N NX =IX 50 X 200 = 10,000 Mean X= N 10,000 50 = Rs 200 Standard Deviation a= N ■-(X)2 Suppose each worker is paid Rs 200. X X^ = (200)^ X 50 workers = 40,000 X 50 = 2,00,0000 a= '20,00,000 50 = ^40,000-40,000 = Ji Thus, Standard Deviation (a) and variance (o') = 1. _ New Series Rise of Rs 20 to each worker to get new series 20 X 50 workers = Rs 1,000 New XX = 10,000 + 1,000 = Rs 11,000 New Mean EX X= N 11,000 " 50 Standard Deviation = Rs 220 a= N Each worker given a rise of Rs 20, i.e., 200 + 20 = Rs 220 XX^ = (220)^ X 50 workers 48,400 X 50 = 24,20,000 a= ^4,20,000 50 -(220)2 = ^48,400-48,400 Thus, Standard Deviation (a) and variance «y') = 1. _ 292 « New . new .ancUrd aeva.on . devanon remain rhe same as rhat ofold series w^ are required .„ ca.cuia,e eoefficienr „, variation ro decide *e uniform.. Old Scries Coefficient of Variation C.V. = ^xlOO - 200 = 0.5% X 100 New Seri^ Coefficient of Variation C.V. = |xlOO 220 = 0.45% X 100 orvaSl^^ro^i-ZCtrtr^ - -- coe^cienr (ni) If each woricer is given 20% h,ke Mean affected: Old + Increase in wages ■•• Rs 200 + Rs 20 = Rs 220 New Mean = Rs 220 - ..e. remain .he same as 5- Relation between JMeasures of Dispersion and -Viation, Mean Devation (a) Q.D. - ^ s.D. (more precisely 0.6745 S.D.) (b) M.D. _ ^ S.D. (more precisely 0.7979 S.D.) (c) Q.D. = I m.D. (more precisely 0.8453 m D ) (d) 6 S.D. = 9 Q.D. = 7.5 M.D. Further, in such distributions • (0 Arithmetic M " ! stnl^TT""''' ^e items. The above relationships of the items, ~ to moderately asymmerric'al for c^t^" *»« Vplied Measures of Dispersion 293 Merits and Demerits and Uses of Standard Deviation Standard deviation is the most satisfactory and widely used measure of dispersion ■ause of the following merits : ■ Merits 1. Based on every item. Unlike the range and location based measures of dispersion, the standard deviation makes use of all the observations in the set of series. That is, it includes every item of the distribution. 2. Correct mathematical process. The standard deviation is the easiest measure of dispersion to handle algebraically and it is the resuk of correct mathematical process. The deviations are calculated from arithmetic mean which is an ideal average. The deviations are squared, so that automatically become positive. Being used on correct mathematical process, it is amenable to further statistical analysis. 3. Rigidly defined. Standard deviation is a well-defined and definite measure of dispersion. It is rigidly defined and its value is always definite and based on all the observations and the actual signs of deviations are used. 4. Sampling fluctuations. Standard deviation is less affected by the fluctuations of sampling than most other measures of dispersion. 5. Mathematical Properties. It is amenable to algebraic treatment and possesses many mathematical properties. It is the only measure for calculating combined standard deviation of two or more groups. It is on account of the properties that standard deviation is used in many advanced studies. Demerits 1. Complex in calculation. Standard deviation is not easy to calculate, nor it is easily understood. In many cases it is more cumbersome in its calculation than either quartile deviation or mean deviation. 2. More weights to extreme items. It gives more weight to extreme items and less weight to those which are near to the mean, because the squares of the deviations which are big in size, would be proportionately greater than the squares of the deviations which are comparatively small. Thus, deviation 2 and 8 are ratio of 1 : 4 but their square, i.e., 4 and 64 would be in the ratio of 1 : 16. Howevei; since standard deviation gives greater weight to extreme items, it does not find much favour with economists and businessmen who are more interested in the results of the modal class. Uses Despite the drawback the standard deviation is the best measure of dispersion and ) uld be used whenever possible. It is widely used in statistics because it possesses most die characteristics of an ideal measure of dispersion, k is a significant measure for aking comparison between variability of two sets of observations to test the significance f various statistical measures of random samples, correlation and regression analysis etc. ' may regard standard deviation as the best and the most powerful measure of lion. 294 (between H-0 tn a u- u Statistics for Economics-Xl reutive measures of or variations Types of Measures of Dispersion/Variation Measure iatiori Range ^Inter-Quartile Range and Quartile Deviation Mean Deviation Standard Deviation 1 re of Variation/ nation Coefficient of Range ^ Coefficient of Quartile Deviation ^ Coefficient of Mean Deviation Coefficient of Standard Deviation (a) Absolute measure • Absol..^^ ^ . the ^mrfda™ ^ f ^^ ^^^ as the data, ^ees. If the data are in kg, the measiV^'e nV Jf " ^e in j d spersxon cannot be used to compare the Matter o ^ absolute St r —■ -- or the coefficient of the absoluteltaC If d ' "" »» P^tcentZ coefficent of dispersion or coeST „f « « caS c«fhcient of range, co^Bc^ro^^^uXtST' -asures ar and coefficient of standard deviation "oo deviation is ca.ed of mean deviation on standard Thus, C.V. - ® = f XIOO measured m the same of var abihty of two or more series wher "uiLts 5 ' " " obtained as percenta;« «- Measures of Dispersion 295 method (lorenz curve)" Lorenz Curve The graphic method of studying dispersion is known as the Lorenz Curve Method. It is named after Dr Max. O. Lorenz who used it for the first time to measure the distribution of weakh and income. Now k is also used for the study of the distribution of profits, wages, turnover etc. In this method of values the frequencies are cumulated and their percentage are calculated. These values are plotted on the graph and curve that is obtained IS called the Lorenz Curve. The greatest defect of this curve is that it does not give a quantitative measure of dispersion. Let us look at the following illustration. Illustration 40. Draw a Lorenz Curve from the following data : Income (In thousand Rs) No. of Persons in thousands Group A 20 10 16 40 20 14 60 40 10 100 50 6 180 80 4 Group B Solution. Income in (Rs) lative Income wm 20 40 60 100 180 _ 20 60 120 220 400 Cumu-laui/ePar^Mt tage sands) hers _____ No. of Pers6ns (stt thou5 15 30 55 100 10 20 40 50 80 Steps 1. The size of items (or if classes are given, then mid-points) are made cumulative. Considering last cumulative total as equal to 100 difference cumulative total are converted into percentages. 2. In the same way frequencies are made cumulative. Considering the last cumulative frequency item as equal to 100, all the different cumulative frequencies are converted into percentages. 296 *!■ r ■ !. of I Measures of Dispersion 297 Solution. as line Illustration 41.. according ro the allt tX" t'^ Profits earned in Rs '000 Area A 6 25 60 84 105 150 170 400 »- Area B 6 11 13 14 15 17 10 14 2 38 52 28 38 26 12 4 Sii inequa nil income After-ta No. 91 9.™^ " . we calculate percentage values as under : Profits (in thousand) CumuCumu- Area A Cumu- No of Cumu- Area B Cumu- No. ab.he fo|Wi„.1 13 30 30 52 92 46 84 175 17.Percentage of Income lorenz curve Curve 5 """ ^T " ot^^'ltSUl'"" ™ ^om 0 to 100 ^ line . Obtaining Lorenz Curve.0 15 59 59 38 158 79 Num.6 6 6 6 2 2 1 25 31 3. cent- .5 14 44 44 28 120 60 105 280 28. and .1 11 17 17 38 40 20 60 . of. of Cumu- lative lative Com.lative lative Profit Profit Profit panies Num.Per- Percen- ber tage age panies cent- ber age 6 6 0. After-m No.Per-.lative lative Com. fa . 5 25. it represents greater inequality in area B as compared to area A. and [c) object of an investigation. We are now m a position to make a there is more inequahty 299 Measures of Dispersion comparative study of these measures. ^t'^ti^tics for Economics-XI _^alculatioi^^ Values After-tax Income 0 . of households : 1348 4^9 » 1892 After-tax income : 10000-20000 20000-40000 No. holds Below 1000 Below 5000 Below 10000 Below 20000 Below 40000 IT 12.-----i^orenz Curve.13 79.150 430 43.0 50.20000 20000 .0 14 100 100 4 200 100 Since curve B is farthest from the line of equal distribution.0 10 86 86 12 196 98 400 1000 100. Illustration 42.1000 1000 .00 of measures of oispersion . 9400 Indian households are classified according to their after-tax income as follows : After-tax income : 0-1000 1000-5000 5000-10000 No.. It would help us in the selection of an appropriate measure of dispersion which depends on-(^) nature of data.5000 5000 .25 94.34 59. of houses holds Cumulative Number of households Cumulative % of house.0 17 76 76 26 184 92 170 600 60. (b) the purpose.10000 10000 . . T .0 1348 4210 1892 1460 490 1 1348 5558 7450 8910 9400 14.0 100. of households : 1460 490 Drawrh.40000 Cunmlatii'c Income r> f Cumulative Percentage of Income No.74 100. 4 Interpretation and application point of view : All the four measures of dispersion are easy to interpret. etc. instead it is calculated from lower and upper quartiles. weather forecasting. This is also easy method. However. in correlation and regression analysis. Quartile deviation is useful when influence of extreme items is minimised as in the study ot social problems.. standard deviation satisfies the most essentials of a goqd measure of dispersion. . Calculation point of view : Range is the easiest and simplest measure because it is the difference between two extreme items. in sampling and other areas of statistical analysis. Standard deviation possesses most of the good characteristics of a measure of dispersion Therefore. it is the most favoured and indispensable measure. quartile deviation and mean deviation. 2. Thus. (ii) It should be rigidly defined. it has precise value. Mean deviation and Standard deviation requires more calculations which are based on the deviations from average. i. in making comparison between variability of two or more sets of data. in testing the significance of random samples. (iii) It should be based on all the observations of a series. as it is not affected too much by the value of extreme items. They are based on every item of the distribution. These essentials are as under : (i) It should be simple to calculate and easy to understand. i.1.. Definite value point of view : All the four-methods of dispersion are rigidly defined and their values are definite.e. Range is highly affected by the extreme item.e. 3 Based on every item point of view : Range and quartile deviation are not based on " all the items of series while mean deviation and standard deviation makes use of all the items. 5 Algebraic treatment point of view : Standard deviation is the best measure of dispersion because of correct mathematical processes as compared to range. Range and quartile deviation are useful for general study of variability. Mean deviation is used by economists and busmess statisticians It is useful in forecasting business cycles and small sample studies. Quartile deviation is superior to the range. Lorenz Curve is a visual aid method but it does not give quantitative measure. It is widely used in statistics. Range is useful for quality control. etc. (v) It should not be affected by fluctuations in sampling. = 2 Relative Measures Coefficient of Quartile Deviation _ QlzQl Qs+Qi Qj = lower quartile Qj = upper quartile Absolute Measure Individual Observations M.D.(iv) It should not be greatly influenced by extreme items.S Q^rtile Deviation Relative Measures Coefficient of range L-S L+S L = largest item S = smallest item Absolute Measure Quartile range = Q^ _ q Semi-interquartile range or Quartile Deviation Q.D. = N . Statistics for Economics-XI of formulae Range Absolute Afeasure Range = L . 300 j„ . (vi) It should be usable for statistical calculations for further or higher order analysis. D.N Measures of Dispersion Relative Measure Coefficient of Mean Deviation Coeff.^Ji^ I £> I = Deviations from median Discrete and Continuous Series M. M. Mean or Median M.D.D. = zfjm N Short-cut (Assuiiied mean) Mediod M. = lll^hilzAmB-^ N Absolute Measure Individual Observation Actual Mean Method a= N Direct Method N ■ ■ y M Assumed Mean Method IN a= ' N ' d = X-A .-- N .D. .D. = M. = -E-X Relative Measure Coefficient of Variation a C.f-on or mean ignoring ± signs Coeff.X 100 301 Step Deviation Method a= d' = N fX~A {N XC Vc C = Common factor "CS Dirt a= a N IfX N . S.D.XorMe R< lative Measure ent nf Standard Dcv.V = . C = Common factor 1V t ■■ r ■■ N .2-(Xf = N 'N[N Assumed Mean Method a= 'N lfd I N \2 Step Deviation Method aIN[NJ Variance or a' = ^^X-Xf ^ N XC a' ^ J^] N N \ i\ J Xe d .——. Give formula for calculating it. Comment.? Do the range and quartile deviation measure dispersion about same value. Statistics for Economics-XI _ Niof + N^cl + + N^dl Here. What are the essentials of a good measure of dispersion.? 9.? 7. Jf + N^dl + N. Which measure of dispersion is the best and how. {b) Three related groups : . and d^ = ^3 -X^ 2.? 5. Define the first and third quartiles. {b) Name the most commonly used measure of relative dispersion.302 Combined Standard Deviation (a) Two related groups : j„ . Some measures of dispersion depend upon the spread of values whereas some calculate the variation of values from central value. A measure of dispersion is a good supplement to the central value in understanding frequency distribution. (a) What do you understand by mean deviation? .? Do you agree? 8.? 3.dl 3 Here. Illustrate the meaning of dispersion with examples. Explain how the quartiles are used to calculate dispersion values.. ^ 6.3 EXERCISES Questions : 1. {a) Name four commonly used measures of absolute dispersion. 4._ + N^ol + Njof + N. 2. Why should we measure dispersion. Why is standard deviation considered to be the most popular measure cf dispersion. . Make a comparative study of various measures. 14. What is meant by absolute and relative measure of dispersion? Briefly explain the concept of 'Lorenz Curve'. Write short notes on: . lo... Explain. Explain ^^ variability? What is the need of calculating a measure 17. 21. 'Coefficient of variation is a relative measure of dispersion'. 15. Define Lorenz Curve.qZt'd^LLT"^""" 10. What IS coefficient of variation? What purpose does it serve? Also distinguish between 'variance' and 'coefficient of variation'. In what way is standard deviation a better measure of dispersion than mean deviation? 13 What is Standard deviation? Explain the uses of standard deviation. . "The standard deviation of heights measured m inches will be larger than the standard deviation of heights measured in feet for the same group of individuals. 20. Otherwise give appropriate explanation of the statement given above. 19. 22. Define coefficient of variation? In what situation would you prefer this as a measure of dispersion. Comment on the validity of the above statement. 16. 303 Measures of Dispersion H2. . 5 -0. May +3. -0. 52.(a) Coefficient of Dispersion (b) Coefficient of Variation (c) Variance Standard Deviation (e) Quartile Deviation (f) Lorenz Curve Problems : Range 1.4 Aug. Marks (Section A) : 20 25 28 45 15 30 Marks (Section B) : 45 52 36 42 28 25 [Section A : Range = 30. Coefficient of Range = 0.1 -1.5 Section B : Range = 27. +1. 72. +2.4 ^ep.55J 2. 36.6 from normal (2002). of Workers :A B C D E F G H I ] Wages in (Rs) : 175 50 50 55 100 90 125 145 70 60 [Range = 125. A and Sec. Coefficient ot Range = 0. 72 304 Jan.1 Feb. 35. Mar. B. Find out range and its coefficient.5 July -0. The daily wages of ten workers are given below. Find range and coefficient of range of the following : (a) Per day earning of seven agricultural labourers in Rs : 60. -1-5 -0. Coetticiem ot Range = 0. Nov.351] 3. Following are the marks obtained by students m Sec. No. Compare the range of marks of students in two sections. Oct.9 . 85. Apr.6 -1. [(a) Range = 50. of Workers : 50 2 70 8 80 12 90 7 100 4 120 3 130 8 150 6 5.67] . . Fir.1 4.oel Find the range and coefficient of range of the following • Age m years : cm &• Frequency : 20 30 40 50 7 3 5 2 [Range = 40.j„ . If o . Xf 10 4 15 12 6. Statistics for Economics-XI June +3. Coeff.^ M . i^vduge = ^u.3 Dec. -6. Coefficient of Range = 0. L. of Range L 0 42 No. Height (in centimetres) Below 162 Below 163 Below 164 Below 165 Below 166 Below 167 Below 168 Below 169 Below 170 No.671 Frequency : 1-5 2 6-10 8 11-15 15 8. Months : 1 2 3 4 5 6 7 8 9 19 11 12 . of persons 1 8 19 32 45 58 85 93 100 [Range = 9 cm.97 Calculate dispersion by Rang.5-10 10 10-15 15 7. Coefficient of Range = 0. 16-20 21-25 26-30 20 10 l^ange = 30. 15-20 20-25 20 5 [Range = 20. ^ ^^ ^^ R^^ge = 0. 03] 'f ■ ^M Measures of Dispersion 305 Quartile lc\ialinn 9. Coefficient of Range = 0.97] :e The following table gives the hei^h. f ^ Method. Calculate Quartile Deviation and its Coefficient of Rajesh's daily income. Coefficient of Range = 0. ^ Income (Rs) : 239 250 251 251 257 258 260 261 262 262 273 275 [Q. = 0.D.016] 13. Calculate lower and upper Quartiles. = Rs 55.687 and Coefficient of Q. Calculate the Semi-interquartile Range and its Coefficient of the following data : Marks : 0-10 10-20 20-30 30-40 40-50 50-60 60-70 No. = 1. Find out Coefficient of Quartile Deviation from the following data : X : 10 15 20 25 f: 6 30 35 40 45 17 29 38 25 14 9 1 [Coefficient of Q. Q^ = 20. = 0.D.304] 12.D.119] 16.D. Coefficient of Q. Interquartile Range and Coefficient of Quartile Deviation of the following series : Height (in inches) : 58 59 60 61 62 63 64 65 66 No.D. Coeff. = 0. = 0. of Students : 4 8 11 15 12 6 3 . Q3 = 8. Coefficient of Q. Coefficient of Q.D. Interquartile Range = 2. = 0. Coefficient of Q. = 0. of Persons : 2 3 6 15 10 5 4 3 1 [Q. Find the Quartile Deviation and its Coefficient from the following data relating to the daily wages of seven workers : .D.25] 15. = 70. = 0.213] 10.2] 14.906. Q^ = 26.875.D. Quartile Deviation and Coefficient of Quartile Deviation of the following series : Values: 5-6 6-7 7-8 8-9 9-10 10-11 Frequency : 5 8 12 15 6 2 [Qj = 6. Daily Wages (in Rs) : 50 90 70 40 80 65 60 [Q.D.D.733.D. = Rs 15. of Q. Find out Quartile Deviation. Calculate Quartile and Coefficient of Quartile Deviation of the following data : Marks : 5-9 10-14 15-19 20-24 25-29 30-34 35-39 Students : 1 3 8 5 4 2 2 [Qj = 15.23] ^ Find out Quartile Deviation and Coefficient of Quartile Deviation of the following items : 145 130 200 210 198 234 159 160 178 257 260 300 345 360 390 [Q. 80- No.43. Q.55.50.33. = 0. Quartile Deviation and Coefficient of Quartile Deviation in the following : Age : 20.D.40.on from mel^'X f^ Xtn: n ■ 12 18 24 30 36 42 Frequency : 4 7 9 18 15 jo 5 24. = 45..on from mean and med. of Members : 3 61 132 154 140 51 2 I [Q3 = 64. Coefficient of Q. Calculate Mean Deviation from median : = = .n.08. Ca.a. Llf .D.D. Calculate the InterquartUe Range for the data given below • Zency : T T T^^ ^ 6 5 4 [I. Q.om « ml". Find out the Q3. = 9. median • ^ P''-^ ms) : 25 28 32 32 36 48 44 45 .e Mean Oe™„o„ f. o„ .a„on f„n.e a™.R.70... -^" = ^ 4 10 9 15 12 7 9 7 . = 12.9] Mcaii Deviation Find Coefflcent of Mean dev.D.17] 'i 1 306 Statistics for Economics-XI 68 M.D.[Q.77] 50 50 18.e devia. = 0.30. Mean Oe™..60. = 12.Q.a. Qj.cn. = 11.an of rS^laTa" ^ ^^ 52 49 45 72 57 47 Ca.337] 17. Coefficient of Q. No. of tomatoes per plant : 0 1 9 cj 4 ^ No. of plants . -y . ^ , , ^ ^ ^ 7 8 9 10 • 2 ^ 7 11 18 24 12 8 6 4 3 Size of Item : 4 ^ „ Frequency =2 4 I ^^ _ 14 3 2 ^^ 16 [M^D from X = 3.32 and Coefficient of M.D. = 0 342 26 Th. • u, "" ^^ " and Coeff,ciem of M D - 0 4051 dev,a,L. ""<1 ""dian and coeffiden, mean s- ^^Mrr r ? r [M.D. from X = 28.56, Coeffidem of M D = 0 228 M.D. from Me = 28, Coefficient of M.D. = 0.233] Measures of Dispersion ^^^ 127 Calculate Mean Deviation from mean of the following data : Class : 3-4 4-5 5-6 6-7 7-8 8-9 9-10 Frequency : 3 8 7 22 60 85 32 [M.D. = 0.915] 28. Calculate Mean and Mean Deviation and coefficient of M.D. for the following distribution : Weekly tvages : 20-40 40-60 60-80 80-100 Workers 20 : 40 30 10 [Mean = 56, M.D. = 15.2 and Coefficient of M.D. = 0.27] 29. Age distribution of hundred life insurance policy holders is as follows : mlrTstVtrthday : 17-19 20-25 26-35 36-40 41-50 51-55 56-60 61-70 Number 16 12 26 14 12 ^_^ Calculate Mean deviation from the median age. :9 [M.D. = 10.73] 30. Find Mean Deviation from median of the marks secured by 100 students in a class-test as given below: Marks : 60-63 63-66 66-69 69-72 72-75 No. of students : 5 18 42 27 8 [M.D. = 2.26] 31. Using Mean Deviation from median of the income group of 5 and 7 members given below, compare which of the group has more variability? Group A : Group B : 4000 3000 4200 4400 4600 4800 4000 4200 4400 4600 4800 5800 [Group A : M.D. = 1240, Coeff M.D. = 0.054 Group B : M.D. = 571.4, Coeff. M.D. = 0.13 Group B has greater variation] Standard Deviation 32. Calculate the Standard Deviation of wage earner's daily earnings : Week Earnings (in Rs) 1 54 2 62 3 63 4 65 5 68 6 71 78 9 10 73 78 82 84 [X =70, S.D. = 9.01] 33. Calculate Standard Deviation of the following two series. Which series has more variability: A : 58 59 60 65 66 B : 56 87 89 46 93 52 75 31 46 48 65 44 54 78 68 [A: X = 56, S.D. = 11.7, C.V. = 20.89% B: X = 68, S.D. = 17.1, C.V = 25.14% Series B has more variability] 308 WaSiSM wem mtM j„ . Statistics for Economics-XI Income (Rs) ; iqo 120 ISO ^ 140 120 150 [X = 133.7S, S.D. = 22.88] 35. 36. : (B) Following are given X variable • 55, 49, 67, 89, 44, 59, 57 (a) Calculate («■) the arithmetic mean («) the standard deviation («/') the mean deviation Also calculate 0) Z(X - 55)^ («) 2 IX - Median! (c) Examine, if («)Z|X-X|>z IX-Median! No. of families 166 552 580 433 268 148 78 37. 9 10 11 12 77 41 20 8 6 1 Calculate Mean and Standard Deviation fron. . n ^^ " Marks (Above) ■ Q jq the followmg data : = 1-76] No. : 150 HO 100 80 80 70 30 ^ Calculate Mean and vana.ce f.r the ~ 2 X 25 4 16 5 21 6 18 7 13 8 10 94 38. 10 3 [X = 5.5; a^ = 3.99] 25-30 16 30-35 8 39. WKMM wm&m Ml 35^0 3 "^^^-o Vanance .. Frequency 2 , ■ ^^-25 7 13 21 Calculate the Coefficient of V^w r . ^^ " = 7.95, 02 ^ 200 workers in a flc.™ lowing distribution of the wagt 2 ^OT : 40-49 en r T fX= 74.45,S.D. = ,5.921.CV...2,.38%J 4: 43 44. Measures of Dispersion 309 40. The following are the scores made by two batsmen A and B in a series of innings: : 12 115 6 73 7 19 119 36 84 29 % \; B : 47 12 76 42 4 51 37 48 13 0 ; Who is better as a run-getter? Who is more consistent? [A : X = 50, S.D. = 41.83, C.V. = 83.66% ■ 70.82%] B : X = 33, S.D. = 23.37, C.V. = (A is better as run-getter and B is more consistent,) 41. The index number of prices of cotton and coal shares in 1998 were as under: Month : Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. Index Number of Prices : Cotton : 188 178 173 164 172 183 184 185 211 217 232 240 Coal : 131 130 130 129 129 129 127 127 130 137 140 142 Which of these two shares do you consider more variable in prices : [Cotton : X = 193.9, S.D. = 23.80, C.V. = 12.27% Coal : X = 131.75, S.D. = 4.815, C.V. = 3.65% Cotton shares are more variable in prices] 42. Calculate the arithmetic mean and the following distribution : standard deviation and variance from Class : 0-5 5-10 10-15 15-20 20-25 25-30 30-35 35-40 Frequency : 2 5 7 21 16 8 13 [X = 21.9, S.D. = 7.99, a^ = 63.97] 3 43. Calculate arithmetic mean and standard deviation and variance from the foSbwing series: Marks : 70-80 60-70 50-60 40-50 30-40 20-30 No. of Students : 7 11 22 0 15 5 [X = 51.67, S.D. = 15.13, a^ = 228.92] ,44. The following tables gives the age distribution of students in a school in 2001 and 2002. Calculate Coefficient of Variation for both the groups. Age :. 17- 18- 19- 20- 21- 22- 23- 24- 25- 2001 : 1 3 8 12 14 14 5 3 2 2002 : 6 22 34 40 32 20 16 9 3 [2001 : X = 21.5, S.D. = 1.703, C.V = 7.92% 2002 : X = 20.91, S.D. = 1.846, C.V. = 8.82%] You are given the following data about height of boys and girls : • Boys Girls 72 , 38 Number Average height (in inches) Variance of distribution (in inches) ■tl 68 9 61 4 310 46. j„ . Statistics for Economics-XI (a) Calculate Coefficient of Variation. (b) Decide whose height is more variable. [{a) Boys : C.V. = 4.41%, Girls • C V - 1 77°/1 r i Ufe No of Years : 0-2 2-4 4-6 6-8 8-10 No of Refrigerators Model A Model B 10-12 ^ 16 13 7 of Wh,chide, has s 4 [Model A : X =5.12, C.V. = 54 9% Model B : X = 6.16, C.V = 36.2% Neck circumference ■. 12 0 12'^ i^n (in inches) ^^.O 14.5 15.0 15.5 16.0 No of Students : 5 20 30 43 60 56 37 16 . crkenon Mean .3 standard deviation. ' ^^^ ^^e [X = 14.01", a = 0.87". The largest size of collar = 16.62". 48 T, u A r I smallest size of collar = 11 4"1 ^^^ --Its : ^ Life ( 000 miles) : 20-25 25-30 30-35 35^0 orand X o c » 12 IS Brand Y 6 in 20 30 15 32 Which brand has a greater variation.? [C.V. (X) = 21.82%; C.V. (Y) = 16 101°/ y U .o. the data he.ow state lih se'r.s'.rcon'.^l'^ 40-45 13 12 45-50 9 0 Variable Series A Series B 10-20 20-30 30-^0 40-50 50-60 60-70 18 22 40_ 32 18 10 32 40 22 18 [Series A : _X = 42.14, S.D. . 14.06,.C.V = 33.36% 50 Th. ( u ■ u, B : X = 37.86, S.D. = 14.06, C.V = 3714%1 50. The following table gives the distribution of wages m the two branches of a falry^ :: — 300-350 " 93 157 105 Measures of Dispersion 82 500 311 gr Find the mean and standard deviation for the two branches for the wages separately. F [a) Which branch pays higher average wages? (b) Which branch has greater variability in wages in relation to the average wages? (c) What is the average monthly wage for the factory as a whole? id) What is the variance of wages of all the workers in the two branches—A and B taken together? [Branch A : Mean = Rs 225, S.D. = Rs 66.20, C.V. = 29.42% Branch B : Mean = Rs 230, S.D. = Rs 62.15, C.V. = 27.02% {a) Branch B pays higher average monthly wages. (b) Branch A has greater variability. (c) Combined Mean = Rs 226.67 (d) Combined Variance = Rs 4215] 51. What percentage of frequencies are there in the range of X ± 3 S.D. X : 60.5 70.5 80.5 90.5 100.5 110.5 120.5 130.5 140.5 f ■ 3 209 81 21 5 21 -78 . 182 305 [X = 100.95, S.D. = 13; Range = 99.12%] 52. Goals scored by two teams A and B in football matches were as follows : No. of Goals in a match : 0 1 2 3 4 No. of matches : A : 17 9 8 5 4 B : 17 9 6 5 3 Find the team which is more consistent in its performance. [Coeff. of variation Team A = 123.6% Coeff. of Variation Team B = 109.0% Thus, Team B is more consistent] 53. Find mean and the standard deviation of the following two groups taken together: Group Mumb^ : ^. ; Mean : , ^ . SOX : - - A 113 159 22.4 B 121 149 20.0 54. [Combined .: X^ z = 153.83, a^ , = 21.8] The number examined, the mean weight and standard deviation in each group of examination by two medical examiners are given below. Calculate mean and standard deviation of both the groups taken together. a iW Medical Exami?ier iiilS^^ MiMm Examined IS^^WMlilli Standard Deviation AB 50 60 113 120 ' y Mean : ^ Weight 6.5 8.2 [Combined : X^ ^ = 116;82, Oj , = 8.25] 312 ^ The following data gives arithmef ^^ Sub-srauh \r.: r77~~~r~-—---—----B'^up. No. of Men AB C {in Rs) 50 100 120 61.0 70.0 80.5 Standard Deviation (in Rs) 8.0 9.0 10.0 c^ A fCombined : X, , , - 7:5 „ 56. A sample of 35 value, h.c o ' ~ ' 2.3 = 11.9] <=0. a group of 50 male worths .he „ '' """ ' = wages are Rs 63 and Rs 9 respSively F?" R^ 54 and Rs 6 tespeaively Fi"d If" group of 90 workers. ' of their weeidy "0 female workers, th^fare '•"> ^^dard deviation for a comLed Coefficient ofvariation of two series are 5S«/ ' = ",, = 91 and What are their ml, TJ" ""'"'on f the coefficient of variation of X series is J4 6»/ 'f f" * = nteans are fOf.a and respe':,!:^^^.^- l^dTZ:^-- No. Of Persons COOO) ^^ 60 iqO i^q 40 50 22.608] Class A Class B ]' 20 80 ^ - li It ll - « « .5 85 „ ' ^ - - « 40 61. xi/St------y^PC- 58. 59. 60. - [rniu-vaiues) No. of students (Eco.) No. of students (Statis.) Chapter 11 MEASURES OF CORRELATION T. 2. 3. 4. 5. 6. Introduction Correlation and Causation Kinds of Correlation Degree of Correlation Methods of Studying Correlation Scatter Diagram Karl Pearson's Coefficient of Correlation (£) Spearman's Rank Correlation List of Formulae iduction In the previous chapters we have discussed measures of central tendency (Mean, Median and Mode), partitional values (Quartiles) and measures of dispersion (Range,' Quartde Deviation, Mean Deviation, Standard Deviation and Lorenz Curve. These are all relating to the description and analysis of single variable only This type of statistical analysis is called 'univariate analysis'. Now, we will deal with problems involving association in two variables. We find that in social as well as natural sciences, where more than one inter-dependent variables are involved, change in one variable brings change in others. For instance, in Biology we know that weight of a person increases with height in Geometry we know the circumference of a circle depends on the radius, in Economics prices vary with supply, cost of industrial production varies with the cost of raw materialsagricultural production depends on the rainfall etc. The relationship between variables is measured by correlation analysis. Thus, 'the term correlation (or covariation) indicates the . relationship between two such variables in which change in the values of one variable, the values of the other variable also change.' This statistical analysis of such data is called bivariate analysis Other Definitions According to Croxton and Cowden, "When the relationship is of d quantitative nature, the appropriate statistical tool for developing and measuring the relationship and expressing it in a brief formula is known correlation." According to L.R. Connor, "If two or more quantities vary in sympathy so that movements in one tend to be accompanied by corresponding movements in other(s) then they are said to be correlated." t 314 H. ^relation and causation j„ . Statistics for Economics-XI 1. Cause and effect : There is a cause and effect relationship between two variables shon w,ves and many .h„„ starred husbands may havt K' ^ r; Measures of Correlation 315 be correlation between price and demand so that in general whenever there is an increase in price the demand falls, and vice-versa. But this does not mean that whenever there is a rise in price the demand must fall. It is possible that with the rise in price the demand may also go up. This is on account of the fact that in economic and social sciences various factors affect the data simultaneously and it is difficult almost impossible to study the effects of these factors separately. Thus, correlation measures co-variation, not causation. It measures the direction and intensity of relationship among variables. s^ds of correlation On the basis of nature of relationship between the variables correlation may be : (1) Positive and Negative Correlation (2) .Linear and Curvilinear Correlation (3) Simple, Multiple and Partial Correlation 1. Positive and Negative Correlation When both the variables change in one direction, that is when both increase or decrease the relationship between the two variables is called positive or direct. But when the change is in opposite directions that is one is increasing and the other is decreasing, the correlation is negative or inverse. For determining the direction of change average values are taken. For example : (I) Positive Correlation (a) . (b) (a) Both variables increasing (il) Negative Correlation (b) Both variables decreasing One variable increasing, the other decreasing One variable decreasing, the other increasing X Y X Y X Y X Y 10 100 70 . 147 15 125 75 110 20 150 60 140 30 110 60 180 30 160 40 135 35 90 40 190 40 190 30 130 40 80 30 200 50 200 15 120 45 75 20 240 60 255 10 90 50 60 10 250 We find that in I (a) the values of X series are increasing so also of the Y series. In I (b) values of X and Y are decreasing. Thus, they are both instance and positive correlation. On the other hand, in II {a) the values of X are increasing and the values of Y are decreasing, similarly in II (b) the values of X are decreasing and the values of Y are increasing. Thus, hey are both examples of negative correlation. 316 Examples : (Positive Correlation)'^"""""'cs-XI 1- Age of husband and age of wife. --easeinheatlllT^^^nofr,. 1- Demand of a commodity mav ^ Increase in the number tfTe ^ "": " 3- Sale of woollen garments ant ly ir "" Yield of crops and Price. " Correlation u the ratio of chance h^n of^rrotstr -ear line. If tbTv:ri:trfmS™=™™bles. Their^SaSnshbt'f "P™ ""-"ncy "f non-hnear "ttnTware ^'phed, rhe bear a constant rari^ ('""'''near), the amount of chale 3. Staple, Multiple and Partial Correlation relationship betwln • T "^"Me or p '' ^^r-ables are IS a study of relaf,V,„,l.- 7 ""her variables from ,1. u , influencing valbles b '' variables Sfllcit ^ ' of tainfalf""stant. For example, 3;,':'' "f other " ™riables correlation. " a certain consrant^Jm:: M Measures of Correlation 317 of correlation The relationship between two values can be determined by the quantitative value of coefficient of correlation which is obtained by calculations. Perfect Correlation : Perfect correlation is that where changes in two related variables are exactly proportional. If equal proportional changes are in the same direction, there is perfect positive correlation betWeen the two values described as +1; and if equal proportional changes are in the reverse direction, there is perfect negative correlation, described as - 1. For example, the circumference of a circle increases in the equal proportionate ratio with the increase in the equal proportionate ratio in the length of its diameter; the amount of electricity bill increase in a perfectly definite ratio with an increase in the number of unit consumed, the volume of a gas varies inversely with the pressure at constant temperature etc. DEGREE OF CORRELATION ive Zero Correlation : The value of the coefficient of correlation may be zero. It means that there is zero correlation. It does not mean the absence of any type of relation between the two variables. Two valued are uncorrelated. However; other type of relation may be there. There is no linear relationship between them. Limited Degree of Correlation : In social science, the variables may be correlated, but an increase in one variable need not always be accompanied by a corresponding or equal increase (or decrease) in the other variable. Correlation is said to be limited positive when there are unequal changes in the two variables in the same direction; and correlation is limited negative when there are unequal changes in the reverse direction. The limited degree of combination can be high (between ± .75 to 1); moderate (± .25 to .75) or low i j I-i 318 (between H-0 tn a u- u Statistics for Economics-Xl Karl Pearson's foLt '"P""'^-"'^ degree of eorrelarion according ,o Dejp-ee of Correlation Perfect Correlation Very high degree of Correlation Sufficiently high degree of Correlation Moderate degree of Correlation Only the possibility of a Correlation Possibly no Correlation Zero Correlation (Uncorrelated) Positive +1 + -9 or more from + .75 to + .9 from + .6 to +.75 from + .3 ro +.6 Less than +.3 0 Negative -1 - .9 or more from - .75 to - .9 from - .6 to -.75 from - .3 to -.6 Less than -.3 0 IhhoD; "on. Somell^ are : SW Coefficient of Correlation {c) Spearman's Rank Correlation scatter diagram tdea about the presence of when it ir^ov^nw^d ^ . We find from the plottings on the scatter diagrams that there is a certain similarity among (a) and {d). 1 Figure [a).measuring X-variable on bokolTL^^apb paper. (e) and show a downward trend—they show negative correlat!on. (e) and (/).scatter diagrams given below : ^"rreiation is negative. (b) and (c) show an upward trend—they show positive correlation. there are differences among (a)."Pward . When the plottedTo „te^ ^^^^^ ^ata are "" P^P^^ called we know that there is some correlation Seen tl ^^^d-upward or downward-the correlat^n is positive. Howe\ er. The chart is prepared by pomt for each pair of observation oTx and y JZTi u'''' ^^ P^^^ plotted m the shape of points. In (a) and (d) the plotted points are . Let us study r=+l Perfect Positive Correlation (a) High Degree ot Positive Correlation (b) Low Degree of Positive Correlation (0 Measures of Correlation Perfect Negative Correlation (d) 319 High Dgree of Negative Correlation Low Degree of Negative Correlation I w V! i r= 0 No Con-elation (9) Fig. (b) and (c) and similar differences among {d). Figure (d). (b) and (e) and (c) and (/). The cluster of ooin^ U the scatter diagram. Y) on graph paper. This kind of scattered diagram shows low degree correlation. From the following pairs of value of variables X and Y draw a scatter diagram and interpret the result. We note that X = 4 and Y = 78 as given first X and Y values. diagram {g) shows such a vast scatter of points that it is impossible to see any trend— this shows no correlation or zero correlation. we will find that the plotted points are very much scattered around the line—^not as near as in the case of {b) and (e). j„ . This kind of scatter diagram shows high degree correlation. We measure 4 on X-axis and 78 lik .5 cm = 2 on X-axis B 64 56 48 40 32 24 16 8 0 320 coordinates of . Illustration 1. Finally. Statistics for Economics-XI scatter diagram Scale : 0. 8 9 10 11 12 13 14 15 54 48 42 36 30 24 18 12 5 72 6 66 7 60 X : 4 Y : 78 Solution. In (c) and (f) if we draw a similar line (regression line).almost in a straight lines—this indicates perfect correlation. We may plot this as point (X.1. the points are near about the line. where X = 4 and Y = 78. In [b) and (e) the plotted points are not in a straight line but if we draw a straight line in the middle of their points (regression line) we will find. __ — — i• — — -- 1 • -4-. -U —1- 02468 10 12 14 16 IS showing more than proportionate an< change I'fl:: m in ---- . The points take the shape of li^e hen tC r 'r "" Weei X aid Rate of Change It is slope of the straight line rwhirh depends on an angle that the str^lghT Hnt makes with the X-axis and is equal to j^Zj rate of change showing almost equal change t-.measure 5 along the x'axis and 72 alongT axis and so on for all d.» .e given X and y Xl from the above scatter diagram we can decide Aat the variables X an" Y "e corre ated. the correlation is negative. 3 We know when the plotted points show some upward trend. the change is exactly in the same proportion as the change in the value of X [Fig. Such relationship will form a curve on graph [Fig. it shows that value Y does not change at all [Fig. (v) Linear correlation exists when the ratio of change between two variables is uniform The relationship is described by the straight line. (c) and (d)]. (iii) If the angle that the straight line makes with X-axis is less than 45".a (i Measures of Correlation 321 showing less than proportionate change showing no change non-linear relationship Fig. Hi) If the angle that the straight line makes with the X-axis is greater than 45° the change in the value Y is more than proportionate to the change in the value of X [Fig. (h)]. (/•) If the straight line makes an angle of 45° with the X-axis. {a) and (b)]. the correlation is positive and when there is downward trend. (e) and (/)] (w) If there is no angle and it is a straight line parallel to X-axis. 322 Ir ! m . In case of non-linear relationship (curvilinear) the amount of change in one variable does not bear a constant ratio to the amount of change in the other variable. the change in value Y is less than proportionate to the change in the value X [Fig. „e t Jcha^^r.X). It is very easy to draw a scatter diagram. 5.»f Statistics for Economics-XI Merits and Demerits of Scatter Diagram Merits : 1. Whether YLTes XorTcau^ —^^ ^oes not tell. not possible to draw a scatter pearson's coefficient of correlation X fgr^BS variable (1867-1936).ItisthemostLerused me^^^^^^^^^ and correlation of coefficient. ^^ ^ of Correlation (r) of two variables « ob aiZ l jfjS^^T. ^-^ffft-ent corresponding deviations of the various 7eZ of the products of the by the product of their standard devZ^Zan JZ ^T Symbolically. ^hen .a. Nxaxxay x=(X. <^evtattons and the number of pairs of observations. y={Y.ca.: t'^nTT" " t?™ r^atX™ — ■'"own . fa case of linear relationship between x lni y lT T ^ donate change in th^ .„ n„„er. r= Ixy Here. It is based on arithmel^ a„d f't'!.t .Y) <yx = Standard deviation of X Series ie ''"\N Measures of Correlation . This is atrcaHedX^^^^ statistician Karl Pearson represented by r. i. N = Number of pairs of observations r = Coefficient of correlation The above formula can be rewritten as under : Txy 323 N r= N.ay The above formula is based on the study of covariance between two series.e.ax. ox oy 1 ^y X 1 )c 1 = ^y " if " -if r= Ixy yjlx^xZy^ or I(X-X).(Y-Y) ^Jiix-xfylm-yf Applying the Karl Pearson's formula Coefficient of Correlation is calculated by following methods : . — X — N Zxy.ay = Standard deviation of Y Series. The covariance between two series is written as follows : N N r= Exy 1 1 —-y. Calculate Product moment of correlation from the following data and interpret the result. Karl Person's coefficient of correlation is also called Product Moment of Correlation. of Students : 10 1 2 3 4 5 6 7 8 9 Marks in Mathematics : 48 15 18 21 24 27 30 36 39 42 Marks in Statistics :25 25 27 27 31 33 35 41 41 45 Solution. Serial No.(a) Actual Mean Method (b) Direct Method (c) Assumed Mean Method (d) Step Deviation Method (a) Actual Mean Method Illustration 2. I 324 II t n^>: i*' Statistics for Economics-XI Cdcula^on of Coefficient of Correlation 15 18 21 24 27 30 36 39 42 48 ZX = 300 -15 -12 -9 -6 -3 0 +6 +9 +12 +18 225 25 144 25 81 27 . 36 27 9 31 0 33 36 35 81 41 144 41 324 45 = 1080 2Y = 330 Steps : 1. ^ = (X . Apply the following formula : r= Ixy Here. Y= r= N ZX 10 330 N Ixy = 30 = 33 Measures of Correlation .X) and y = (Y _ y) Let us calculate arithmetic means of X and Y series : 300 X= Now we get. Calculate arithmetic means of X and Y series 7. we get Ixy EY _ m N " 6 = 22 . Y = Applying formula. Illustration 3.Here. there is high degree of positive correlation. 33 24 1971 35 27 1981 30 24 Death rate Solution.98 >/l080x480 >/518400 720 Hence. Calculation of Coefficient of Correlation Actual Mean Method : EX X= 180 N 30 . Calculate Karl Pearson's coefficient of correlation between birth rate and death rate from the following data : Year Birth rate 1931 24 15 1941 26 20 1951 32 22 1961 . Zxy = 708. Zx^ =1080 and ^ 439 708 708 708 325 r= = 0. 920 Birth Death X-X rate Y.r= Here. there is high degree of positive correlation. yJlx^x-Ly^ Ixy = 81.4 6 35 27 +5 25 +5 '25 25 30 24 0 0 +2 4 0 EX = 180 81 EY = 132 Ex^ = 0 Ex = 90 Ey = 0 Pi' t-i \vi I. 326 Statistics for Economics-XI X data compute product moment correlation between No of items Arithmetic Mean Square of deviation from Mean X series 15 25 136 Y series 15 18 138 c ----------138 If = 86 Exy = .y rate X ■y xy 24 15 -6 36 -7 \ 49 42 26 20 -4 16 -2 4. Zx^ = 90 and = 86 81 81 81 r= V90x86 V7740 87.9772 Hence. 8 32 22 +2 4 b 0 0 33 24 +3 9 +2 . = 0.-. we are given the following Morr^^^ ^ "" = 136. ^ 138^ ^^ ^ ^^2 Applying formula Now.5. r= Ixy Now.996 Hence.Summation of product of deviations of X v o r means = 122. means.533 . we get N X ax X ay r= 420 420 = 0. Y series from their respective "cviauons ot A and Solution.5 and Zxy = 420 Applying formula. we get r= r= Ixy 122 122 122 = 0. Regarding deviations of the values in X anH v t .891 >^36x138 V18768 136. there is high degree positive correlation between X and Y their arithmetic means be 420. cx = 4. ay = 3. Ld the coSf^fl^^^^^^^^^ IXe^^S Solution. Given N = 50. coefficient of correlation (r) = 0.A . we get r= N. Covariance of X and Y = ^ .6 Variance of Y (af) = 81.28 and their covariance is 7.8 = y/l3:s = 3.6 and variance of Y series is 81.28 Covariance of X and Y = Lxy = 7. Now. +12.83 Hence.71 Measures of Correlation Variance of Y (a/) = 16. If the covariance between X and Y variables is3 X and Y are respectively 13 8 and 16 4 Vir.90.ay Exy 1 1 —-X — x — N ax ay r = 12.50x4. Given.71 4.5x3.05 327 Applying formula.ax.90 .5 ^rl^'^Z ^ -cl Y.25 = 0. ^^ation 6.l ? ^ Variance of X (ax^) = 13.3 X 0. there is high degree of positive correlation between X and Y.4 oy = Vl^ = 4.27 X 0.u t between them. Solution.3 and variances of " ^he Karl Pearson's coefficient of correlation Solution. Find the standard deviation of X series if coefficient of correlation between two series X and Y is = 0.05 = 12.5 787.3 X —X ^ 3. Illustration 7. We are given. r= Ixy yjlx^xly^ or 0.6 X xN ax ay 1 or ax 9.8. and Ix^ = 100. standard deviation of Y = 5. Calculate the number of items for which r = + 0.90 = 9.ay = V81. Ixy = 200. we get r= lay N.05 Applying formula. Solution. Now.05 ax = 7.28 = 7.ay 2jcy 1 1 x — x — 0.ax.8 = 200 Now.05 0. where x and y denotes deviation of items from actual mean.28 X 9.6 7.534 = 2.6 or 2.99 approx 3 Therefore.6 ax = 2. we get . Applying formula. variance of X (ax^) = (3)^ = 9 Illustration 8.534 ax = 7. sr = or 0. 328 Statistics for Economics-XI V 2 40000 64 =625 Now.. ll In 625 number of items is 25.64 = 100 xZy or 0. = or 5 = VN 1625 VN or /CX2 625 (5)^ = ^ or 25 = or 25 N = 625 'N = Hence.^O.64 X 100 x = 40000 64 Ey^ = 40000 yJlOOxZy^ 40000 100 xZy^ i ^Iv i>t f Ii . = 25 625 N 25 ~ . (b) Direct Method J:XY-N r= in. fr— the following where.. Y = ZY N N ~(Y) v\2 r= X. In j n nJ • EXY = -p.N. 1XY-N.{Y} X ^V X— r= . r= X= lix' N {Xfx ZX N .{X).Kf /ix-* NVN nixy~zxxy y/NEX^ • -iZXf Xy/NZY^ -{ZYf ^ " . 2. • 10 10 11 12 12 2:x = 55 5 6432 ZY= 20 100 100 121 144 144 ZX' = 609 25 36 16 9 4 lY' = 90 Steps : 1. Months Price (in Rs) Quantity (in kg) 1 10 5 2 10 6 3 11 4 4 12 3 Solution. EX^ 3. Calculate arithmetic means of X and Y series. EY^ . Square the values of X series and obtain the total.e. i.e. Calculate the product moment correlation (Karl Pearson's coefficient of correlation) between price and quantity and comment on its sign and magnitude. i.. Calculation of coefficient of correlation. Square the values of Y series and obtain the total. The data of price and quantity purchased relating to a commodity for 5 months are given below..Measures of Correlation 329 Illustration 9. N(Yf Let us calculate arithmetic means of X and Y series X= EX 55 EY 20 N 5 ~ ^^ ' ^ .324 .949 >/4xV10 2x3. ZY^ = 90.X. ZXY = 214. ZXY. EX^ = 609. Apply the following formula : 5 12 2 xy 50 60 44 36 24 EXY = 214 r= IXY-N. Multiply X and Y values and find out the total.162 6.Y VeX^ .N(Xf x ^zy^ . 5..4. i.e. X = 11. we get 214-5x11x4 =4 r= ^609-5(11)^ xV90-5(4)2 214-5x44 Ii ■ [i 1 330 214-220 V609-605x>/90-80 -6 -6 -6 Statistics for Economics-XI = -0. Y = 4 and N = 5 Now.N 5 Here. Interprete the result and comment on their relationship. Illustration 10.Hence. fx. In other words. Draw a scatter diagram and calculate Karl Pearson's coefficient of correlation between X and Y. X : 1 3 4 5■7 8 : 2 6 8 10 . 14 16 Solution. purchase (demand) decreased due to increase in the price of commodity. there is high degree of negative correlatiori between price and quantity purchased relating to a commodity of 5 months. ^ ¥ % u scatter diagram Scale: 1 cm = 1 on X-axis 1 cm = 2 on Y-axis -Q lo v ■SA 1 in 1V -Q(V) U —A-At 2 u )■1 J. t ( 17 ! < . = 656 and N = 6 r= 6 . and it goes up from left bottom to right top then there IS perfect positive correlation between X and Y. 4 From the above scatter uiagram we can decide that variables X and Y are correlated. lY = 56. IXY = 328. fon equ in s con negt Measures of Correlation 353 331 X y X' 1 2 1 4 2 3 6 9 36 18 4 8 16 64 32 5 10 25 100 50 7 14 49 196 98 8 16 64 256 128 X = 28 XY' ZY = 56 IX' = 164 ZY' = 656 ZXY = 328 Applying formula. we get r= N V N N Here.Fig. The points take the shape of line. IT. 2X = 28. IX' = 164. If equal proportional changes are in the reverse direction.(28)^ r rr-f 1164 56{56r 328-261.334 X VI 33. there is perfect negative correlation (r = -1).ado„ Statistics for Economics-XI I on their relationship. In such situation. resulting to r = +1.67 = +1 5. Y values are exactly double than the corresponding values of X movuig in same direction (upward). correlation results to perfect positiJe correlation.67 V33.666 X V656 . f/ ' fiPl IN 332 -fflcient of c„„e. comment -3 9 .67 r = +1 There is perfect positive correlation by scatter diagram and even by Karl Peaison^ formula.522.67 66.774x11.666 66.547 66. between X and Y and X Y : Solution. We observe from the illustration the changes in tw^o values X and Y are exactiv in equal proportion.334 66.33 V164 -130. -2 4 -1 1 24 39 -3 -2 -1 1 2 3 IX = 0 l^jgilftjon of Coeffident of Correlation 941149 xy = 28 941149 = 28 81 16 1 1 16 81 ly^ = 196 XY -27 -8 -1 1 8 27 Zxy= 0 r= IXY- N ■2 (ZYf fzy^- . Idxdyr= Ux. we can use the assumed meat. Calculate Karl Pearson's coefficient of correlation of the following data of height of fathers in inches (X) and their sons (Y).N =0 yf^x^/65.334 = 0 5. method.291x8. Correlation coefficient can be obtained by the following formula. Interpret the result.Zdy N_ W(Zdyf N Illustration 12.083" rhey Measures of Correlation 333 (c) Assumed Mean Method When actual mean is not a whole number. To avoid such tedious calculations. the calculation by actual mean method and direct method will involve a lot of calculations and time. Calculation of Coefficient of Correlation 65 66 57 67 68 69 . Height of fathers (in inches) : 65 66 57 67 68 69 70 72 Height of sons (in inches) : 67 56 65 68 72 72 69 71 Solution. but a fraction or the series is large. we get r = -j 8 48 + 25 >/i56-12. i. Ikixiiy. ILdx. Calculate the deviations of Y series from an assumed mean (65) and denote them by dy and find out the total.. ZJx = -10. Zdxdy = 48./. 1.e.e. Applying the following formula. 4.e. i.. Multiply t/x and f/y and find out the total.e. ZJ^a ^ .. i. W = 156. we get 334 Statistics for Economics-XI Here. N = 8. Uy. I^y = 20. Square the deviations of Y series and obtain the total.. Square the deviations of X series and obtain the total. I^y^. 3.5XV244-50 .70 72 -3 -2 -11 -1 0 +1 +2 +4 Zdx = -10 9 4 121 1 0 1 4 16 Z^^ = 156 67 56 65 68 72 72 69 71 +2 -9 0 +3 +7 +7 +4 +6 ldy = 20 4 81 0 9 49 49 16 36 Zdy" = 244 -6 +18 0 -3 0 +7 +8 +24 Idxdy = 48 Steps : 1.e. 5..44 Now. 6. Zdx^. i. Calculate the deviations of X series from an assumed mean (68) and denote them by dx and find out the total. 1563 + 2.i [2.2220 = -0.3587 = -0.1 [log 143. Calculation of Coefficient of Correlation X-60 X-60 .2878] = 1.3587) = Antilog T.8633 .• 335 Measures of Correlation I unaffected by the change of origin and change of scale of X and Y.Price (Rs) X dx' (kg) 20 dr Y ■ y-700 dy^ drdy I /W t*jf dy 100 .5 + log 194] = 1.4378 = 0.s) : 40 Supply (in kg) lo^ .3587 (+1) .6413 r = 0. Solution.1 [4. log 73 .2. we apply the same formula of assumed mean method.8633 .0.5xVm log r . The data on price and supply relating to a commodity for 7 months are given below : ^ ^ ".97x13.8633 . "" by u Stq .4441] = 1. After changing these deviations.73 73 73 11. VM3.-1 + (1 .o : 400 200 500 1000 400 1100 1200 Calculate product moment of correlation between price and quantity and comment on its sign and magnitude.438 {d) Step Deviation Method convenient common factor to redZ „ T .o . Illustration 13.92 ~ U^ "" the™: " —l>etwee„ height of fathers and We can simplify the above calculations by using log tables : Taking Logarithms 73 Hence. e.CZdy) N_ Zdx. i. Applying formula. i. Zd-f. Idxdy. 6.e.. 80 +20 +1 1 1000 +300 +3 9 3 100 +40 +2 4 400 -300 9 -6 120 +60 +3 9 lioo +400 +4 16 12 140 +80 +4 16 1200 +500 +5 25 20 ZX = 560 97 lAxdy= 40 Tdx = 7 -3 Zdx^ = 35 ZY = 4800 Zdy = -l Idf = Steps : 1. Idx.e. 3. Uy. Idx\ 4. Square the step deviations of X series and obtain the total. Denote them by dx and find out the total. Multiply dx and dy and find out the total.. Square the step deviations of Y series and obtain the total.e..e. 5. i. i.dyr= [Zdxf N Zdy^(Zdy)^ N 336 . i.. 2. Denote them by dy and find out the total. Calculate the deviations" of Y series from an assumed mean and divide them by common factor. we get (Zdx). Calculate the deviations of X series from an assumed mean and divide them by common factor.20 -40 -2 4 400 -300 -3 Q 6: 40 -20 -1 1 200 -500 -5 25 s 60 0 0 0 500 -200 -2 4 o.. ^ Calculate^coeffic^^^^^ SolL^W^rmultSTLes^lOa^r.dy = 40.24 ^n.— The following example would illustrate the poim • Illustration 14.Here.84 52."^^^ ^^^^^ values of X and Y would L ^ ^ ^y ^o that the X:1 ■ ^= 5 6 t . Zdx = 7. W = 35 and Ldy^ = 97 ^_ . N = 7.29x9.05 commodity.— 41 y/Bxy/96-857 41 41 = + 0.4 and 3 7 lev ci\ 1 T J'l ^^^^ these values to he 12.^'' . 10 11 J3 ' ' ' Calcularion of Coefficient of Correlation . supply mcreases due to mcrease in the price of Change of Scale in the Calculation of r valu" of ~ ff ^^ ^^^ we take the values as T 2 ani s thrva L T Z ^^ ^ ^00 and values of Y-series are 1 i 2.787 _ 5. Idx. ner words. Uy = _ r= 40-1^ '7 Statistics for Economics-XI 1. Uxdy = 46. Linear relationship : If two variables are plotted on a scatter diagram. there is high degree of positive correlation. .997 V28x47 Hence. So there is a linear relationship between the variables.Measures of Correlation 359 Applying formula 337 "Ldxdy — r= (ldx)(Uy) N Here.997. Assumptions of Karl Pearson's CoefiScient of Correlation Pearsonian coefficient is based on following assumptions : 1. Ux = 0 Idy = 0. we get +46r= (OHO) . If the original values of X and Y were used the result would still be the same and r would be +0. EJx^ = 28. it is assumed that the plotted points will form a straight line. Zdy^ = 76.7 f^xf^ + 46 sBmm LIBRARY = 0. N = 7 Now. affecting the distribution of items in two series. They may. 3. MerU^^d Demerits of Karl Pearson's Coeffident Demerits 3 process is time consuming. Variables like indices of price and supply. Ji ■ "i 338 Mathematical Properties of th^ r Th^ ( 11 Statistics for Economics-XI Coefficient of Correlation are muIdpUed or divided by so™ ^rant ^ f'"" * ^-ri' consr^r . Causal relationship : Correlation is only meaningful. 5. and need not necessarily be indepe^detit Uncor^L ^ """elated variables and y Stnyly implies the absence of Itoear rZI^ \ ™™bles X however. (See Illustration 3. whether such 4. which form a normal distribution. Proper grouping : It will be a better correlation analysis if there is an equal number of pairs. ages of husbands and wives. Correlation lies between ± 1 Thi ^ i . heights of fathers and sons. subtracted or added frorivl®'f^^'/v""®" "" '^at a " and 14). of X and Y series.2. Error of measurement: If the error of measurement is reduced to the minimum the coefficient of correlation is more reliable. because the factors that affect these variables are not common. The calculated coefficient of correlation of such series is usually termed as ''non-sense or spurious^ correlation. The converse of the dieorem r=n i. the weight of an individual during the last ten years may show an upward trend and his income during this period may also show similar tendency but there cannot be any correlation between the two series because the forces affecting the two series are entirely unconnected with each other. price and demand are affected by such forces the normal distribution is formed. • relaZ'Sp'™ between the variables. 4. if there is a cause and effect relationship between the force. There is no relationship between rice and wheat. Similarly. if there is no such relationship. be related in some other fo™between them. It-is meaningless. Normality : The correlated variables are affected by a large number of independent causes. a British psychologist developed a formula in 1904 which consists in obtaining the correlation coefficient between ranks of N individuals in the two attributes under study called coefficient of correlation. It is the Product Moment Correlation between the ranks. When rk = -1. because under ranking method original values are not taken into account. then there is complete disagreement in order of ranks and they are in opposite direction. If rk = +1. Let us examine by following example : 1 23 1 23 000 000 niy = 0 rk = \- . then there is complete agreement in the order of ranks and the direction of the rank is also the same. the differences of corresponding rank vaiues are calculated and following formula is used : rfe = 1 N^-N where.otherwise it may be misinterpreted. After assigning ranks to the various items. The result we get from this method is only approximate one. rk = Coefficient of rank correlation ZD' = the total of squares of the differences of corresponding ranks N = the number of pairs of observations Like Karl Pearsons. ' interpretation. This method is applicable only to individual observations rather than frequency distribution. ^Rww^ ____________ under consideration Measures of Correlation ^^^ Charles Edward spearman. by rank differences. the value or rk lies between +1 and -1. (c) When Ranks are equal or repeated. i. The problems are of three types of calculation of rank correlation : When Ranks are given. Ir i (If: . 1 •2 3 321 -2 0 2 4 04 =8 rk = l=16ZD^ N^-N 6x8 3^-3 = 1 . i..2 = -1 Perfect negative correlation.e.e. there is complete disagreement. there is a complete agreement. (fc) When Ranks are not given..=1ZD^ N^-N 6x0 3^-3 =1-0=1 Perfect positive correlation. 340 (a) When ranks are given : Illustration 15. j„ . In a hah Entry Judge I Judge II Calculate the rank correlation coeLent!" Solution. Statistics for Economics-XI A B CDE 1 2 345 2 3 164 G // I / K 7 8 9 10 11 8 7 10 11 9 F65 ^^^ff^^ffident of Correlatic A B C D E F G H I J K N= 11 . Apply the following formula : rk=l- . Calculate the difference of two ranks /e Rn u 2.12 3 4 5 6 7 89 10 11 2 -1 3 -1 1 +2 6 -2 4 +1 5 +1 8 -1 7 +1 10 -1 11 -1 9 +2 Steps : 11441111114 ZD'= 20. Square these differences and find o^ 'Z tok] 3. 1. N^-N n'-n where. Statistics : 36 56 20 65. Calculation of Rank Coeffident of Correlation 36 4 50 -5 -1 ' 1 56 8 35 2 +6 36 20 2 70 8 -6 36 65 10 25 1 +9 81 42 5 58 6 -1 1 33 3 75 1? -6 36 44 6 60 7 -1 1 53 7 45 4 \ +3 9 .Now we get.909.42 33 44 53 15 60 Economics : 50 35 70 25 58 75 60 45 89 38 Solution. there is high degree positive correlation. ZD^ = 20 and N = H Measures of Correlation 363 341 rk=l6x20 11^-11 =1120 1320 = 1 .e. From the following marks obtained by 10 students in Statistics and Economics. i.0.909 Hence. {b) When ranks are not given : Illustration 16. two judges are agreeing to the degree of 0.091 = 0. calculate Spearman's coefficient of rank correlation. It indicates that judges have fairly strong likes and dislikes so far as ranking of the babies are concerned.. 15 1 89 10 -9 81 60 9 38 3 +6 36 N= 10 = 318 Steps 1.3025 x 73850^30^ -765 . N'-N ^D. R^ .-927 . Square these differences and find out the total TB^.. (55) iiOx 2260-3025 V3850 . next to its second rank and so. 342 Applying the formula.-0.e. the next to its second rank and so on or smallest item the first. 3. on. Ranks can be given by allotting the biggest item the first rank. 2^x^226-(55). Any one of the above method of ranking must be followed in case of both the variables.R^) and denote these differences by D. Statistics for Economics-XI rk = 1 Here. Apply the formula. Assigns ranks to given data. . we get j„ .= 318 and N = 10 rk = 1 ^x318 =11908 990 th uu = 1 -1. 4. 2. Find the difference of two ranks (i.927. Entry No. which is same as before 343 Measures of Correlation Blustration 17. Ten entries are submitted for a competition. : 1 2 3 4 5 6 7 8 9 10 Judge A : 9 3 7 5 1 6 2 4 10 8 Judge B : 9 1 10 4 3 8 5 2 7 6 Judge C : 6 3 8 7 2 4 1 5 9 10 Calculation of Rank Coefficient of Correlation - ■A •a 1 9 9 6 2 3 1 3 3 7 10 8 4 5 4 7 5 1 3 2 6 6.0. Tbree judges study eacb Ranks given by : Calculate the appropriate rank correlations to help you to answer the following questions : (a) Which pair of judges agree the most? (b) Which pair of judges disagree the most? Solution. 2 5 9 10 7 9 10 8 6 10 N = 10 .927.^^ X ^JS25 = . 8 4 7 2 5 1 8 4. 1575 990 ID.0.88 . we get rk = lN-' .N rk (between Judges A and B) rk = 1 6x48 10" -10 = +0.29 290 = 1-^=1.•4 +4 16 -3 9 +1 1 +4 16 +2 4 -1 " 1. -3 9 +3 9 +1 1 ' -2 4 +2 4 -2 '4 -4 16 SD^ = 48 ID' = 26 Applying the following formula.71 rk (between Judges A and C) rk = I 6x26 10-^10 = +0.-3 9 -2 4 -1 \-1 +1 1 -2 4 +2- .8425 = I _ ^ = 1 _ 0..IjF^ 0 0 +3 9 +3 9 +2 4 0 0 -2 4 -3 9 -1 1 +2 4 +1 1 -2 4 .. 9 + 49 = 122.5 = 0.344 rk (between Judges B and C) Statistics for Economics-XI rk = l6x88 10^-10 ^ ^-e ^e nearest approach (*) ^mce is n^nimun. we get 0.0. therefore.5 . .5 990 6 ED2 = 0.(3)^ + (7)2 = 82. Find the corrit coefficiem'jrnLl:!::^™^" "" rk=l- N^-N Substituting the values in above formula. of the pair of judges ^ and C.5 = 1 - (10)^-10 = 1 .5 x 990 2£>2 ^ 0-5x990 6 = 82.5 6x122.5 Corrected ID^ = 82. they disagree the in and" ^^rr/t^S -"^nts in ranks in t»o subjects obtained W L of 1 ' <liff"ence of 7.5 .5 Corrected rk = I =1- ^ ^^d Solution. 25 40 24 7 10 -3.(10)^-10 735 990 rk = + 0.5 2.1.0 1.0 1.00 25 9 5 4 +1.5 1 +2.2576 (c) When ranks are equal or repeated 13 13 24 15 20 19 Measures of Correlation ■ Solution.5 12. Calculation of Rank Coefficient of Correlation 345 X Y 48 13 8 5.5 57 19 9 8 +1.5 +0.5 +2.00 15 6 2 2.5 -0.25 33 13 6 5.5 6.25 65 20 10 9 +1.5 .25 16 15 3.0 9.5 7 -3.5 6.oo" N = 10 Steps : 0.5 .25 _ ZD^ = 39.0 i.25 16 4 3.00 9 6 1 2.5 0. 754 . i. In case.3 rk = 1ID^ + ^(m^ -m) +. they are assigned average ranks. 2.754 990 rfe = 0. there are more than one such group of values with common rank 1/12 (m^ . Assign the ranks to given data.5 and in Y series value 13 are given the rank ^^ = 5. m represents for number of times whose ranks are repeated.e. rk 6{39.0 990 246 = 1 . For example. N^-N Now we get.5) 990 =110^-0 6x41.5 + 1..246 = 0. the Obtaining ID^ apply the formula. When equal ranks are assigned to same of entries and adjustment is made in the formula of rank correlation...m).5 + -2) + -2) + -2)} = 1 ~ in3„n ta : =1=16(39.1. in X series value 16 repeated twice and they are each ranked ^^ =3. When two or more items are of equal value.5 and so on.. adding — {m^ -m) to the value of SD^ Here.0. is added as many times the number of such groups. The adjusted formula is as under : . ay N ^ ax ^ ay (b) Direct Method 4 NN Ixy xZ/ ZXY-N r= In N N.o Karl Pearson.'- degree of correlation. there is high degree of positive correlation.ax. 346 j„ . De^r. 'N In .Hence. Karl Pearson's Coefficient of Correlation (a) Actual Mean Method r = ^y _ Zxy J_ J N. Statistics for Economics-XI Merits and Demerits of the Rank Method Merits : M^hT " » . grouped frequency dis^ibnrion (bivtiaSs^l^n' "' OF FORMULAlt 1.compared . y = (Y . dxdy = Multiplying the deviations taken from assumed mean of X series with the deviations taken from assumed mean of Y series. ay = Standard deviation of Y series. X = (X . dy = (Y .Y).N _N N N N1XY-EX. deviations taken from actual mean of Y series. Spearman's Rank Coefficient of Correlation .IN . Measures of Correlation (c) Assumed Mean Method and Step Deviation Method r= 347 N N Explanation of Symbols r = Karl Pearson's Coefficient of Correlation. deviations taken from actual mean of X series. of pairs of observations dx = {X . 2.A) deviations taken from assumed mean of Y series.IY zxy-n.x. ZX = Sum of the values of X series ZY = Sum of the values of Y series ZX^ = Sum of square of the values of X series ZY^ = Sum of square of the values of Y series ZXY = Multiplying X and Y values and obtaining the total N = No. ax = Standard deviation of X series.y I------------^^ ± —1\.X). deviations taken from assumed mean of X series.A). (a) Positive and negatives correlation ib) Lmear and non-li„ear correlation. 11 12. does it mean that variable . m = Number of times the value repeated. 348 exercises j„ . 6. partial and multiple correlation Give three examples of perfect correlation. 13. 14.ro. Questions .^^ T™ ^^ — Distinguish between : '^o-elation from coeffidem of variation. D = Difference of ranks. 7. ~ " fea What is meant by correlaHon? wi. 5.!^^::. N = Number of pairs of observations.^^. ' ^e^pl^r between X and y is . Statistics for Economics-XI 4.^^ t » Distrnguish betv^e. 10.rk = \=1N^-N N^-N rk = Spearman's rank correlation.' and ic) Simple. What kind of relationship exists between X and Y? 17. 9.^/^.He scaler Distinguish between covariance and variance. negaL or no J (/) Sale of woollen garments and the day temperaturedCflllrrtfou^^^^^^^^^^^ /J" 'he po. 26 35 57 68 8 12 9 11 349 Me^swres of Correlation 15 What are the advantages of Spearman's rank correlation over Karl Pearson's correlation coefficient? Explain the method of calculatmg Spearman s rank correlation coefficient.. ia) Define Spearman's rank correlation.„« .d^r Smir born and export over last correlation): correlations (positive.legate product moment r . Write short notes on : (a) Spurious correlation. (a) How is Karl Pearson's coefficient of correlation defined? (b) What are the limits of the correlation r? (c) If r = +1 or r = -1. 16.we .correlation be equal to the val^^^? ^^ ^ -d Y. and . ib) What are the limits of rk ! ' ^aSl. will this 8. Height im inches) : 72 60 63 66 70. draw an estimating hne. 11 9 6 12 11 3 Following are the heights and weights of 10 students in a class. Construct the scatter diagram of the data given below and interpret it Average value (in Lakhs Rupees) Year Cotton (import) Cloth (export) : 1990 : 47 171 1991 64 1992 100 1993 97 : 70 100 111 133 85 {expuri/ 103 139 1994 126 . ib) Do you think that there is any correlation between profit and capital employed. Problems : 1. Is it positive or negative? Is it high or low? (c) By graphic inspection. 75 58 78 72 62 Weight (m kg) : 65 54 55 61 60 54 50 63 65 50 4. Capital employed (in crores of Rs) (X) Profit (in lacs of Rs) (Y) (a) Make a scatter diagram. 2 Plot the following data as a scatter diagram and comment over the result : ' X ■ 11 10 15 13 10 16 13 8 17 14 y : 6 7 9 9 7 .[b) Positive and negative correlation. Draw a scatter diagram of the following data : 50 40 60 28 1995 203 1996 . 32 20 " 24 36 6. Draw a scatter diagram for the data given below and interpret it X: 10 20 30 40 y. Give the following pairs of value of variables of capital em^ployed and^profit^. Draw a scatter ■ diagram and indicate whether the correlation is positive or negative. 70 48 80 44 XY 15 7 18 10 30 17 27 16 25 12 23 13 30 9 f! 350 • -. Statistics for Economics-XI X Series Y Series Arithmetic Mean Square of deviations from Arithmetic Mean ■ -------J__ Summation of products of deviatioZl^lTZTV'^---means = 122 ^ ^nd Y series from their respective Number of points of values = 15 15 25 18 25 39 41 . . ^ 11 20 12 15 73'' ^^ ^ certain shop for 12 months. and ^ .^o 550 « 16 17 20 19 19 20 25 27 11. Ten students got the following percentage of b ■ c • f'" = Ser. ^ ^ =Vl4 8 18 10 22 9 3 . n 13 14 .^^ .24 27 30 36 27 31 33 35 Calculate product moment of correlation between X and Y ■ . Calculate correlation coeffident between X the nnn.- . Calculate product moment correlation ^^ ^^ ^^ ^^ ^^ 60 13.S 2 2 .98] 91 95 49 40 SliL" Z r ''' whea. the number of rain coats solrl in ot rainy days per month Interpret the results.al No. • ^^ = " 3ri Y.K . 1 ^ 'J' ^^ Statistics and Mathematics Statistics : gO 60 51 76 58 J J ' ^ 10 Mathematics 45 yj ^^ 62 64 72 56 58 . (/« Rs '000) ^ 15 14 13 j3 12. .42 41 48 45 [r = 0.. Calculate Karl Pearson's coefficient nf i l ^ '' = 0-1621] the following 10 firms : ^^ correlation between the sales and expenses of Firms 12 3 ■™---JJ Expenses . Ky .2 -2 -1 < Calculate Karl Pearson's coefficient of correlation and interpret the resu.^ 3 4 7 10 11 29 [r = . Students Marks (in Maths) Marks (in Statis. 9 8 10 12 11 13 14 8 16 16. 4 -3 -2 -1 0+1+2+3+4 : j -3 -4 0+4+1 .931 Calculate the correlation coefficient of the marks obtained by 12 students m Mathematics and Statistics and interpret it.) A 50 22 B 54 25 C 56 34 D 59 28 £ 60 26 F 62 30 G 61 32 H 65 30 3 .0.t. ^^ ^ ^^ 15 Find the product moment correlation of the following data : ' X • 1 4 5 6 7 2 Y .67] Measures of Correlation 351 ^ f The deviations from their means of two senes (X and Y) are given below . 9 15 [r = +0. '" «''«.783] 67 68 68 72 69 VO 71 73 69 70 [r = 0.-'' Costofltvtng : 98 99 [r = +0. and expenses of the following 10 firms. A 18 Find Karl Pearson's coefficient of correlation from the following index numbers and 'f.47] 17. The height of fathers and sons are given below : Height of fathers (in inches) : 65 66 67 Height of sons (in inches) : 67 68 64 Calculate Karl Pearson's coefficient of correlation.85] 19. . 50 13 3 55 14 4 60 16 5 65 16 6 65 15 .7 7 K ^ 67 71 71 74 28 34 36 40 [r = + 0. Firms Sales Expenses 1 50 11 2. Find the product moment correlation between sales. 82] 21 Find suitable coefficient of correlation for the following data : . 23. I6OO 1 Ton lonn ^ ^ 10 12 Co^MmRs) 1800 1900 1700 Calculate coefficient of correlation K.797] Calculate the coefficient of correlation for the following ages of husbands and wives in years at the time of their marriage.99] 352 22.836] population and death rate^ ^^^^^^ between the density of C 400 14 . Statistics for Economics-XI « age of ca„ and annua.7 65 15 8 60 14 9 60 13 20. Age of husbands : 23 27 28 28 29 30 31 Ageofu^ives : 18 20 22 27 21 29 27 33 35 36 29 28 29 [r = +0. matoenance Age of cars (years) . P. p . ^^ = +0. 2 4 Annual maintenance 2100 2000 ■.^iFertUizers used ^ tons) : 15 18 20 24 30^ 35^ Productivity (in tons) ■ ^^ 40 50 150 160 (r = +0.. 10 50 13 [r = +0. 8288 series : Arithmetic Average = 65 •• Standard deviation = 23 33 r series : Arithmetic Average = 66 Standard deviation =14 9 Rauk Correlation 26.^ ' coefficient of correlation. ir = +0.78] Calculate Spearman' r .988] Total of the deviation of X = -170 Total of the deviation of Y = -20 Total of squares of deviation of X = 2264 lotal of the squares of deviation of Y . F 300 13 [r = +0." n Judge X Judge Y 20 22 20 15 14 .Citites Density Death rate A 200 10 B 500 16 D 700 20 E 600 17 24. The following are the marks obtained fout nf mm u t'" = +0-143] emp oyment interview held by two Ldependem . ^n^ependem judges separately Calculate the rank Candidates ■■ A n n r^ '' . „on from the following data : ^ •• 15 10 I '' 20 25 40 ^ 16 25 12 8 27. 13 13 24 6 15 4 20 9 6 11 12 11 1 [r = -0. 12 96 2 3 4 10 354782 5 6 7 8 9 10 Calculate rank coefficient of correlation. 29 Calculate rank coefficient of correlation of the followmg data : ' X : 80 78 75 75 68 67 60 Y : 12 13 14 14 14 16 15 ^^ _ ^^^^ 30. 7 8 9 10 11 ^7 9 8 2 1 3 12 11 4 10 6 futeB : ^ : 5 3 11 2 12 10 5 9 7 Calculate Spearman's rank correlation.454] 59 17 12 5 8 [r = +0. 31 Calculate the coefficient of rank correlation from the following data X ■ 48 33 40 9 16 16 65 25 15 y .721] 353 Measures of Correlation 28 Two judges in a beauty competition rank the 12 entries as follows X: 1 y.10 8 11 12 13 9 [r = 0.73] . They were ranked by two judges as under : ^ . Twelve entries were submitted in a flower show competition.86] 57 19 [r = +0. 3.32. Persons Years of Service Efficiency rating A 24 66 B 30 51 C 12 84 D 25 66 E 29 45 F 19 81 G 16 72 H 10 97 IJ 11 7 92 70 [r = -0.78] 33. From the following data calculate coefficient of correlation by the method of rank 95 70 60 80 81 150 115 110 140 142 XY 75 120 68 134 50 100 [r = +0.93] 1. 2. Calculate rank coefficient of correlation between years of service and efficiency rating. . 8. 6. 5.ces of price of some commodities may'alat illustration : ^ ^^^ » 'T -™°''it. in measures of rp r ion-ab j^^^^^ Po-'ional values : Deviation.^l^tZSieThlt: rtTp^t I P^. Mean Deviation and tondat Devia.l"t"" two variables. 11. ^^^^ "s examine from the following Commodity Prices of ve^ 2000 ?etable oil and tea . Chapter 12 IWTRDOCUTION TO INDEX NUMBERS Introduction Definition Types of Index Numbers Problems in Construction of Index Numbere Methods of Constructing Index Numbers Consumer Price Index (CPI) Index of industrial Production (IIP) General Uses of Index Numbere Inflation and Index Numbers Limitations of Index Numbers List of Formulae li'ftSWf INTRODUCTICm ■ have r'r' we Quantities. "-is "" """"nary measures of change in a group of related ail . 7.4. 9. From the above tookAe s™ ' association of chapter we will learn how to obtS sum jrr"^ ""T? variables.es. 10. Vegetable oil (per litre) Rs Tea (per kg in Rs) 40 100 ^yjyjj 80 150 ti c Introduction to Index Numbers 355 We can measure the change in the prices of vegetable oil and tea in two ways : (a) Actual Difference (b) Relative Change (Price Relative) (a} Actual difference. The actual difference in price is the difference between the current year price and the base year price.100 = Rs 50 Relative change = Actual difference Base year Price Relative change in vegetiable oil : 80-40 or 1Current year Price Base year Price For vegitable For Tea 40 . From the above example : 80 . The relative change in prices is the actual difference in prices relative to the original price. From this. it appears that the increase in price of tea is more than the increase in price of vegetable oil. (b) Relative change (price relative).40 = Rs 40 150 .Base year price Current year : 2005 Base year : 2000 Difference in : Vegetable oil (per litre) Tea (per kg) We find that the rate of vegetable oil is increased by Rs 40 and of tea by Rs 50 from the year 2000 to 2005. Actual difference = Current year price . Current Year Price Base Year Price iL Po 100 100 356 P^ . if we calculate the rise in percentage taking 2000 as the base year. we. However.150-100 100 =1 or 1= 0. find that the rise is 100% of vegetable oil and 50% in case of tea.5 or 1 40 150 100 =1 = 0. Symbolically.5 This change can also be expressed in percentage : For vegetable oil : 1 x 100 = 100% And for tea : 0.price of the current year (2005) Pa = price of the base year (2000) Statistics for Economics-XI Vegetable oil : x 100 = 200 Tea 40 m 100 = 150 .5 x 100 = 50% The ratio of prices in two years is called price relative which is a pure number and this price relative for a single commodity even may be called an index number of that commodity. e. iv " i of measurement. Tley are usually IXJIZI^Z^ . Index numbers are the bafo^rlX^^VrrnX:'™''''''"■ definition orhif Je-chats^—trd r Tllnr IrT " "r » commodtty to another.M Increase of vegetable o. sales.75 it c!:. export.. ™ " "" (b) The relative comparison o o^e T : ^000 = 100. Now.00 = 1. 80 150 + 40 100 2 + 1. Similarly. "Sncultural and industrial economy.s .i hat Dearness Allowance of Governlm eij^ » ".5 = 1.dex goes up. important than just the As measurement of veeerahip nil . "" "Ot be combined.. change in pncrrrll ^"-at of tea. actual difference in prices.l is 100°/Th f .v. I.5. wages «c Thev ar7 f ."seeher . prices. .. mcreasedby27%. But relative changes or price relattons are pure numbers namely for vegetable „„ Rs 80 Rs150 R740 ^^^ tea Rs 100 be combined to obtam the arithmetic mean of price relatives. we come acroTs rdex ^b '"T"''-production..nthe!ame„ay cltsi'XL''"" "f . Thus..5 . their absZe duffel l "" . hospitals etc. Index numbers are expressed in terms of percentages so as to show the extent of relative change. ttpes of index numbers The usefulness of any index number lies in the types of questions it can answer. Wholesale Price Index (WPI) 2. production of various agricultural crops. percentage sign (%) is never used. The technique of index numbers is utilised in measuring changes in magnitude which are not capable of direct measurement due to composite and complex character of the phenomenon. they can broadly classified as under : 1. "Index numbers are devices for measuring differences in the magnitude of a group of related variables". Index number are called SpeciaUsed type of averages in the senses that they help us in comparing change in series which are in different units. Averages like mean." According to Croxton and Cowden. geographic location of other characteristics. 'cost of living'. median and mode can be used to compare only those series which are expressed in the same unit. Examples of such phenomena or magnitudes are price level. Each index number is designed for particular purpose. Consumer Price Index (CPI) or Cost of Living Index . persons. and it is the purpose that determines its method of construction There are various kinds of index numbers. Index numbers are relative or comparative measurement of group of items. However. prices of specified list of commodities. 4. Thus. 2. They compare changes taking place over time or between places or like categoriesschools. In economics and business. "An index number is a statistical measure designed to show changes in variables or a group of related variables with respect to time. Changes in business activity in a country are not capable of direct measurement but it is possible to study relative changes in business activity by studying the variation in the values of same such factors which affect business activity and which are capable of direct measurement. 'business or economic activity' etc. volume of production in different sectors of an industry. the characteristics of index numbers can be heighhghted as under : 1.Introduction to Index Numbers 357 According to Spigel. 3. we have . Sensex is a useful guide for the investors in the stock marter If rh. It does not include the items pertaining to services like repairing charges. Wholesale price index (WPI) is used to measure the general price level where we are required to obtain the wholesale prices of industrial. If the object is to measure relative change in industrial production. Purpose : Every index number has its own particular uses and hmitations. some point of reference is to be decided. working class (labour) or agricultural workers. In the above illustration about prices of vegetable oil and tea. number of Agriculn.3. If the purpose is to measure cost of living of middle class families. WPI is used to eliminate the effect of 358 Statistics for Economics-XI the Priva" str" "he public and 4. Producers Price Index /PPT^ Introduction to Index Numbers 359 in construction of index numbers Following are the important problems which must be well defined for the construction of index numbers : 1. in a particular region or city. Index number of Agricultural Production (L\P) 5. Sensex 1. agricultural and other products from wholesale market. If the purpose is to measure the general price level. Selection of base period : When comparison is to be made between different time periods or different places. •■ appropnate time for mvestment. then wholesale price index number is used.ral Production (lAP) is used to study the rise and fall of the yteld of pnncpal crops from one period to other period 5. then consumer price mdex number is used. 2. The first and foremost problem in the construction of index numbers is in regard to the objective or the purpose for which they are required. This is called base. barber charges etc. Index number of Industrial Production (IIP) 4. then index number of industrial production is to be used. It is important to know what is to be measured and how these measures are used. The rise in sensex at the highest level reflects the base''??S'valut'oft°hT'°™ Bombay Stock Exchange Sensitive todex with 1978-79 as iiuinoer will replace wholesale price index. fashion. However sometimes chain base method is used. (b) The base period should not be too near or too far : This is because people usually prefer to compare present conditions with conditions in base or reference period that is not too far back time. However. Due to introduction of new commodities. (c) The base period should be normal and representative period : Base period should be free from all sorts of abnormalities and random or irregular fluctuations like earthquakes. Naturally. (d) It should be a period for which actual data is available Fixed base and chain base : If the period of comparison is kept fixed for all current years. in economy many commodities may go out of use. 3. since there is a large variety of goods and prices. labour strikes. it is nof^"s bk « include all commodities in construction of index numbers be considered : . it is called fixed base period. floods. For making the comparison over a period of time it must be remembered : (a) The base period should not be either too short or too long : It should be neither less than a month nor more than a year from calculations' point of view. Care also must be taken that data from unrelated commodities or periods are not grouped together for the calculation of price index. the chain base method gives a better picture than what is obtained by fixed base method. On the other hand. The base is assigned the value of 100%. In such situation it becomes necessary to shift the base period. famines. change in habits. wars.taken year 2000 as the base year and 2005 as current year for our calculations' of index numbers. taste. Selection of items : Collection of data is a special problem in constructing index numbers. '' ^^^^ calculations. in which the changes in the prices for any given year are compared with prices in the preceding year and not with the fixed year. a choice 360 Statistics for Economics-XI of some representative items has to be made. If the base period is too far the comparison becomes meaningless. lockouts. If the number of the commodities is too large. inclusion of too few Items would make the index number unrepresentative of he ™ mcrdfal. economic boom and depression. much would depend upon the purpose of constructing the index. 8. therefore. it is called value weighting. In the same way for coLtruction ofZsle ^^ — which-sm^nt toT W ^^^ ^^^ be neither too ot data.be either implicit (arbitrary) or explicit (actual).also I Introduction to Index Numbers 361 6. in const uction of wholesale price mdex number to know the general price level. Choice of an average : For constructing an index number any average such as mean. we sZw nclude wholesale prices of some major mdustrial and agricultural ^olmoS and Other goods and services. mode. System of weighting : In order to allow each commodity to have reasonable influence on the index it is advisable to use a suitable weighting system. Unweighted index numbers are those where all commodities are given equal importance.^be changes m mdustrfalToI we must collect the prices relating to production of various goods of factories For nlrh^'. 7. Weighting may be done according to : (a) Value or quantity produced. w! can re y onT^^^^^^^ published market reports by business houses. weights are assigned to the various items. For example. The method of weighting used would depend on the purpose of index. Chamber of Commerce aS L utilised " Pl-. and (c) Value or quantity sold. The geometric mean and harmonic mean are difficuh to calculate hence. Though with the development of the use of electronic computers. When the quantity is the basis of weight it is called quantity weighting and when the value is the basis. But in most cases different commodities are given different degrees of importance. ' ^^ ^^^ consideration. the use of geometric mean is also becoming popular.(«) Commodities selected should be relevant and representative of the group according to the purpose of mdex number. (b) Va ue of quantity consumed. Choice of method : There are various methods of calculating index numbers such as the aggregative method or the price relative method. arithmetic mean is used. Weight may . Various methods have been proposed for calculation of weighted index number such as Laspeyre's . median. Ihere is always a chance of getting spurious or misleading results if the w ^^ r . geometric mean and harmonic mean can be used Frorn the practical point of view median and mode are unsuitable because of their being Latic. the data available and the resources at the disposal of the person or organisation constructing the mdex number. Marshall Edgeworth's method. In this method.method. . Fisher's method is considered as ideal for constructing index numbers. No single formula can be said to be appropriate for all types of index numbers and as such the choice of a formula wi 1 have to be made taking into account the object of index numbei. » constructing indei The methods of construction of index numbers are given below : Methods _ Price 362 Statistics for Economics-XI Unweighted Index Numbers I. Kelley's method and Fisher's method. Dorbish and Bowley's method. total of the current year prices for the various commodities is divided by the total of base year prices and the quotient is multiplied by 100. Symbolically. Simple Aggregative of Actual Price Method This is the simplest method of calculating index numbers. Price Index Po^ = Quantity Index ^01 = ^Po X 100 100 where. Paasche's method. Calculate price index number for 2005 taking 1995 as the base year trom the following data by simple aggregative method. Commodities Prices in 1995 (in Rs) Prices in 2005 (in Rs) Solution.Poi = Current year price index number ^01 = Current year quantity index number Zp^ = Total of current year prices for various commodities ^Po = Total of base year prices for various commodities = Total of current year quantities for various commodities = Total of base year quantities for various commodities Illustration 1. I A B C D E 100 80 160 220 40 140 120 180 240 40 Construction of Price Index Number Commodities A 100 140 B 80 120 C 160 180 D 220 240 E 40 40 Zp^ = 600 Price in 199S (Rs) Price in 2005 (Rs) Xpj = 720 It: exi Introduction to Index Numbers 363 Steps . = = ^ X 100 Zpo Here.1 Aggregate the current year prices of various commodities (Ep.. p = Price index number of the current year (2005) Zp = Total of current year prices for various commodities Zp .^. we get = ^xlOO = —X 100 = 120 thus. ISrelate the base year prices of various commodities W 3.) 2.Total of base year prices for various commodities Now.:: Number for 2004 wirh base 1995 from Ae^followmg Commodities ■■ ^ ^^^ jj 5 80 60 20 10 2 6 1995 Quantity (kg) 2004 Quantity (kg) Calculation of Quantity Index Number Commodities ABCDE Total Quantity in 1995 (in kg) (qp) ___ _ 40 10 52 ■Quantity in 2004 (in kg) Iq^} •^ ^ . Apply the following formula : Po. the pncr^. we get ^^ . 364 j„ .48% as compared to 1995.100 ^01 E^o Here. Zq.Uo = 107 80 60 20 10 6 24.48 ^01 107 extent of 64. Statistics for Economics-XI Illustration 3. 1996 14 1997 16 1998 20 1999 22 2000 24 Calculation of Index Numbers (Base year 1995) Price Index Numbers ^ ^xlQO Pff 1995 10 100 1996 14 ~ X 100 = 140 10 1997 16 16 jQ X 100 = 160 . = 176 _ IZi X 100 = 164. = 107. of 2004.. 101 = Quantity index no. data (Base Year 1995). ^ 1996 to 2000 from the following Year : 1995 Price : 10 Solution. Compute index numbers for the vears 1996 innn ( u r . = 176 Applying formula. and Zq. 'a : : Price Relatives SL X 100 . Construct pure index number for 2005 taking 2000 as the base year from the following data by simple average of price relative method.140 100 B 80 120 120 „ " X 100 = 150 80 . Construction of Price Index Number Commodities Price in 2000 (in Rs) Price in 2005 (in Rs) Prices in 2000 Commodities Prices in 2005 . 'A price relative is the price of the current period expressed as a percentage of the price at the base period'. :: Pov . ABCD L 100 80 160 220 40 140 120 180 240 40 " Solution. Index IS influenced by the items with the large unit prices II. wiiixi^iiL ycai Po = Price of the base year Limitations 1. Simple Average of Price Relatives Method X 100.1998 20 1999 22 20 jQ x 100 = 200 24 10 2000 Hprp -h — 22 ^ X 100 = 220 24 c ^u 24 Y^ X 100 = 240 » 1 ----'I'* ^.. No weight is given to the relative importance of items 2. Index number by this method is the arithmetie mean or median or geometric' 365 Introduction to Index Numbers Illustration 4.ixv. ^ A 100 140 ^^^ X 100 . Calculate the price relatives of current year |LxlOO 2. In other words.C 160 180 D 220 240 X 100 = 109.32 N~s m th^nrVt' T" 2005 is 122. = Price index number of current year (2005) 366 j„ . p^^ = ^xlOO {Po 611. Apply the following formula : >1 Po^ = Po xlOO N Here.6 = 122. 4.100 40 Total • Z^x 100 F 611.32.6 Po Steps : 1. Divide the total of price relatives by number of commodities. Aggregate the calculated price relatives. there is net increase m the prices of commodities m the year 2005 to the extent of 22. Statistics for Economics-XI ^ X 100 = Price relatives of current year N = Number of commodities Now we get. 3.1 220 E 40 40 "^^x 100 .32% as compared to . ^ 2. Index number is not influenced by extreme items. Paasche's method. This assumption may not be always correct. This assumption ^od appropriate pportance in the of assigning ndex numbers r and Rowley's hod. There is a problem of selecting a proper appropriate average. Fisher's ' syllabus of ucting index ' "" " ' ^ '--'■'a 'O ger quantity index n . Marshall Edgeworth's method. Weighted Aggregative Method Weights are assigned to the various items. There are various methods of assigning merk r J Tn Laspeyre's method.Merits 1. Equal importance is given to all LfxC? ItvlliS* Limitations 1. According to syllabus of Class XI. The relatives calculation are assumed to have equal importance. Kelley's method and Fisher's method Fisher's constructing index numbers. Weighted Index Numbers weil?*''''^ '' 1" appropriate weights are assigned to various commodities to reflect their relative importance in the I. we are discussing here Laspeyre's and Paasche's method of constructing index Laspeyre's Method Paasche's Method Price Index fo: = X 100 Zpolo Quantity Index = X 100 ZqoPo X 100 Quantity Index = X 100 ZqoPi Price Index = v^^ for Economics-XI j Introduction to Index Numbers jthere is net increase 2% as compared to ^nce is given to all fquoted or absolute !. Dorbish and Bowley's method. T'''' ''' Laspeyre-s metltod is very widely used" Ti. ha The which are not changed front one^yert^n « S„ Te r^ T ^ V' Value Index Number = Base year values V = ^lilyinn •^01 xioo or = XlOO where XVo Poi = Price index number ^01 = Quantity index number Vgj = Value index number Pi = Current year price pQ = Base year price = Current year quantity = Base year quantity V. and' the'^na^..denoted as . „ ^t™ ylJlt^^^. = Current year value (Zp^q^) Vp = Base year value {Lp q ) Commodities ABCD 1996 Base Year Price 10 8 6 4 Quantity 30 15 20 10 2005 Current Year Price 12 10 66 . = quantity of current year (200J) Po = price of base year (1996) . (Price Index nuntM " otrC.Quantity 50 25 30 20 368 Solution.'™' ^ ""^hts. ° quantity of base year (19961 (A) Laspeyre's Method. Construction of Price Index Numbers Statistics for Economics-XI Year (ZOOS) 1 Quantity It PSt Ptlo 50 25 30 20 300 120 120 40 120 Pitr 500 200 180 80 360 150 120 60 600 250 180 = 580 = 960 = 690 = 1150 where 1.00. The above steps give us Laspeyre's Price Index Number Symbolically. In this merhoH K Steps. contntodtties „.^^ urease m the Laspeyre's Quantity Index : 18-96% as compared to 1996. . X 100 690 = J^ X 100 = 118.96 Thus the price index number of 2005 is 1 IS v pnces of commodities m the year 2005 to the ex.^^ "" —« wtth base year weights and TptfL^t' ' * "-'ent by .th base year weights and oti^i. 52 JOU Thus. Symbolically. Paasche's Method : p 3. Steps. Paasche's Quantity Index : _ = ^fiPL X 100 ZqoPr 1150 690 xlOO = 166.67 .ty of commodittes n.ant. Mukiply current year prices of various commodities with current year weights and obtain Spj^j. Divide by JLp^q^ and multiply the quotient of 100. taken as weights. In other words. there is net increase in prices of commodities in the year 2005 to the extent of 19.. the year ^OO^l'^^^irroH^I^t™ Introduction to Index Numbers 391 (B) Paasche's Method : In this method current year quantities are.= |»xlOO ^oPo 960 ^01 = 580 XlOO = 165.79 Thus.79% as compared to 1996. Multiply the base year prices of various commodities with the current year weights and obtain 3. (Price Index Number) 1. 2.79. the price index number of 2005 is 119. the quantity index number of 2005 is 165 .^x100 1150 960 X 100 = 119. Value Index Number : Zpo^l 1150 580 X 100 = 198. Weighted Average of Price Relatives Method Illustration 6. Statistics for Economics-XI Price 2004 20 15 10 5 4 Price 2005 Commodities ABCDE Weights (w) % . In other words.67.28% as compared to 1996.28 Thus.67% as compared to 1996. there is net increase in quantity of commodities in the year 2005 to the extent of 66. the quantity index number of 2005 is 166. n. Calculate weighted average of price relative index number of prices for 2005 on the basis of 2004 from the following data : 370 Commodities AB CDE Solution. there is net increase in value of commodities in the year 2005 to the extent of 98. the value index number of 2005 is 198.28. In other words.Thus. Weights 20 12 8 4 6 j„ . nr fPl/l or [PV] . calculated pereentages for each ttem hy value weights.20 12 8 4 6 Steps : of Price Index Numbers 35 18 11 5 5 Price 2004 20 15 10 5 4 Price 2005 35 18 11 5 5 Value weights (PolJ [V] 400 180 80 20 24 Pi ^xlOO ypo fp] IV = 704 175 120 110 100 125 [PV] 70000 ■21600 8800 2000 3000 IPV = 105400 Calculate the price relatives of the current year fexiool ■ Item of the period for which the in^ "Jexpressmg each of the same item in the tse pL^" ca4>ated as a percentage p. People consume different types of commodities. They measure the change in the cost of living of a particular section of society due to change in the retail price. Consumer price index numbers are designed to measure the average change over time in the price paid by ultimate consumer for a specified quantity of goods and services.75J. The general index number fails to reveal this. i.71 tr in n CO im fin coi pu] are Introduction to Index Numbers 371 somer price index (cpi) The wholesale price index numbers measure the changes in the general level of prices and they fail to reflect the effect of the increase or decrease of prices on the cost of living of different classes or group of people in a society. Consumer price index numbers are also called (/) Cost of living index number. People's consumption habit is also different from person to person.Po./I -n ./l /o as compared to 2004. Construction of Consumer Price Index . or (ttt) Price of living index numbers. = — XlOO \Po ^PoqoZV . or ^^ _ 105400 = 149.e.= i^y.ere . middle class and poor class. or («) Retail price index number.. place to place and class to class.s a net mcrease ^^'. So there is the need to construct consumer price index. A change in the price level affects the cost of living of different classes of people differently. richer class. the consumer price index for industrial workers is by far the most popular index. government employees etc.■ cCect tetai. The CPI for industrial workers is increasingly considered as the appropriate indicator of general inflation. (ii) the urban non-manual workers.. e. city. rural area. Pnce must relate to a fixed list of items for a fixed quality. 2. CPI measures changes in retail prices of goods and services covering 260 items of consumption from 70 centres. The enquiry is done on a random sample basis.The following are the steps in construction of consumer price index : (1) Determination of the class of people : Consumer price index numbers are constructed separately for different classes of people or groups or sections of the society. Retad price must be the price which is given by the consumer. urban wage earners. which shows the most accurate impact of price rise on the cost of living of common people. (/■) Food («) Clothing (Hi) Fuel and Lighting (iv) House Rent (v) Miscellaneous the "--i-s is very in. This is constructed on monthly basis with lag of one month.'and also for different geographical areas like town. hilly area and so on. and (Hi) the agricultural labourers In India. as also prices at which they are purchased are noted down.pona„t in to shop and W 17. then we have to decide about the low paid or high paid government employees as their consumption pattern differs. The quantity of the commodities consumed. When we talk of government employees. industrial workers. and their family budgets are 372 Statistics for Economics-XI Jt''™^ "" - groups. (2) Conducting family budged enquiry : Family budget enquiry is held with a view to find out how much an average family of this group spends on different items of consumption. The major groups of consumers for whom the consumer price index numbers have been constructed in India are : (/■) the industrial workers.g.^^^ 1. The class for which the index number has to be constructed must be as far as possible homogeneous from the point of view of income and habits. agricultural labourers. The base year of CPI (IW) is 1982. The pHncpies ^tSroTtr. priee. The group has to be clearly defined. . Some famihes are selected from the total number by lottery method. urban area. Calculate by : . necessitating an upward adjustment in the wages and salaries of employees The rise of wages or salaries is equal to the amount of percentage it exceeds 100. M drscount U given to all customers. it means the cost of hving has decline by the balancing percentage between 100 and calculated index number. First we musfSaToecw" >e"ts or mail must be instructed properly anTK listTtem . Illustration 7. W = Weights Introduction to Index Numbers (ii) Aggregative Expenditure Method or Aggregative Method : Consumer price Index = ^^ x 100 Zpo^o This is based upon Laspeyre's method. What is the cost of living index of 2004 as compared with 1995. If the calculated cost of living index number is more than 100. An enquiry into the budgets of the middle class families in a certain city gave following information. it means a higher cost of living.3. it can be taken into account. If the calculated index number is less than 100. R = ^x 100 for each item Po. In ttcTt'oft ° T""' """ " """ OP'" price questiomtaires. According to this method.'^^^^ " be conducted by "cU pLngC^thllhSm? for rhfferenr classes <ff Peo-pl^Tltrnd^^r^ ^r^^^ Methods of Construction (t) Famdy Budget Method or Weighted Relatives : IWR Consumer Price Index = IW Here. the various items are given weights on the basis of quantity consumed in the base year. A (R) Weighted Relatives (WR) Food 35 1400 1500 107.) Price Relative . and (ii) Aggregative expenditure method. (i) Constructing Cost of Living\Index Number (Family Budget Method) Food Fuel Clothing Rent Misc.(i) Family budget method. Expenses on items Price (in Rs) 2004 Price (in Rs) 1995 Solution. 374 .9 Cost of living Index for 2004 CPI = LWR 13449.5% in prices of 2004 with that of 1995.9 3000 IWR = 13449. there is increase of 34.499 ZW 100 Thus.9 = 134. 20 250 400 160 3200 10 ZW = 100 Price (in Rs) 2004 3749.14 Fuel 200 .250 125 1250 Clothing 20 500 750 150- Rent 15 200 300 150 2250 Misc. 35% 20% 15% 20% 1500 250 750 300 400 1400 200 500 200 250 10% Expenses on item Weights (%) (W) Price (in Rs) 1995 (Po) (P. 10. Weights 35 lb 20 15 20 (ii) Aggregate Expenditure Method Statistics for Economics-XI Price (in Rs) 1995 Po 1400 200 500 200 250 Consumer Price Index for 2004 Price (in Rs) 2004 Pi P<Ao Pi'Jo 1500 • 250 750 300 400 49000 2000 10000 3000 5000 8000 ^Po^o = 69000 52500 2500 15000 4500 = 82500 CPI .^Pi'^o 82500 100 " « ^ 100 = 119. 200 25 20 40 65 Solution.565 items are 75. and 4 rewrtvdv Pr^r ""^^ts of these hving for 2005 with 1980 as JS hSe """ber for eost of Items ' Price in 1980Price in 2005 Food Clothing Fuel and lighting House rent Misc.Expenses on items Food Fuel Clothing Rent Misc. _-^Consfructing Cost of Living Index Number 100 20 15 30 35 . 6. 5. The consumer price index for June 2005 was 125.47 100 = 184. The food index was 120 and that of other items 135. 100 = Wj + We are given consumer price Index =125 IWR .47 Introduction to Index Numbers Cost of living Index for 2005 CPI = 375 IWR IW 18459.(1) CPI = 125 = IW 120 Wi +135 W2 .71 15000 1250 666. IW = 100 135 W. there is increase of 84. Items Index Weights (I) Food 120 WI :m w.33 185.Items Weights W Price 1980 (Pol 100 Weighted Relatives (WR) Price 200S (PJ Price: Relative R Food Clothing Fuel and lighting House rent Misc. 100 .84 ZW = 100 IWR = 18459.6% in prices of 2005 with that of 1980„ Illustration 9.65 799. What is the percentage of the total weight given to food? Solution. W^ = Food and W^ = other items Hence. IWj == 120 Wj + 135 W^ Let the total weight = 100.594 Thus. Other items 135 120 Wj w. 75 10 5 6 4 100 20 15 30 35 200 25 20 40 65 200 125 133.98 742.33 133. solving equation (1) and (2). percentage of total weighs given to food = 66.67 = 33.67 W. + 135 W^ 1000 = 15 Wj W.or 12500 = 120 Wj + 135 W^ •(2) Now.67 33. 100 = 12500 = We get 13500 = 12500 = .66.33 Hence..33 IW = 100 W. 376 Verification Statistics for Economics-XI Items Food Other items Index (I) 120 135 Weights (W) 66. X (135) Wj + W^ 120 Wj + 135 W^ 135 Wj + 135 W^ 120 W. = 100 .67% and for other items = 33. . = iff^ = 66.33% M..67 Now using equation (1). . we get 100 = Wj + 100 = 66. 25 F I 11 £ purchasmg power of rupee in 2004-05 Rs 32T:I-ch w" tr P" n-onth was 100 and for 2004. = 12500 CPI = IWI IW 12500 100 = 125 Consumer Price Index No.s wages^T^elrl^Telltagt""" """""" Real wage = Money Wages Consumer Power Index For 2000. . the consumer price index was 400 in 7(\[\a n^ u 100 m 2000-01.01 : ^ X 100 = Rs 3250 For 2004-05 : ^ X 100 = 1250 Introduction to Index Numbers 377 However the monthly money wage was raised from Rs 3250 to 5000 in 2004-05. Then a rupee in 2004-^5 wol be ^^uTto ' " "" 100 400 = 0. The worker has not gained.0J was AmwTlnA A Consumer pnce mdex for 2000-01 was by rise of h. price index is'called J prici^dX^fri^Ze ' {a) Purchase Power of Money = _ ^_ Consumer Price Index (fe) Real wages = Consumer Price Index Suppose. is 125 as given in question. Uses of Consumer Price Index Consume. The real wage of the worker is Rs 1250 in 2004-05 as compared Rs 3250 in 2000-01. In fact his real wage has gone down.8000 4500 IW. rice. due to shortages. wage policy and general economic policies. his salary rise should be of Rs 10.000 (16. The government (central or state) and many big industrial and business units use consumer price index numbers to regulate the Dearness Allowance (D. We are given : Year ^ Salary CPI Base CurrentRs 4000 Rs 6000 100 400 When 100 is the CPI of Base year. If the prices of some important essential commodities (like wheat. Solution. This compensates them for increased cost of living due to price rise.A. They are the best measures of economic progress in any country.) or grant of bonus to employees. If the salary of a person in the base year is Rs 4000 per annum and the current year salary is Rs 6000.000 Hence.) increase. 2.6000 = Rs 10.000) in current year to maintain the same standard of living. cloth. Consumer price index numbers are used widely in wage negotiations and wage contracts. Costs of living index numbers are used for deflating value series in national accounts. As such the first step in the construction of such index numbers is to find the level of output of various . bis salary is Rs 4000 400 CPI of current year. his salary should be 400x4000 100 = Rs 16. 5. 3. They are used for automatic adjustment (increase) of wages corresponding to a unit increase in the consumer price index. They are used by the government for the formulation of price policy. the government may decide to provide them through fair price shops or rationing. They tell about the relative increase or decrease in the level of industrial production in a country in relation to the level of production in the base year. by how much should his salary rise to maintain the same standard of living if the CPI is 400.Example 12.000 . 4. These indices can be constructed by studying variations in the level of industrial output. sugar. etc. mdex of industrial production (iip) Index numbers of industrial production are fairly common these days. In India. automobiles. aluminum conductors. railway wagons. earth moving equipment. cement machinery. caustic soda. Index of Industrial Production is published by Central Statistical Organisation (CSO). The old series of Index of Industrial . cars. aluminum. If the variations in the value of output are to be studied. mdices of mdustrial production are constructed either by studying changes in the quantum of production or its value. II. not in ^^ Statistics for Economics-XJ values. radio receivers. Thus. Petroleum.'aatlT''. electric fans. data about the value of mdustnal output have to be used for the purpose of constructing such index numbers. crude steel. crude (off-shore and on-shore) Iron ore. semi-finished steel. Mining Industries : Coal (inc.9 162. paper and paper bond. ' IV. ' but now new series is pubHshed with the base year 1993-94. petroleum refinery products. phosphatic fertihzer (P^Oj). cotton textile machinery. lignite). The general index of industrial production is the most popular among these.8 Usually important data about production are collected under following major heads: I. Electrical Engineering Industries : Power transformers. vitamin A. (commercial vehicles. land rovers). Mechanical Engineering Industries : Machine tools. III. jeeps. An abstract from Economic Survey 2005-06 is as under : General Index of Industrial Production Base 1993-94 = 100 1993-94 1998-99 2004-05 100 1999-00 2000-01 2001-02 2002-03 2003-04 145.6 189.industries of the country. Chemical and Allied Industries : Nitrogenous fertilizer (N). . . automobile tyres. soda ash. streptomycin.0 176. pig iron). steel castings. power driven pumps. It should be remembered that these index numbers throw light on changes in the quantum of production. chloramphenical powdei.6 167. agricultural tractors. bicycles sewing machines. Index of Industrial Production in India A number of index numbers of industrial production are compiled in India by official and non-official agencies. electrical lamps. bister copper. Industrial Statistics Wing. diesel engines. MetaUurgical Industries : Hot metal (inc. V. penicillin. bicycle tyres.2 154. electric motors.0 204. cement. salt. Weights are usually assigned on the basis of the relative importance of different industries. The weighted arithmetic average or geometric mean of the relatives give the index number of industrial production. staple fibre etc. Many other criteria of relative importance can also be laid down. Method of Constructing Index of Industrial Production Formula : Using simple arithmetic : Index No. Introduction to Index Numbers 379 The data relating to the production of the above mentioned industries are cq^lected either monthly. VII. soap etc.2 79. we find that the growth performances of broad Industrial categories differ.47 155. The production of the base year is taken as 1^0 and the current year's production is expressed as a percentage of the base year's production.0 From the above table. spuny and filament yarn. .36 222. The relative importance of industries is usually decided on the basis of capital invested. quarterly or yearly. in May 200S Mining and quarrying Manufacturing Electricity 10.17 General Index 213. the gross value of productions. coffee. Usually weights in an index number of industrial production are based on the values of net output of different industries. TextUe Industries : Jute textiles.7 10. Electricity Generated : Related to utility. Miscellaneous : Glass. These percentages are multiplied by the relative weights assigned to various industries. The following table shows broad industrial grouping and their weights. mixed/blended cloth.VI. cloth (cotton cloth). turnover. IX. tea. Usually important data about production are collected under above major heads. of Industrial Production (IIP) = where. Such index numbers can be constructed both for gross output as well as net output. vanaspati. VIII. Food Industries : Sugar. Broad groupings Weight in % Index No. net output etc. 50 1928.50 133.iL Uo W IW q^ = Current year Quantity produced q^ ^ Base year Quantity produced W = Relative importance of different outputs. Mining 120 160 20 2.66 3437. Textile 3. Statistics for Economics-XI Solution.57 2666. Electrical 1996 la nm 120 80 70 80 90 87.IuslnI ■ Milling 2.33 137.99 XW = 100 12220. Chemical 80 70 25 5. Mechanical Engineering 4. Mechanical Engineering 4. of Index Number of Industrial Production In. Industry Output (Units) 1996 2004 Weights 1.50 1999. Chemical 5. .33 -__ W ^xlOO % / \ 'SL ! w 160 110 90 70 120 20 25 15 25 15 133.20 . Construct Index of Industrial Production for 2004 from the following information. Textile 80 110 25 3. Illustration 13.50 128. ^ . Electrical 90 120 15 70 90 15 380 .55 2187. reserve bank deposits^r"row 1on T "" activities of a country and these md cesTan t u" could act as an economic blromeJer business -^ex which 2.Industrial Production Index = X iL W ZW 12220.„ Infme"r„gfc7:a7uf ^ nature into / Introduction to Index Numbers " 'be .20 22. To measure comparative chanirfe ■ tu measure relative temporal or crol^eS^n^^.20 100 = 122.20% increase of industrial production in 2004. general uses of index numbers ItTor^b™^^ can be summarised as follows : as barometers to find th^Ta^td^^^^^r T"" ^^ act barometers which are used'm physlTrmr" ^ L^ke measure the level of economic andTsmesTac ™ ""^bers barometers' or 'barometers of econorr ^ ^ ^'economic exchange.r''™ ""'"bers is to compared with same base figure Inde^^^^^^^^ ' "^"^^le or a set of variables time to time. among differentXe^and T " comparison of changes from m the phenomena like prTce £ cost ^^ ^ measured with the help'of ''' ^^^ ^^^anges ^^^^^^ -f --^g complex variable through time or space -- --red . This helps in determining the wage policy of a country. imports. 5. To help in study of trend : Index numbers are very useful in the trend or tendency of a series over a period of time. industrial production. With the help of index numbers of prices. Very often statements are made that purchasing power of the Indian rupee in 2000 is only 20 paise as compared to its purchasing power in 1990.) help in economic and business policy making. a business executive is in a better position to take decisions about whether a new product should be launched or whether there is scope for exploring new markets or whether the existing pricing and production policies need a change. sociologists may speak of population indices. income etc. or to adjust wages for cost of living changes and thus be transformed into real income and nominal sales into real sales through appropriate index numbers.. wages. relative wholesale and retail price index numbers are the output (volume of trade. It is not in the field of business and economics that index numbers are used as a basis for policy frame but even in disciplines like Sociology and Psychology their utility is immense. prices. 4.381 3. national income and variety of other phenomena. Health authorities prepare indices to display changes in the adequacy of hospital facilities and educational research organisations have devised formulae to measure changes in effectiveness of school systems. It is easy to find out the trend of exports. psychologists measure intelligence quotients which are essential index numbers comparing a person's intelligence score with that of an average for his or her age. For example. For example. They help in framing suitable policies : Index numbers are indispensable tools for the management of any government organisation or an individual business concern for efficient planning and formulation of business policies. 6. It means that a person who was having an income of Rs 1000 per month in 1990 should have an income of Rs 5000 to maintain the same standard which he was maintaining in 1990. lo measure the purchasing power of money : Index numbers are helpful in finding out the intrinsic worth of money as contrasted with its nominal worth. demand. For adjusting National Income : Index number are vfery helpful in deflating (adjusting) national income on the basis of constant prices to enable us to find out whether there is any change in the real income of the people. It is also useful in forecasting future trends. inflation and index numbers . industrial and agrxultural production etc. balance of payments. They are used to adjust the original data for price changes. n L T movements .„ the 17 7 """^ers which (WPI).™ tL movement m prices of important in modern ecoTo^c " tTa ' ifa/T T P™es is very activity WPI is the most commo::i.mdex and ^ or week. They are : ^ measured by two types of index (a) The Wholesale Price Index (WPI).:. haircuts. T" T" to be cateeorised a. The variationsin gen rarprfcel" 1 m ^^^^ ^^^^ numbers.:dt:ar^f Wholesale Index Number in India Eco^dvt: Mti^r^^^^^^^^ published every week by the office of was constructed m 194ny he S <" -^h index number .s aL. rents.-PP^X that keeps the IS expressed as percentarrist P. cakes. 1 382 Statistics for Economics-XI inflaS. According to Ackley "Inflation can be defined as a persistent and appreciable rise in general level of average prices"..According to Samuelson— By inflation we mean a time of generally rising prices for goods and factors of production—rising prices of bread. and (b) Consumer Price Index (CPI) pnce at which a commortv^ oU fn " ® '"'n'"^ P"« 'te measure the general p^e it . /„ Pe'^P'ible and persistent over time with rismg prices The essence of '' Tl pnce levef rLg ove ' t me Jnfl^^ " f -.n a comprehensive way h ' n ifdl:ri„?T commodities in all trade and transact. „ . rising wages. etc.r. 0 14. Power.I"" base year 1993-94. Light and Lubricants Manufactured Goods AU .^ published with the of c^dilX'atT''" «> "ver large number --- Introduction to Index Numbers 405 383 Category Weifihts % 22. of items 98 19 318 (a) Primary goods (b) Fuel power.2 63. Cunnnodirics Last week of 1989-90 1990-91 '''''' .1952-53 as base year in mrTre ^ T^uT Government of India has tl base st 98.8 ■No. lag of two weeks I^is due to rhe-^ ^^ '' ^^ """ ' Table 1 Index Number of Wholesale Prices Year Primary Articles Fuel. light and lubricants ^_(c) Manufactured goods Source : Economi^ Survey ^05-06 Inflation and Wholesale Price Index Number WPI IS the only price index m ndt wS ^^^ ^^^de and transactions. 1991-92 1992-93 1993-94 1994-95 1995-96 1996-97 1997-98 1998-99 1999-00 2000-01 2001-02 2002-03 2003-04 2004-05 Average of weeks 2005-06 October-Nov. (Provisional) 167 196 225 .8 214 217.1 254 258.3 121 125 136 142 153 159 162 168 178 181 183 198 199 Base 1993-94 = 100) . 232 259 (Base 1981-82 = 100) 165 189 214 246 278 175 171.8 231 233.1 190 191. 2 144 161.9 144 159.109 115 130 148 153 193 223 231 256 263 290 313 312 117 ■ 116.2 126 128.8 152 172.5 172 197.6 135 141.3 162 180.8 129 134.3 .7 139 150.8 173 198.3 169 189.9 123 122. let us understand the uses of WPI under the following heads: 1.3 in 1993-94 . viz.Statistics for Economics-XI Uses of Wholesale Price Index Numbers From the above Table 1 of Wholesale Price Index Number obtained from Economic Survey. Price trends in India : Ever since independence the price trends in India have varied between sharp to moderate increases. Estimation of demand and supply 5. Forecasting future prices 4. Determining real changes in aggregates 6.. molesale Prices. The rising trend in wholesale prices. Uses in planning 1. but it assumed alarming dimensions since 1972-73 after the first oil shock of 1973 when" OPEC nations affected a manifold rise in oil prices. 2005-2006. bas continued ever since 1960-61. 1952-53 and 1954-55 when prices showed a moderate decline. with 1970-71 as base 100) increased to 175 in 1974-75 and further to 256 in 1980-81 thus showing two and a half fold . Measuring rate of inflation 3. almost the entire period of over five decades since 1950-51 has shown persistent rise in prices. Price trends in India 2.increase in price in just one decade. OPEC again increased petroleum prices in 1978 that adversely affected our To-^o'?! ^ u ^^^ """^ber of wholesale prices. With the exception of some years of the First Five-Year Plan. The base year for WPI was changed to 1981-82 = 100 under the new index which rose to 258. as shown by the Wholesale . WPI of previous year' WPI of Picvious year "189. Using WPI of 2003-04 and 2004-05 for all the commodities from the table given above.3' 180.1 per cent at end March 2005. savings and wealth etc.3 or Rate of inflation = XlOO = 5.1% Introduction to Index Numbers 385 Thus.3 X 100 = 5. which was followed by a softening trend until August 27. Table 1 shows the movement of wholesale prices of various commodity groups since 198687 2 Measuring rate of inflation : WPI is used to measure the rate of inflation.7 per cent on April 2.3 per cent.6 per cent at end March 2004 to 5.1% WPI of current year . While the rate rose steadily . One can also calculate inflation rates for different commodities or commodity groups as required for policy purposes. The rate of inflation is useful to know the real value of income. the rate of inflation can be calculated as under : Rate of inflation = WPI of current year WPI of previous year XlOO -100 = [189. Economic Survey 2005-' Annual point-to-point inflation rate in terms of the Wholesale Price Index (WPI) increased from 4. the annual inflation rate during 2004-05 was 5.5-180. The Wholesale Price Index stood at 189.1% in case of all commodities. 2005.showing another two and a half fold rise in price in a little over one decade The base year was again changed to 1993-94 = 100 under the current series of Wholesale Price '''''' P"ce level between 1993-94 and 2004-05. The year 2005-06 started with an inflation rate of 5.5 in 2004-05.5 " L180. 2005 when it reached a trough of 3. it remained below 5 per cent. National income is defined as the value of goods and services produced in a certain year. national income. National income at current prices can be obtained after calculating the value of goods and services according to prices prevailing in the same year. Average WPI inflation decelerated from 10. Forecasting future prices : From the above time series data of WPI understand that the wholesale price level has increased in 2004-05 for primary articles by 83%.5%.700).5 per cent on January 21. Thus. 3. At 4. The real change in national income can be calculated as : = 11^x780 150 = Rs 728 crore Here. The real change in national income can be calculated as given below : Real Change of National Income WPI of Base year WPI of current year X National income of current prices For example.7 per cent during 200102 to 2004-05. while actual monetary increase is Rs 80 crore (780 . light and lubricants 191%. An increase in the national income at the current prices may be due to : (a) an increase in the general price level. 5. WPI can be used to forecast the increase in future prices.4 per cent recorded a year ago. Suppose the WPI increased to 150 in the year 2002 as compared to 2001 WPI as 140.thereafter. suppose the national income of the country in 2001 on the basis of current year prices amounts to Rs 700 crore which is increased to Rs 780 crore in 2002. for fuel power. 4. capital formation etc. Determining real changes in aggregatives : WPI are useful to determine the real changes in aggregates like. 2006 it was significantly lower than 5. for manufactured products 69% and for all commodities by 89.6 per cent in the first half of 1990s to 4. Estimation of demand and supply : One can use an appropriate model to estimate the future demand and supply as the prices affect both the demand and supply. national expenditure. WPI therefore is useful for analysing and forecasting trade situations by interpreting the present trend in supply and demand conditions. or .700). the real increase in national income of Rs 28 crore (728 . Statistics for Economics-XI amotJ^rerpeSra^d'tr^^^^^^^^^ lajches number of projects wh.as services labourers or non-n^JTurlTZXye^ Workers (CPI-IW Base 1982 .e I"dex for Industrial Changes in cost of living of rur 1 aretfl^^^^^^^^^ (CPI-AL Base 1986-87 f 100) while CW for TIrt m ^^ -^-tnal workers. bVd. The origfnal e«imatKo« ofT' the real cost of such indicated by Wpf is con rred Ss tfelTe „7 cannot be same every '' ^ to revise the annual cost of projects On this b'^^^^^ " governmem of funds for various schemLr^ Inflation and Consumer Price Index in India (CPI r. 386 j„ .e " " " " P"™'™ year due to rising costs of nroiecr ti projects.(b) an increase in the real output. 100r£s cha^.^J .rs I'a.'y — —ty the food price will have gSam ™ ^^^^^^ the rise in price increase will not be inflationary Table 2 Government gives the statement as oil . "" ^S"cultural labourers does It for urban non-manulltorkS measure of price rise of inflation and is used for de^erm nL ^ '' considered a good governmem employees as well as otLHlr T ^ allowance (DA) of monthly basis and is availabTe after Tw of r " ^ ^^^ '' and agriculture labourers are publ^sheJ bVLa^^^^^ Organisation published the CpCmber of lib. ^^^ Statistical because their typical consumptrbtll^o^^^^^^^^^^ groupAslivt trZ^^rJ^TtJ^'-' .ch require huge for rhese profecrs in . 2006.67 16. Housing 5.27 6. Pan.58 23. the impact of rising prices on urban non-manual employees was a little less than that on tbe industrial workers. Table 3 shows the movement of prices in India as shown by the All India Consumer Price Index. Misc. Supari.00 Base 2001 (Weight in %) 86. 3. The''cotrrC'^'drS Introduction to Index Numbers 387 commodities and regarded as an index of changes in cost of hving of industrial workers This index shows that there has been a five-fold rise in retail prices and cost of living of industrial workers since 1982.43 15. This index has gone up to 548 in Oct. 2006 as against 100 in base year 1982. 87) ' thus^ea^'res t ST^m ^th'atg Sr of""^ — ^^ that the consumers pa^.36 100.67 8.15 6.Major Group 1. bedding and foot wear 6. Fuel and lighting 4. 2005-06 (p.27 6. General Index for Urban Non-Manual Employees showed over fourfold rise between 1982 and Oct. tobacco etc. Thus.28 8.19 2. Clothing.26 100. Food 2. . Economic Survey. Prices in rural areas also showed a rising trend but the extent of rise in prices this case was lower than that in urban areas as is indicated by the the general index for agricultural labourers.00 3. group Base 1982 (Weight in %) Grand Total 57.00 Source . Table 3 All India Consumer Price Index Numbers Industrial Workers Urban Non-Manual Agricultural Labourers (Base 1982 = 100} Employees (Base 1984-85) Last Month of up to 1994-95 Base 1960-61 = 100 1996-97 onwards = 100) Food General (Base 1986-87 = 100) Index Index General Index 1986-87 141 137 115 572 1987-88 154 149 126 629 1988-89 168 163 136 708 1889-90 177 173 145 746 1990-91 199 193 161 803 1991-92 230 219 183 958 1992-93 254 240 202 1076 1993-94 272 258 216 1114 1994-95 304 284 237 1204 1995-96 337 313 259 234 1996-97 369 342 283 256 1997-98 388 366 302 264 1998-99 445 414 337 293 1999-00 446 428. 352 306 2000-01 453 444 371 305 2001-02 466 463 390 309 2002-03 477 482 405 319 2003-04 495 500 420 331 General Index . 2004-05 506 520 436 339 Oct. 06 538 548 460 356 Source : Economic Survey. Wage Price Spiral (i. Increased Wages and Salaries Supply Side Factors 1. Slow Pace of Industrialisation 3. Deficit Financing 5.e. we have to look^tn L tr t understand the various price or inflation Demand Side Factors 1. It also puts pressure on interest rates. 388 Statistics for Economics-XI Causes of Rising Prices (Inflation) in India. Increase in Petroleum Prices 4. and adversely affects both savings and investment. High inflation hurts the poor with their incomes not indexed to prices. Slow Growth of Agriculture 2. Growth of Black Money 6. Faster Growth in Money Supply than the Growth Rate of National Income 3. Increase in Money Supply 2. Changes in Administered Prices 5. workers demanding higher wages) limitations of index numbers 4 Introduction to Index Numbers > list of formulae . 2005-2006 (p.. Because of its implications for the poor and its possible destabilizing effects on macro economic stability. containment of inflation is high on the Government agenda. S-63). Massive Increase of Government Expenditure 4. ^ ^ X 100 ZqoPo . Weighted Average of Price Relative Method Z ^xlOO x(Poqo) [Po J Poi = 389 Quantity Index . Unweighted Simple Average of Price Relative Method X^xlOO VPo J Po.^ x 100 ^Po Quantity Index ^ = ^ x 100 y V. X100 Zpo% 2. 100 ^oPi Consumer Price Index A. Paasche's Method p. Weighted Aggregative Method : Price Index A. = ^^ x 100 ZPoqo 4.1. Value Index V^j = ^ x 100 evq ^ im. Laspeyre's Method = ^^ x 100 B. Unweighted Simple Aggregative Method Price Index p . = 3.. Aggregative Expenditure Method or Aggregati CPI = ^ X 100 . XW. = iL ZW W 390 j„ . What are the problems in construction of index numbers? 4. Why do we need an index number. Briefly explain the importance of index numbers in the study of Economics. What are the desirable properties of the base period? 9.B.? 3. Family Budget Method CPI = Index of Industrial Product ZPV Method :ive 'ZWR " or _ XW . Distinguish between Laspeyre's method and Paasche's method of constructing index number. What is the difference between a pure index and quantity index? 7. "XW7" Industrial Production Index No. Is change in any price reflected in a price index? 11. 5. Distinguish between actual difference and relative difference in prices. Distinguish between 'weighted' and 'unweighted' index of prices? 10. State the general uses of index numbers. 6.Define index numbers. What does consumer price index for industrial workers measure? . 8. 2. What is index number? Discuss briefly the uses of index numbers. Statistics for Economics-XI exercises Questions : 1. Write short notes on : (a) Base year Index of Industrial Production c Value Index (J) Consumer Price Index .Define Consumer Price Index number. . 16. Can CPI number for urban non-manual employees represent the changes in cost of hvmg of President of India? 20. 22. Discuss the limitations of index numbers. Distinguish between 'Wholesale Price Index' and 'Consumer Price Index'. 19. Prices in Rs 2001 Prices in Rs 2002 50 80 40 60 10 . 14. Commodities :A B CD E 12. Why is it essential to have different CPI for different categories of consumers? 18. 5 20 10 6 2 . Explain the uses of consumer price index numbers. Problems : 1. What are the uses of Wholesale Price Index numbers? 15. (e) Wholesale Price Index. 13. 17. Try to list the important items of consumption in your family. Explain Index Numbers of Industrial Production. Construct the Index Number for 2002 with 2001 as base from the following prices of commodities by simple (Unweighted) aggregative method. What do you mean by inflation? How the wholesale price index numbers are useful for measuring the rate of inflation? 21. 30 0.36 0.32 0. Item Coal Producers Price V- 2000 2001 2002 (Rs) 5 3 4 (Rs) 2 3 Crude oil ----"c.28] 3.00 4.50 Cheese per Tin 18. 2002 = 114.00 Milk per Litre3.[Index Number = 164.00 3.00 22. Calculate the index number for 2002 with 2001 as base from the following prices of the commodities by simple (unweighted) aggregative method. Using the following data and 2002 as the base period.30 0.39 B 0.50 [Index Number = 116.47] 4.33 0.80 Eggs per Dozen 4.24 0. Commodity Price Price and unit ' (2001) (2002) Butter per kg 20. Calculate Quantity Index Numbers from the following data by simple aggregative method taking quantity of 1998 as base.25 0.30 .48] Introduction to Index Numbers 391 2. compute simple aggregative price indices for the two fuels.00 4.00 19.71. [Index Number : 2001 = 85. Commodity Quantity (in tons) 1998 1999 2000 2001 2002 A 0.36 0.80 Bread per kg2. 4 5. 50 2. Commodities -.u 1993 1994 1995 1996 1997 (in Rs) 75 50 65 60 72 1998 1999 2000 2001 2002 Statistics for Economics-Xl .50 2. Bricks Timber Plaster Board Sand Cement 10 20 16 21 6 3 5 2 7 14 [Index No.40 2. = 147] Construct the index number for 2000 taking 1990 as base by price relative method using arithmetic mean.25 0.3.32 0.60 (Quantity Index No.1. = 117.30 D 2.1.5] 392 (between H-0 tn a u. Items Prices (1991) Prices (2002) 6.C 0. 127. = 147. 130.28 0.00 2.5) Calculate index number for 2002 on the base prices for 1991 from the following by average of price relative method.20 0.A B C D Price (1990) : 10 20 30 40 Price (2000) : 13 17 60 70 [Index No. 125. . 2001 Cummodtty ABC Pnce _ 4 3 Quantity 2002 Price .2491 m^hor Tt^ ^^^^ data by weighted aggregative method using : a Lasoevre's method ^ u aggregative Commodity Price (2001) Quantity (2001) ABCD 4325 in v^r^rr.3 624. 2002 and L th'e wLr'"^'' ""-ber of wholesale prLes Weights ^^^ Index Food Article Manufactures Industrial Raw Material Semi-Manufactures Miscellaneous 31 30 18 17 4 473.2 403.. I'aascncs : 158... [WPI = 449.99] aZtitv in? ^"^u price index and quantity index numbers wn-li onoi__j quantity index numbers with base 2001 and interpret. ^u J .6 390.2 510. XJ/. Price (2002) Quantity (2002) 20 15 25 10 6 5 3 4 10 23 15 40 L^^-^f^j-^o .4 Q 1 1 .//.Wholesah Prices (in Rs) 7« 69 75 84 80 wholesale prices in India for second week of Sept. 0 3. ' Bread Meat Tea 6 4 0.3] 13.02.2 80 C 2. and {ii) Paasche's Method. Also calculate value index number and interpret them.00 24.0 40 B 3.92.25] Commodity Price Base Year (in Rs) A 6. Commodity Base year Current Year Quantity lbs.5 paise 50 paise 40 paise Price per lb. Quantity Index = 143 18-Paasche's : Price Index = 69.s 252 _ 624 Quantity 31 6 [Laspeyre's : Price Index = 76.0 3. Prepare consumer price index numbers from the following data for 2000 and 1999 taking 1998 as base. (ii) 81.84. Group 1998 1999 (Price in Rs) 2000 A 20.00 . Price per lb. Calculate weighted aggregative of actual price index number and quantity index number from the following data using (/) Laspeyre's Method.00 21.0 20 Price Current Year Quantity Base Year (in kg) [Index Number = 122.0 8. Quantity lbs.5 30 [Index Number = (/") 86. Quantity Index = 130] Introduction to Index Numbers 415 11. 40 paise 45 paise 90 paise 7 5 1. Collection and Organisation of Data. transport in land and foreign trade and different business activities.00 C 5.12 [Index numbers.. government activities etc. .B 1.00 2. 1999 = 127.43] From the data given below construct the consumer price index number Commodity Price Relatives Food 250 45 Rent 150 15 Clothing 320 Weights 20 Fuel and Lighting 190 5 Miscellaneous 300 15 [Index Number = 253. business.25 2.00 D 2. Productivity Awareness In the previous Unit 1. Questionnaire for Dealers . we have studied the Meaning of Economics. 2000 = 107. Scope and Importance of Statistics in Economics.5] UNIT 4 DEVELOPING PROJECTS IN EC Chapter 13 PREPARATION OF A PROJECT REPORT Introduction 2. in Unit 2. and in Unit 3.25 1. These tools are Very important in our daily life to analyse different economic activities such as consumption.00 8. Uses of Project Report 3. 5. About the Various Statistical Tools. Consumer Awareness 4. distribution.25. For example.00 8. Reports are prepared to give information about the development of institution. production. In this chapter we will learn the method of developing a project report which will help us in understanding the application of statistical tools to analyse the various types of business activities.50 1. product. In the international context. Chamber of Commerce. In the light of above examples it is very clear that project reports help in understanding the requirements of shareholders. roads to construct in the light of changing population of a respective area. Such surveys are conducted by non-government organisations. 3.) in war. consumers. hfe saving drugs. Reserve Bank of India plans the opening of new branches of commercial banks. namely. Central/state governments prepare reports for future development in priority areas such as road. for this purpose the government conducts surveys to know about likely requirement of primary health centres and schools for basic education. 2. power. etc. health. Such surveys are conducted by manufacturing organisations. price and uses of product in changing environment and technology.1. preference for landline phone or mobile phone. Similarly government decides the requirement of power (Mega Watts). e. Shareholders may be interested to know about the earning of organisation and possibility of getting dividend while holding the shares of the company. 4. For example. Preparation of a Project Report 395 Federation of Indian Chamber of Commerce and Industry (FICCI). etc. teleconununication. drought. Central and State Governments. cooperative banks or agricultural banks in the light of increasing credit requirement of population on the basis of survey reports. uroject Uses of project report can be highlighted as under : 1.^ detergent powder or detergent cake. United Nations Organisation (UNO) plans humanitarian help (food. Those organisations who ignore the changing requirement of the consumers or population may fail in achieving their goals and objectives. education.. . Reserve Bank of India and financial institutions and national and international bodies to plan their activities for future operations. etc. societies. To make aware individual groups about the present environment conditions of business/government. etc. Consumer may be interested in knowing the quality. etc. earthquakes and such other natural calamities based on survey reports. Confederation ofIndian Industry (CII) conduct surveys of abroad to know the business opportunities arising out of economic development of respective nation.g. fully automatic or semi-automatic washing machine. 5. 9. 7. technological aspects of national and international significance. 3. government agencies. economical. To invest in those securities that provides higher rate of interest/dividend to shareholders. To exploit opportunities in the national and international markets by trade associations: 8. Conclusion . etc. lesser weight. Preparation of Questionnaire 3. such as right to basic needs.. Any consumer is exploited on this ground can approach to the appropriate authorities to seek compensation or replacement of goods. international. To pay competitive prices for irequired goods by the consumer to take the real value of the price paid to sellers. 1986 has provided various rights to the consumers. For this consumers may be made aware about their rights and informed about proper agencies. redressal. The Indian Consumer Protection Act. provide poor quality. 5. To direct the efforts of organisation in given objectives based on opportunities provided in the changing environment. Collection of Data 4. 6. Analysis and Interpretation 5. defective product. To provide food. manufacturers charge higher price.2. representation and healthy environment. To help in the pohcy formation about the economic and social development of the country. It helps in conducting research on various issues such as political. 4. social.. board of directors and national and international agencies. ^^imers iuvareni Consumers may be exploited by manufacturers. medical help to badly affected areas due to any natural calamities by national. choice. social and non-government organisations. safety. To pin-point the weaknesses of organisation so as to overcome such weaknesses. to the consumers. ^^^ Statistics for Econotnics-Xl There are five steps in preparing a project report for consumer awareness : 1. information. which they can approach for grievances. e.g. education. Identification of Problem 2. .... Preparation of Questionnaire To know more about various aspects of air-conditioner in a more systematic manner.. we must design a questionnaire covering all the aspects discussed above........ Let us take the example of air-conditioner where we are interested to know from dealers about the performance of airconditioner with respect to price.. cooling technology... 8-Worst) (/) Videocon {iv) National {vii) Voltas {ii) Carrier {v) Samsung {viii) Others {Hi) Amtrex {vi) LG .. quality......... Q.. Which brands of AC's do you currently deal in : (/■) Videocon (//) Carrier (f) Samsung {viii) Others {iv) National {vii) Voltas {Hi) Amtrex {vi) LG Q 3.... keeping in view other air-conditioners' manufacturers product available in the market in competition. Q......... {ii) ...... Address :. availability. car... Which brand AC would you recommend to the customer? (rank them) (1-Best..... questionnaire for dealers Name:.. after sales service etc...... computer etc................ air-conditioner.. {iv) ............. refrigerator.......... 1.. Please recall some air-condition brand name : W ....... 2..Identification of Problem We want to know about consumers'/dealers' knowledge about the product of a company manufacturing namely............ warranty. washing machine.... {Hi) ... scooter.. colour TV..... Phone No.............. .....................— Q........ Which AC company provides the highest margins to their dealer? Name of the company : ......... 6..... If No................... (mention the name)? Name . 4......... 7... Rank the companies in regularity of supply (1-Best........................ Some time..... Which is most demanding brand of AC................ 8-Worst) : (iv) National (/■) Videocon (ii) Carrier (v) Samsung (vi) LG (Hi) Amtrex (vii) Voltas (viii) Others Q....... Does the brand name influence the customer? Yes ......... ................ Which brand of AC has least customer complaints (mention the name)? Name : .. 5. 8-Worst) (ii) Quahty (Hi) Performance (v) After sale service (viii) Warranty 397 (vi) Authentic (i) Price (iv) Availability (vii) Technology Q................ No................... (1-Best.............. Rank the customer conferences while buying an AC............................................Preparation of a Project Report Q................................. 8-Worst) (ii) Quality (Hi) Performance (v) After sale service (viii) Warranty (vi) Authentic (i) Price (iv) Availability (vii) Technology Q.......... then what else influences him (specify) : Q....... Reason behind above recommendation (rank the factors) : (1-Best.......... 8.... 10..... Q....................... 9..................................... ................ Do you agree that huge advertisement campaigns are the most responsible factors for the changing market scenario and increasing demand? (i) Agree very strongly (Hi) Agree (v) Disagree (vii) Don't know Q.....Specification :........................... 12. Warranty period offered by the companies (kindly tick) : Brand name (i) Videocon (ii) Carrier (iii) Amtrex (iv) National (v) LG (vi) Samsung (vii) Voltas (viii) Others Warranty 6 month-1 yr................ 398 j„ . 14................. 15...................................... Which AC company do you feel is the most aggressive in giving discounts and scheme (please specify)? Name of the company :............. 3 yrs................. Q.. Which brand of AC in your opinion is having most advance technology? Name the brand : .. Statistics for Economics-XI Q....................... Generally what short of problem do you face while doing a sale? Specify : .... 11. 1-2 yrs.......... 13...... 2-3 yrs........ and above ......Q.............. Q. Average No. Window Projected Last yr. nf sales (Monthwtse) 2004 2005 . 17.Q. of units sold per month from your counter. May and June in last three years : {For the brand which you deal in) Company name 200 i No. (Please specify brandwise) (i) Videocon fl («) Carrier Q (Hi) Amtrex (v) LG [j^ (vi) Samsung (viii) Others (iv) National (vii) Voltas n 1 Brand name ___ Lastyr. -----—» '/f*"" ^^^ uiaiiu. 19. Split Projected (/■) Videocon (ii) Carrier (iii) Amtrex (iv) National (v) LG (vi) Samsung (vii) Voltas (viii) Others — Preparation of a Project Report 399 Q. Please give the sales break-up for the month of April. What is the market size of the area you are dealing in' (i) 0-1000 Machines — (ii) 1000-2000 Machines (iv) 3000-4000 Machines (iii) 2000-3000 Machines (v) 4000 and above Q. 16. ...... graphs and diagrams....................... Kolkata... May Jme (i) Videocon (ii) Carrier (Hi) Amtrex (iv) National (v) LG (vi) Samsung (vii) Voltas (viii) Others Q. {ii) Enhancing your infrastructure and sales persons' team.... To get a substantial growth in your present sale which of the following would you prefer? (/) Having a better brand name If......... Delhi. pie-diagram etc....... where we want to position our product... The number and geographical areas depend upon our requirement.. name the brand ....May Apr.. Chennai and other capital cities of states.... Collection of Data The above questionnaire with the help of investigators using sampling method will be filled in by the dealers...................... bar diagrams...................... namely....... 20... viz.. We can also collect the information from government and industrial publications to know about the growth of air-conditioner industry and future government policy in this respect. Analysis and Interpretation Data collected through questionnaire will be classified and presented in the form of tables. May June Apr..... For rigorous analysis 400 Statistics for Economics-XI standard deviation and eoeffident oTva^t ^ OnV^K"" "" ....... „e of view of : ^ customers prefer to buy air-conditioner from the point . data. Table 1 Consumer Awareness about Air-conditioners Atvareness Brand Present Availability Price After Sales Service Technology Conclusion (U O< ai o cr UJ CL consumer awareness about air-conditioners Scale : 0.5 cm = 10 percentage on V-axis Brand Present Availability Price After sales Services xxxxxxv xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx Technology E3 Videocon □ Amtrex H Samsung EHJ Voltas m Carrier g National m LG ^ Others Observation obse?ve bar diagram.also pro. are g.ven beiow .eet fntnre demand tWongh ^- Blustration: Table and dtagram (based on hypothetiea. Coefficient of Variation (C. the Air-conditioner company will come to know about the brand. After Sales Service : LG or Videocon 5. Analysis Let us analyse the given data by applying different statistical tools (Mean. Mean : N 2. Through this observation the company will be in a position to decide regularity of supply the number of units to be produced and to improve after sales service as per requirements of future consumers. present availability.1. after sales service. Standard Deviation : a= nx-xf N N 3. Present AvatlahtUty : Voltas or other brand 4 Preparation of a Project Report 401 3. price. Brand : Either Vidiocon or Samsung 2. Standard deviation and Coefficient of Variation) using the following formulae : 1. technology etc.) 100 a =J" Table 2 Consumer Awareness about Air-conditioners . Price : LG or Videocon 4.V. Technology : Videocon or LG Thus. 8 8 Brand 24 101 3 10 44 Mean : X % 20.44 -5.8 Observation Considering brand. after sales service and technology.2% and hence will prefer to buy Videocon air-conditioner.2 10.04 +0.24 +0.24 +3.2 1.24 -0.2 0.62 6.2 6 Samsung 17 17 21 12 8 17 14 10 97 60 49 15 19.2 M.(Figures in percentages) Awareness Name of Companies Videocon Others Cjirrier Amtrex National 8 10 20 18 8 9 Present Availability 12 9 7 11 14 13 Price 25 5 12 28 9 5 After Sales Service 18 12 6 6 17 Technology 22 10 7 8 12 IX 34 40 75 8.2 67.8 +1.64 +1.44 Z(X XV 112.8 33.8 14.2 4.2 Af . average percentage of customers of Videocon air-conditioner is the highest as 20.8 6.8 46. price. present availability.04 +1.8 23.-8.8 3.2 1. 402 Videocm Jf-f fX- Carrier X- (X~ X- X Si/^ R +3.8 a 10.84 +1.8 -2.2 .84 +0.4 12 LG Voltas 9.44 -0. 68 61.84 jf/' M 25 -1.8 1.67 24.04 2.76 -4 0.96 1 -6.6 2. we get j„ .C.04 5.64 48 125.4 40.96 -3 9 -4.2 51.8 3..22 6.72 Thus.19 7.099 63.24 9 -2.35 8.93 11.4 +2 4 +0.61 Name of Companies Videocon Carrier Amtrex 54 78.LG Vbit» t Othm 1 X .8 25 +7.44 0.57 77.58 0 0 -1.04 0.64 0.—-.88 46.74 +5 -1 -3 +2 -3 .04 4 +1.25 90.56 +2 +3 -3 -2 0 4 9940 26 5.2 0.8 .1 /X - X-X flf-X)^ X -X 16 -0.2 57. Statistics for Economics-XI Calculation of Standard Deviation and Coefficient of Variation Name of Companies 1.4 1.V 52.6 73.96 +5 9 +8.8 23.64 0. 61 Observations ruJr!'^of variation is the highest for other brands as 90.88 Coefftcient of Variation 52. Requirement to gyJhTLT'"''^ dealers/consumers to fill in the questionnaire .56 63.2 57.35 8.74 46.2 Standard devtatton 8.72 24. hence the customers will not prefer to buy other branded air-conditioners.67 5.57 77.93 11.19 7.National Samsung LG Voltas others Mean » X 20.68 61.84 1.099 6.62 6.8 6.61%.8 10.8 8 15 19.25 90.4 12 9. raTned\^o' " ^^ ^he information reoZd Ar Potential dealers/customers. structure of foreign trade. savings and capital formation etc. we say X company is more productive than Y company. Productivity varies from company to company. by Further. r Preparation of a Project Report 403 ivity Productivity is the ratio between input and output of an organisation.V. price. In this case the labour productivity of Y company is better.500. For example. -4 Identification of Problem We want to know productivity awareness amongst the enterprises of the following economic problems.V. capital etc.V. External factors include growth of agriculture and industrial production. Therefore. is less by Rs 1. organisation structure. For example. the cost of labour of X company to manufacture colour T. ability of the firm to substitute different inputs. graphs and diagrams etc. X company manufactures a colour T. students should be asked to analyse and interpret the data collected 5. growth of bank deposits and credit. In this case X company is more productive than Y company because X company's manufacturing cost of colour T.000. They may also suggest the future course of action for the company. they may be asked to prepare required tables. We can identify the problems like : [a) Industrial production {b) National budget . In addition to this productivity. etc. for Rs 10. Productivity is determined with internal and external factors. Internal factors are technology.000 and that of Y company is Rs 2.V. 4. for Rs 11. while Y company manufactures the similar T.000. although overall productivity of X company is better as compared to company Y. we can also be able to calculate productivity of different factors of production such as labour. managerial ability.000. composition and growth of GDP. is Rs 3. Statistics for Economics-XI Table 1 Annual Growth Rates of Industrial Production in Major Sectors of Industry (Base : 1993-94 = 100) Period Weights 1995-96 1996-97 1997-98 1998-99 1999-00 2000-01 2001-02 2002-03 2003-04 . 404 Illustration 1.(c) Population growth {d) Gross national product (e) Financial assistance by All India Financial Institutions Collection of Data Different ministries and departments of Central and State Governments publish regularly current information alongwith statistical data on the number of subjects. P. Statistical Tables and Analysis Following are few illustrations for analysis. This information is quite reliable for related studies. We can collect data about identified problems from Newspapers/Economic Surveys/RBI Bulletin/Government Budget of the State or the Nation/Census Reports/NSS Reports/Annual Survey of Industries/ Labour Gazettes/Agriculture Statistics of India/Indian Trade Journals etc. 8 1.1 2.1 8.4 6.2 4.1 7.2 5.4 5.4 (In per cent) Manufacturing Electricity Overall 79.8 1.7 4.0 2.7 -1.3 4.1 6.9 6.1 7.7 5.47 9.9 -0.9 4.6 8.) Mining and Quarrying 10.4 9.7 6.0 14.9 3.1 13.6 6.0 2.0 5.0 7.8 economic survey : 2005-2006 (p 132) loofotconr' from the second quarter of ^-'^^very that commenced .0 6.0 9.7 7.) 2005-06 (April-Dec.2 5.8 5.2 8.5 4.4 8.2 5.17 100.3 6.0 3.1 0.2004-05 2004-05 (April-Dec.36 10.3 4.1 7.4 5.8 7.7 6.2 6. roads and transport. Table 2 Trends in Deficit of Central Government (As per cent of GDP) Year Revenue Primary s Deficitbefkit y Deficit 1990-91 3. (iv) bigh oil prices. (//■) lack of domestic and external demand' Preparation of a Project Report 405 (Hi) lack of reforms in land and labour markets.n Ae^Xondmg^erd'of'zm^^^^^ Impressive performance of the mannfacuring sector which grew at 8 9 n! mwsmmsm ^ . Illustration 2.8 6.3 2.o a gro«h of 8.7 1992-93 2..7 4.6 4.5 0.vear^mctde" during the current H) normal business and investment cycles.8 2. (viii) continuing high real interest rates.7 1 .1 0.4 4.6 per Li .4 1994-95 3. Bscat .5 0. (v) existence of excess capacity in some sectors..8 1993-94 3.2 6. (vii) infrastructure bottlenecks particularly power.per cent compared .6 1991-92 2. ----------iiivcstmen capacity additions and contributed to this shortage . (vi) business cycle. 1 5.7 5.6 4.5 (Provisional) 2005-06 (BE) 4.8 0.6 0.7 5.5 4.0 4. The fiscal deficit excludes the transfer of States' share in the small savings collections.2 4.2 2002-03 4. leading to a marked improvement m the quahty of deficit Th^ available ar..4 1.1 1999-00 3.1995-96 2.1 1997-98 3. Department of Expenditure.8 1998-99 3.7 0. Anlaysis : The Fiscal Responsibility and Budget Management Act (FRBMA). declined from 6.5 0. The ratios to GDP for 2005-06 (BE) are based on CSO's Advance Estimates GDP at current market prices prior to 1999-2000 based on 1993-94 series and from 1999-2000 based on new 1999-2000 series.5 0.2 1996-97 2.1 0.6 2001-02 4. 2003 continued to provide a strong institutional mechanism for making sustained progress at 406 Statistics for Economics-XI demand on «t^tt^lr^^^^^ a proportion of GDP..3 1 * Provisional and unaudited as reported by Controller General of Accounts.1 2. ^ .5 2004-05=^ 2. Economic Survey—2005-06 (page 24). 2. Notes: 1.9 2003-04 3. Ministry of Finance.5 6.0 0.4 1. Source : Budget Document.3 2000-01 4.0 4. t itsTsi o " cSrS^^^^^ A^dr^i^ l^rov.<>' G^P .4 -0.5 0.ha.6 per cen.9 5. .Budeet for 7005 n^ u.^ ' 7 .S Requirement 2. The 5S. TTiey can also be asked ro make presentation of snch problems ItotTr" " 1t ""rpretation of data of the stable they have . respectively..dj. ^ ^ P^'" cent. 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