Time Series000

ETENCA“TIME SERIES” Of the most important tasks before economists and businessmen these days is to make estimates for the future. For example, a businessman is interested in finding out his likely sales in the year 2008 or as a long-term planning in 2020 or the year 2030 so that he could adjust his production accordingly and avoid the possibility of either unsold stocks or inadequate production to meet the demand. Similarly, an economist is interested in estimating the likely population in the coming year so that proper planning can be carried out with regard to food supply, jobs for the people etc. However the first step in making estimates for the future consists of gathering information from the past. In this connection one usually deals with statistical data which are collected, observed or recorded at successive intervals if time. Such data are generally referred to as time series. Thus when we observe numerical data at different points of time the set of observations is known as time series. For example, if we observe production, sales population, imports, exports etc, at different points of time, say over the last 5 or 10 years, the set of observations formed shall constitute time series. Hence in the analysis of time series, time is the most important factor because the variable is related to time which may be either year, month, week, day, hour or even minutes or seconds. A few definitions of time series are given below: 1:- “ A time series is a set of statistical observations arranged in chronological order.” 2:- “ A time series consist of statistical data which are collected, recorded and observed over successive increments of time.” 3:- “ A time series may be define as a collection of magnitudes belonging to different time periods of some variable or composite of variables, such as production of steel, per capital income, gross national product, price of tobacco, or index of industrial production/” 4:- “When quantitative data are arranged in the order of their occurrence, the resulting statistical series is called a time series”. 5:- “ A time series is a set of observations taken at specified times, usually at equal intervals”. Mathematically a time series is defined by the values Y1, Y2……. Of a variable Y ( temperature, closing price of a share, etc.) at times t1, t2,….. Thus Y is a function of t symbolized by Y =F(t)”. It is clear from the above definitions that timeservers consist of data arranged chronologically. Thus if we record the data relating to population per capital income, prices, production, etc.. for the last 5, 10, 15, 20 years or some other time period, the series so emerging would be called time series. It should be noted that the term “ time series” is usually used with reference to economic data and the economists are largely responsible for the development of the techniques of time series analysis. However, the term “time series” can apply to all other phenomena that are related to time such as the number of accidents occurring in a day, the variation in the temperature of a patient during a certain period, number of marriages taking place during a certain period, etc. In this text, our discussion shall be limited to time series of economic and business data. But three techniques can also be applied to any of the other natural or social sciences. The problem of time series analysis can best be appreciated with the help of the following example: Page 1 The following are the figures of sales of refrigerators of a firm in thousand units: Year Sales of firm A(‘000) Year Sales of Firm A(‘000) 2000 40 2004 43 2001 42 2005 48 2002 47 2006 65 2003 41 2007 42 If we observe the above series we find that generally the sales have increased but for two years a decline is also noticed. The statistician, therefore tries to analyse the effect of the various forces under four broad heads: (1) Changes that have occurred as a result of general tendency of the data to increase or decrease, known as secular movements”. (2) Changes that have taken place during a period of 12 months as result of change in climate, whether conditions, festivals etc. Such changes are called’ seasonal variations”. (3) Changes that have taken place as a result of booms and depressions .Such changes are classified under the head cyclical variations”. (4) Changes that have taken place as a result of such forces that could not be predicted like folds, earthquakes, famines, etc. Such changes are classified under the head “irregular or erratic variations”. (5) These are called components of time series and shall be discussed in detail. Utility of TIME SERIS ANALYSIS The analysis of time series is of great significance not only to the economist and businessman but also to the scientist, astronomist, geologist, sociologist, biologist, research worker, etc,, for reason given below: 1:- It helps in understanding past behavior:- By observing data over a period of time once can easily understand what changes have taken place in the past, such analysis will be extremely helpful in predicting the future behavior. 2:- It helps in planning future operations:- Plans for the future cannot be made without forecasting events and relationship they will have. Statistical techniques have been evolved which enable time series to be analyzed in such a way that the influences which have determined the form of that series may be ascertained. If the regularly of occurrence of any feature over a sufficiently long period could be clearly established then, within limits, prediction of probable future variations would become possible. 3. It helps in evaluating current accomplishments:- The actual performance can be compared with the expected performance and the cause of variation analyzed. For example, if expected sale for 2007-08 was 10,000 refrigerators and the actual sale was only 9,000, one can investigate the cause for the shortfall in achievements. Time series analysis will enable us to apply the scientific procedure of “ holding other things constant” as we examine one variable at a time. For example, if we know how much is the effect of seasonality on business we may devise ways and means of ironing out the seasonal influence or decreasing it by producing commodities with complementary seasons. 4. It facilitates comparison;, Different time series are often compared and important conclusions drawn theretrom. However one should not be led to believe that by time series analysis one can foretell with 100 per cent accuracy the course of future events. After all, statisticians are not foretellers. This could be possible only if the influence of the various forces which affect these series such as climate customs and traditions, growth and decline factors and the compiles forces which produce business cycle would have been regular in their operation. ?However, the facts of life reveal that this type Page 2 of regularity does not exist. But this When such analysis is coupled with a careful examination of current business indicators one can un doubted improve substantially upon guestimates(i.e. estimates based upon pure guesswork) in forecasting future business conditions. COMPONENTS OF TIME SERIES It is customary to classify the fluctuations of a time series into four basic types of variations. Which superimposed and acting all in concert account for changes in the series over a period of time. Those four types of patterns, movements, or, as they are often called, components or elements of a time series are: 1:- Secular Trend 2:- Seasonal variations 3:- Cyclical variations 4:- Irregular variations It may noted that any or all of these components may be present in any particular series. The following graph gives the sale of Cola for the year 1993 to 2007: 20 18 Sale of Cola B 14 12 0 1993 1995 1997 1999 2001 2003 2005 2007 The original data in this grapes is represent by curve (a). the general movement persisting over a long period of time represented by the diagonal line (b) drawn thought the irregular curve is called secular trend. Next, if we study the irregular curve year by year, we see that in each year the curve starts with a low figure and reaches a peak about the middle of the year and then decreases again. This type of flotation, which completes the whole sequence of Changes within the span of a year and has about the same pattern year after is called seasonal variation. Furthermore, looking at the broken curve superimposed on the original irregular curve ,we find pronounced fluctuations moving up and down every few years through the length of the chart . These are known as biasness clues or cyclical fluctuations . they are so called because they comprise a series of repeated sequence just as a wheel goes round and round. Finally, the little saw-tooth irregularities on the original carve represent what are referred to as irregular movements. In traditional or classical time series analysis , it is ordinarily assumed that there is a multiplicative relationship between these four components, that is it is assumed that any particular value in a series is the product of factor that can be attributed to the various components. Symbolically, Y=T x S x C x I Where Y denotes the result of four element : T= trend; S= seasonal component ; C = Cyclical component; I=Irregular Component. Another approach is to treat each observation of a time series as the sum of these for components. Symbolically Y=T+S+C+I To prevent confusion between the two models it should be pointed out that in the multiplicative model S,C and I are indexes expressed as decimal percent can be expressed as seasonal, cyclical and irregular in nature. Page 3 Example. If in multiplicative model, T= 400 x S=1.5, C = 1.2 and I = 0.8 then: Y = T x S x C x I = 400 x 1.5 x 1.2 x 0.8 = 576 If in the additive model, T = 400, S = 120, C = 20 and I= - 40 Then Y = 400 + 120 + 20 – 40 =50 The additive model assumes that all the components of the time series are independent of one another. For example ,it assumes the trend has no effect on the seasonal component, no matter how high or low this value may become .Further , it assumes that the business cycle has no effect on the seasonal component. If the index for December is typically 1.50 or 1505, this per cent will not be affected by either prosperity or recession. White the additive model may work well within limits, it is doubtful if one always can rely on the independence of components that it assumes. In the multiplicative model. it is assumed that the four components are due to different causes but they are not necessarily independent and they can affect one another. There is little agreement amongst experts about the validity of the different assumptions—some feel that the given classification is too crude and that there are more than four types of movements. Nothing specific is really known 55About how the components are related. How they combine to produce particular effects, or whether they are really separable. The effects of the various components might be additive, multiplicative or they might be combined in any one of indefinitely large number of other ways. Different models (assumption or theories) will lead to different results. Although the multiplicative assumption characterizes the majority of economic times series. Consequently, the multiplicative model is not only considered the standard of traditional assumption for time series analysis but is more often employed in practice than all other possible models combined. For this reason, we shall use only the multiplicative model in our subsequent discussion. The task of performing a time series analysis, just like the analysis of a chemist in breaking a substance into its constituent parts, is to operate on the data in such a way as to bring out separately each of the components present. 1:- Secular Trend:The term trend is very commonly used in day-to-day parlance. For example , we after talk of rising trend of population, prices etc. Tend also called secular or long term trend, is the basic tendency of production, sales, income, employment, etc.. to grow or decline over a period of time. The concept of trend does not include short-range oscillations but rather steady movements over a long period of time. Secular trend movements are attributable to factors such as population change technological progress and largescale shifts in consumer tastes. The presence of more people means that more food, clothing , housing are necessary. Technological changes discovery and exhaustion of natural resources, mass production methods, improvements in business organizations and governmental intervention in the economy are other major causes for the growth or decline of many economy time series. In some cases growth in one series involves decline in another, For example the displacement of silk by rayon, the bullock-carts by other modes of transport like trucks tempo etc. Similarly better medical facilities improved sanitation diet etc. on the one hand, reduce the death rate and on the other contribute to a rise in birth rate. There are all sorts of trends: some series increase slowly and some increase fast, others decrease At varying rates, some remain relatively constant for long periods of time, and some after a period of growth or decline reverse themselves and enter a period of decline or growth, Broadly speaking the various types of trends are divided under two heads: 1:- Linear or Straight Line Trends: and 2:- Non linear Trends Page 4 For a proper understanding of the meaning of trend , the reader’s attention is directed to the following two points: (1) When we say that secular trend refers to the general tendency of the data to grow or decline over a long period of time, one may be interested in finding out as to what constitutes a long period of time. Does it mean several years? The answer is “no”. On the other hand, whether a particular period can be regarded as long as not in the study of secular trend depends upon the nature. MEASURENT OF TREND:Given any long-term series, we wish to determine and present the direction which it takes—is it growing or declining? there are two important reasons for trend measurement: (i)To find out trend characteristics in and themselves. In studying trend in and of itself, we ascertain the growth factor for example, we can compare the growth in the textile industry with the growth in the economy as a whole or with the growth industries, or we can compare the growth in on firm of the textile industry with the growth in the industry as a whole . Moreover , we can compare through trend characteristics the growth of the textile industry in India with that of other countries .the growth factor also helps us in predicting the future behavior of the data. If a trend can be determined, the rate of change can be ascertained and tentative estimates concerning future made accordingly. (ii) To enable us to eliminate trend order to study other elements. The elimination of trend levels us with seasonal , cyclical and irregular factors. We can then, in two or more series, compare or use the impact of these three relatively short-term elements divorced from the long-term factor. The various method that can be used for determining trend are • • • • Freehand or graphic method. Semi-average method . Moving average method. Method of least squares. Question 1:-. Fit a trend line to the following data by the freehand method: Year Production of steel Year Production of steel (million tonnes) (million tones) 1999 20 2004 25 2000 22 2005 23 2001 24 2006 26 2002 21 2007 25 2003 23 ---------------------------------------------------------------------------------------------------------------------------------------------- Question 2:- Fit a trend line to the following data by the freehand method of semi-average: of semi-average: Year 2001 2002 2003 Sales of a (thousand unit) 102 105 114 year 2005 2006 2007 Sales of firm a (thousand unit) 108 116 112 Page 5 account for the difference between the two figures. Year Sales (Rs. By joining these two points. we get a trend line which describes the given data. SOLUTION:CALCULATION OF TREND VALUES BY THE METHOD OF SEMI-AVERAGES Month January February March April May June Sales in M. tonnes 230 1.The sale of a commodity in million tones varied from January 2007 to December 2007 in the following: 280 300 280 280 270 240 230 230 220 200 210 200 Fit a trend by the method of semi-averages.650(Total) of 280 First Six months 300 280 280 270 240 Average of the first half= Month July August September October November December Sales in M.2004 110 ---------------------------------------------------------------------------------------------------------------------------------------------- Question 3:-. 305 Trend by the Method of Semi-Averages 290 275 Trend Line 260 Page 6 .e.290 (Total) of last six month 230 220 200 210 200 = 275 tonnes Average of the second half = = 215 tonnes These two figures namely 275 and 215. lakhs) Year Sales (Rs. 520 lakhs. shall be plotted at the middle of their respective periods. at the middle of March-April and that of September-October.5 2002 444 2006 490 2003 454 2007 500 ---------------------------------------------------------------------------------------------------------------------------------------------- Question 4:. lakhs) 2000 412 2004 470 2001 438 1748/4 =437 2005 482 1942/4=485. i. tonees 1. Fit a trend line by the freehand method of semi-average: to the data given below Estimate the sales for the year 2008 if the actual sale for that year is Rs. 2007. and industries in a failure in a country during 1992to 2007: Year No.0 17.8 7.8 17. Sep.8 23.1 9.6 15.4 11. Dec.6 11.1 10.8 9.6 11. Nov.9 20. of failures year No.4 12. Months ---------------------------------------------------------------------------------------------------------------------------------------------- Question 5:.8 11.245 230 215 Actual Data 200 Jan Feb March April May June July Aug.0 -- Page 7 .4 11.8 7 yearly moving total ---153 140 123 108 87 81 81 78 71 63 -- 7 yearly moving average ----21.Calculate 5-yearly and 7-yearly moving average for the following data of the numbers of commercial. of failures 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 23 26 28 32 20 12 12 10 9 13 11 14 12 9 5–yearly moving total 129 118 104 86 63 56 55 57 59 59 69 39 5 yearly moving average --25.0 11.CALCULATION OF 5-YEARLY MOVING AVERAGES Year No.6 11. oct. SOLUTION:.2 12.2 11.6 20. of failure 1992 23 2000 9 1993 26 2001 13 1994 28 2002 11 1995 32 2003 14 1996 20 2004 12 1997 12 2005 9 1998 12 2006 3 1999 10 2007 1 Also plot the actual and trend values on a graph. Value 100 82 65 49 34 20 7 SOLUTION:.ESTIMATING THE TREND VALUES Page 8 .2006 2007 3 1 --- --- --- --- 35 Trend by the Method of Least Squares 30 25 20 15 10 5 0 1992 1994 1996 1998 2000 2002 2004 2006 YEars ---------------------------------------------------------------------------------------------------------------------------------------------- Question6:.Estimate the trend values using the data given by taking Year 1994 1995 1996 1997 1998 1999 2000 Value 12 25 39 54 70 87 105 year 2001 2002 2003 2004 2005 2006 2007 a four-yearly moving average. If we are calculating 4 yearly moving average.0 2005 34 34.75 1997 54 188 47.75 316 79.YEar Value 12 25 4 yearly moving totals --- 4 yearly moving average --- 1994 1995 1996 39 130 32.0 1999 87 84. YEar Value 4 yearly 4.75 There is another method of centering the moving averages.75 230 57.00 374 93.0 54.75 168 42.yearly 4 yearly moving total moving moving avergae average centred 250 62.5 2004 49 49.5 2000 105 92.5 4 yearly moving cebtred 39.75 352 88.lbs) Page 9 . we will then take four-yearly totals and of these totals we will again take 2-yearly totals and divide these totals by 8.5 2006 20 ---2007 7 ------------------------------------------------------------------------------------------------------------------------------------------------- Question 7:.lbs) year Production (m.0 2002 82 81.00 296 74.Assume a four-yearly cycle and calculate the trend by the method moving average from the following data relating to the production of in India: Year Production (m.0 2003 65 65.5 1998 70 70.75 362 90.75 110 27.5 2001 100 90. 50 2326 ---- 581.5-yearly and 7-yearly moving average and plot the data on the graph : Year 1993 1994 1995 1996 1997 1998 1999 2000 2001 Cyclical fluctuations +2 +1 0 -2 -1 +2 +1 0 -2 Year 2002 2003 2004 2005 2006 2007 Cyclical fluctuations -1 +2 +1 0 -2 -1 ---------------------------------------------------------------------------------------------------------------------------------------------- Solution:.50 467 2027 2002 --491.1998 1999 2000 2001 2002 464 515 518 467 502 2003 2004 2005 2006 2007 540 557 571 586 612 SOLUTION:. From the following data.50 571 586 612 4 yearly moving average centered --- 572. calculate 3-yearly .62 511.75 502 2066 2003 500.50 2006 2007 542.50 540 2170 2004 506.50 557 553.75 2002 2001 500.62 529.CALCULATION OF THREE-YEARLY FIVE-YEARLY AND SEVEN-YEARLY MOVING AVERAGES Year Cyclical 3 yearly moving 5 yearly 7 yearly moving fluctuations average moving average Page 10 .00 563.50 ---- ---- IIIustration 8.00 2254 2005 516.) 4 yearly 4.CALCULATION OF TREND BY THE MOVING AVERGE METHOD YEar Production (m.yearly moving total moving average 1998 1999 464 515 ---1964 2000 518 495. lbs. 00 -0.1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 +2 +1 0 -2 -1 +2 +1 0 -2 -1 +2 +1 0 -2 -1 -+ 1.00 -0.1.33 + 0.43 -0.28 -0.43 + 0.33 -1.00 -0.67 +1.14 -0.67 +1.43 +0.28 -0.33 +0.00 average --0 0 0 0 0 0 0 0 0 0 0 --- ---+0.14 -0.14 -0.43 ---- 3 Trend by the Method of Moving Average 2 1 0 -1 -2 1993 1995 1997 1999 2001 2003 2005 2007 Year Page 11 .33 -1.00 -0.00 -0.33 . 3 we can obtain other trend values. first trend value need be obtained and then if the value b si positive we may continue adding the value of b to every preceding value. production in million tones.Below are given the figures of production (million tones) of a sugar factory: Year Production (m. However. Plot these figures on a graph and how the trend line. 1.2. YEar Trend values Yc 84 86 88 90 92 94 96 ∑Yc =630 Since ∑X = 0: a = .-----------------------------------------------------------------------------------------------------------Question 8:. For 2002 it will be 84 +2 = 86. SOLUTION:. For X = -1. Yc = 90 +2 (-2) = 86. N=7. then instead of adding we will deduct. If b is negative . since the value of b is constant. tones) Y 2001 80 -3 -240 9 2002 90 -2 -180 4 2003 92 -1 -92 1 2004 83 0 0 0 2005 94 +1 +94 1 2006 99 +2 +198 4 2007 92 +3 +276 9 N=7 ∑Y =630 ∑X =0 ∑XY = 56 ∑X2 = 28 The equation of the straight line is Yc = a + b X. X units one year. (iii) The graph of the above data is given below: Trend by the Method of Least Squares 100 Trend Line 95 Page 12 . For X = -3. Yc =90+2 (-1) =88. Yc 90+ 2(-3) =84 For X = -2. and so on. ∑X2 =28 A= = 90. 2004. Y units.tonnes) (i) (ii) 2001 80 2002 90 2003 92 2004 83 2005 94 2006 99 2007 92 Fit a straight line trend to these figures. for 2003 it will be 86 + 2 = 88.FITTING THE STRAIGHT LINE TREND’ Production X XY X2 (m. ∑XY =56.b= ∑Y = 630. and b = =2 Hence the equation of the straight line trend is Yc = 90+ 2X Origin. Similarly by putting X =0. 90 8 Actual Data 80 75 0 2001 2002 2003 2004 2005 2006 2007 Years If instead of middle year as origin. Estimate the value for 2012: Year 2001 2002 2003 2004 2005 2006 2007 Page 13 . Y=84. we take first year as origin the solution would be as follows: Yc Year Production X XY X2 (m.946 ∑X2 =91 ∑Yc=630 Yc = a+b X. tones) 2001 80 0 0 0 84 2002 90 1 90 1 86 2003 92 2 184 4 88 2004 83 3 249 9 90 2005 94 4 376 16 92 2006 99 5 495 25 94 2007 92 6 552 36 96 N=7 ∑Y = 630 ∑X = 21 ∑XY = 1. For X =0.---------------------------28b = . ---------------------------------------------------------------------------------------------------------------------------------------------------------------- Question 9:. ∑XY = a ∑X + b∑X2 Substituting the value. (i) by 3 1890 = 21 a + 63 b 1946 1 a + 91 b --. However. In the first case 2004 was taken as origin whereas in the second case 2001 was taken s origin. production in million tones.56 or b = 2 Substituting the value of b In Eqn. trend values are the same. one year: Y units. Note: The difference in the two equations is because of the difference in origin. 630 = 7a + 21 b 1946 = 21 a + 91 b Multiplying Eqn.Fit a straight line trend for the following series. (i) 630 = 7a +21 (2) 7 a =630 -42 = 588 or a = 84 Thus. ∑Y = Na+b ∑X. the equation is Yc = 84 + 2 X Origin 2001: X units. a= = 80 85 95 X2 9 4 1 0 1 4 9 ∑x2= 28 = 76. X will be +8 Y2010 = 76 + 4.86.857 Hence Y = 76 + 4.Fitting Straight Line TRend Year Production of Deviations XY Steel (m.Production of steel (m. Thus the likely sales for the year 2012 is 114.857 (8) = 114. lakhs) Page 14 .86m. ---------------------------------------------------------------------------------------------------------------------------------------------- Question 10:.tonnes) 60 72 75 65 Solution:.857 X For 2012. assuming that the same rate of change continues . from 2004 tones) Y 2001 60 -3 -180 2002 72 -2 -144 2003 75 -1 -75 2004 65 0 0 2005 80 +1 +80 2006 85 +2 +170 2007 95 +3 +285 N=7 ∑Y= 532 ∑X=0 ∑XY=136 The equation of the straight line trend is Y = a+b X Since ∑X=0. tones. what would be the predicted earning for the year 2009? Year 2000 2001 2002 2003 2004 2005 2006 2007 Earning 38 40 65 72 69 60 87 95 (Rs. And b= = = 4.Fit a straight line trend by the method of least squares the following data . 5 2003 72 -0.50 12.5 2001 40 -2.25 2003 72 -0.25 2004 69 +0.667X For 2009. The same result will be obtained if we do not multiply the deviations by 2.75 + 40.00 2.25 N= 8 ∑Y = 526 ∑X =0 ∑XY = 308 ∑X2 = 42.5 2007 95 +3.SOLUTION:.5 +34.FITTING OF STRAIGHT LINE TREND BY THE METHOD OF LEAST SQUARES Year Earning (RS.337 = 106.667: Y = 65.5 -97.5 2005 60 01.5 +100.25 2006 87 +2.75 +3.087.087 lakhs.25 2002 65 -1.5 +332.667 (11) = 65.50 0.00 A= = = 65.50 6. Lakhs) Y Deviations from 2003.333 Page 15 .5 2004 69 +0. Y wll be Y = 65.50 2.75 B= = = 3.5 -36.5 -133.5 2000 38 -3.75: b = = = 7. But in that case our computations would be more difficult as could be seen below: Year Sales in Deviations XY X2 thousands of from 2003. X will be + 11 When X is 11. 106.5 +217.5 +90.5 N=8 ∑Y = 526 The equation of the straight line trend is A= = Deviations multiplied by 2 X XY X2 -7 -5 -3 -1 +1 +3 +5 +7 ∑X =0 -266 -200 -195 -72 +69 +180 +435 +665 ∑XY = 616 49 25 9 1 1 9 25 49 ∑X2 = 168 = 65.25 2005 60 +1.5 2006 87 +2.5 2002 65 -1.00 12.25 2007 95 +3.25 2001 40 -2.5 rupees Y 2000 38 -3.00 6.00 0.75 +3. Thus the estimated earnings for the year 2009 are Rs. 803 -3. as in the first method discussed above.183 +0.(i) The equation of the straight line trend is Yx = a + b X Since ∑X is not zero. we get 1113 = 7 a +357 b Deducting equation (iii) from (i) = 1. Eliminate the trend.38 -490 = -356 b or b = Substituting the value of b in equation (i) 623 = 7a + 1. tones) 2001 2002 2003 2004 77 88 94 85 2005 2006 2007 91 98 90 (i) (ii) (iii) Fit a straight line by the least squares method and tabulate the trend values. What component of series are thus left over? What is monthly increase in the production.577 2004 85 0 0 0 88.423 +6.437 2007 90 +5 +450 25 95.703 2 N =7 ∑Y = 623 ∑X =1 ∑XY = +159 ∑X = 51 ∑Yc = 623 ∑(Y-Yc) =0 623 = 7 a+b 159 = a+51b Multiply the second equation by 7.The advantage of this method is that the value of b gives annual increment of change rather than 6 monthly increment.703 -5.38 Page 16 .957 2003 94 -1 -94 1 87.563 +6.043 +1. It is clear from the above illustration that in the first case the value of b is half of what we obtained from the second method ( b was 3.803 2005 91 +1 +91 1 90. tones) 2004 as Yc origin 2001 77 -4 -308 16 83.283 -6.333 in the second case. tones) Year production (m.283 2002 88 -2 -176 4 86. Hence we will not have to double the v alue of b to obtain yearly increment. the value of a and b will be obtained directly by solving the following two normal equations: ∑Y = N a + b ∑X ∑XY = a ∑X + b ∑X2 Year Production Taking XY X2 Trend value (Y – Yc) (m.817 2006 98 +2 +196 4 91. Answer:.Below are given the figurer of production (in million tines) of a sugar factory : Year production (m.67 in the first case and 7.) ---------------------------------------------------------------------------------------------------------------------------------------------------------------- Question11:. 38 = 621.38X5) = 95. Y = 88.803 SO the equation of straight line trend is Y = 88.38 X When X = -4.5.563 When X = +5.38 = 87. Y = 88.717 Page 17 .38 = 90. we solve the following normal equations: ∑Y = Na + b X + c ∑X2 ∑XY = a ∑X + b ∑X + c ∑X3 ∑X2 Y = a ∑X2 + b ∑X3 + C∑X4 Year 2002 Prices X X2 X3 X4 XY X2Y Trend (Rs. Fit a parabola y = a + v X + c X2 TO THESE DATA PRICE OF THE COMMODITY FOR THEYEAR 2008: Year Price Year Price 2002 100 2005 140 2003 107 2006 181 2004 128 2007 192 Solution:. Y = 88. Y = 88.703 (ii) After the eliminating the trend we are left with cyclical and irregular variations The monthly increase in the production of sugar is b/12.423 When X =0.To determine the values of a.803 + 1.e.38 X2) = 91. tones ---------------------------------------------------------------------------------------------------------------------------------------------------------------- Question12:.The price of a commodity during 2002-2007 is given below.803 + (1. When X = +1.38 (-4) = 88.803 + (1. b and c. 1.7a = 623 – 1. i.803 .803 + 1.803 – 2. Y = 88.52 = 83.183 When X = +2.803 – 1.803.38 (-2) = 88.803 + 1. Y = 88.) Value Y (YC) 100 -2 4 -8 16 -200 400 97.38 /12 = 0.803 + 1.283 When X = -2 .043 When X = -1 . Y = 88.115m.62 or a =88.76 = 86. 002 115 =771 848 = 6a + 3b + 19 c 771 = 3a + 19 b +27 c 3.352 = 280 b + 168 c Multiplying equation (iv) by 8.857 N=6 ∑Y = 848 ∑X = 3 ∑X2 = 19 ∑X3 = 27 ∑X4 = ∑XY ∑X2Y=3.2003 107 -1 1 -1 1 -107 107 110.885 2007 192 +3 9 +27 81 +576 1728 196.401 2004 128 0 0 0 0 0 0 126.099 = 19 a + 27 b + 115c Multiplying the second equation by 2 and keeping the first as it is we get 848 = 6a + 3 b + 19 c 1542 = 6a + 38 b + 54 C -- -- -- -- ----------------------------------694 = -35 b – 35 c Or 35b + 35 c = 694 Multiply Eqn.657 2005 140 +1 1 +1 1 +140 140 146.485 2006 181 +2 4 +8 16 +362 724 169.552 Page 18 .552 Solving equations (iv) and (v) 280 b + 280 c = 5. we have 280 b + 280 c = 5.099 ∑Yc=848.649 = 57 a + 361 b + 513 c 9297 = 57 a + 81 b + 345 c --------------------------------------5. (iii) by 3. (ii) by 19 and Eqn. we get 14. 144 = 97.786 Substituting the value of C in Eqn.786 ( -2)2 = 126.352 -- -- -- -------------------------112 c = 200 or c = 1.042 and c = 1.5 = 631. b = 18.657 + 18.042 X + 1786 X 2 When X = .657 – 36.042 (-1) + 1.786 (-1)2 = 126.126 + 33.042 (-2) + 1. Y = 126.857 Price for the year 2008 Page 19 .657 .786 (3) 2 = 196.18.042 + 1.042 (3) + 1.657 + 18.280 b + 168 c = 5.786 = 110.042 (2) + 1.657 A = 126.2 Y = 126.485 When X =2 Y = 126.786) = 6 a + 54.657 + 18. Y = 126.717 When X = -1 Y = 126.042 848 = 6a + 3 ( 18.94 or a = 126.885 When X = 3. 5 or b = 18.401 When X = 1 Y = 126.084 + 7. (iv) 35 b + (35 X 1.786 (2) 2 = 169.657 + 18.657 + 18.042) + 19 (1.786 = 146.042 + 1.657 .934 6 a = 759.786 Thus Substituting these values in the equation.786 _ = 694 35 b = 694 – 62.657 + 18. 227.For 2008 X would be equal to 4.786 (4)2 = 126.657 +72.The sales of a company in lakhs of rupees for the years 2001 to 2007 are given below Year 2001 2002 2003 2004 2005 2006 2007 Sales (Rs. Putting X = 4 in the equation Y = 126.401 Thus the likely price of the commodity for the year 2008 is Rs.154X Estimated sales for 2008. lakhs) 32 47 65 92 132 190 275 Find trend value by using the equation Y=a bx and estimate the value for 2008.41 approx. Logy = 1.9lakh Page 20 .042 (4) + 1.168 + 28.657 + 18. The graph of the actual and trend values is given below:200 180 Trend Line 100 140 Actual Data 120 100 0 2002 2003 2004 2005 Years 2006 2007 ------------------------------------------------------------------------------------------------------------------- Question13:. Rs.385.576 = 227.9704+0. 1206 1 +2.154 X For 2008X would be +4.9 lakhs.for example.5864 = 385.8129 1 -1.When data are expressed annually there is no seasonal variation.5576 2007 275 +3 2. Movement and concededly data frequently exhibit strong seasonal movement and considerable industry attaches to devising a pattern of average seasonal .8129 2004 92 0 1.9707 + .3442 2003 65 -1 1. log Y will be Log Y = 1. when X =4. log Y (RS. 385.154 Hence log Y = 1.5153 2002 47 -2 1. 154 (4) = 2. if we Page 21 .9704.7926 ∑X2 =28 ∑X log Y = 4.4393 9 +7.1206 2006 190 +2 2.5051 9 -4.2788 4 +4. ---------------------------------------------------------------------------------------------------------------------------------------------------------------- Seasonal variation: Most of the phenomena in economics and business show seasonal patterns .5864 Y = AL 2. Lakhs) 2001 32 -3 1. if we observe .9 Thus the estimated sale for the year 2008 is Rs.SOLUTION:.FITITNG EQUATION OF THE FORM Y = ab X YEar Sales(Y) X Log Y X2 X.9704 + 0.3179 N=7 ∑Y = 833 ∑X=0 ∑log Y = 13.6721 4 -3. = = 0.9638 0 0 2005 132 +1 2.3237 Log a = Log b = = = 1. we shall be able to answer a very basic question namely. Are 75 per cent of these of the average month. 3. The method should be designed to meet the following criteria: (1) It should measure only the seasonal forces in the data. It should not be influenced by the forces of trend or cycle that may be present. The following are some of the method more popularly used for measuring seasonal variation : 1. 2. Monthly or Quarterly). If we know by how much the sales of this quarter are usually above or below the previous quarter for seasonal reasons. Seasonal. Ratio-to Trend Method. Page 22 . Or a number of years. the June quarter always contributes so much more or so much less to the series. i. Ratio-to Moving Average Method. we can calculate. That has taken place typically in each month. Purchases or whatever our data happen to be. In order to analyses seasonal variation . In index form . (2) It should modify the erratic fluctuations in the data with an acceptable system of averaging. are maximum. (3) It should recognize slowly changing seasonal patterns that may be present and modify the index to keep up with those changes. Before attempting to measure seasonal variation certain preliminary decisions must be made. Indexes are given as percentages of their average . To obtain a static description of a pattern of seasonal variation it will be desirable to first free the data from the effects of trend .e. each month is represented by a figure expressing it as a percentage of the average month. For example. Orders. One these other components have been eliminated. Method of Simple Average (Weekly. Sales. it is necessary to assume that the seasonal pattern is superimposed on a series of value and is independent of these in the sense that the same pattern is superimposed irrespective of the level the series . this means that for the month of January. If a seasonal index for January is 75. Thus a second index may be specific or typical. i.It is thus a generalized expression of seasonal variations for a series.e. for example it necessary to decide whether weekly quarterly or monthly indexes are required . For monthly data. this will be decided in the light of the nature of the problem and the type of data available seasonal adjustments help avoid misinterpretation. Any acceptable modern method for computing such an index probably will be programmed for a computer solution. Thus the measures of seasonal variation are called seasonal indexes (per cent). cycles and irregular variation. a seasonal index consists of 12 numbers. one for each month of year. A typical seasonal index is obtained by averaging a number of specific seasonal . A specific seasonal index refers to the seasonal changes during a particular year. There are many techniques available for computing an index of seasonal variation. was this due to an underlying up what tendency or simply because this quarter is usually seasonally higher than the previous quarter .a measure of seasonal variation which is usually referred to as a seasonal index. Many of the simpler methods were devised prior to the development of electronic computers and were designed to sacrifice precision for ease of computation.observe the sales of a bookseller we find that for the quarter July-September (when most of the students purchase books) sales. 4 262. Link Relative Method.Consumption of monthly electric power in million of kWh for street lighting in a big city during 2003-07 is given below: YEAR JAN.33 82.609 1.61 91. 325 342 367 389 422 DEC.05 Page 23 . (ii) Find the totals of January.2 285. if we are given monthly data for five years than.450 1.44 80. FEB. (i) Arrange the unadjusted data by years and month (or. March April May June July August (2) 318 281 278 250 231 216 223 245 (3) 342 309 299 268 249 236 242 262 (4) 367 328 302 287 269 251 259 284 (5) 392 349 342 311 290 373 282 305 (6) 420 378 370 334 314 296 305 330 Monthly total for 5 year (7) 1.353 1.6 254. 302 321 345 364 396 NOV.14 103.we will the same result Question14:.645 1. 347 364 394 417 452 CONSTRUCTION OF SEASONAL INDICES BY THE METHOD OF MONTLY AVERAGE Month 2003 Consumption of monthly electric power 2004 2005 2006 2007 (1) Jan.79 90.4. Fab.57 85.8 329.426 Five yearly average (8) 367. quarters if quarterly data are given).8 290. (v) Taking the average of monthly average as 100. Solution :- JULY 223 242 259 282 305 AUG.0 321. The following step are necessary for calculation the index. APR. 269 288 309 328 356 OCT.0 270. compute the percentages of various monthly average as follows: Seasonal index for January = monthly average for January / average of monthly average X 100 If instead of the average of each month the total of each month are obtained . (iii) Divide each totals by the number of years for which data are given. Method of Simple Averages:This is the simplest method of obtaining a seasonal index. (iv) Obtain by average of monthly average by dividing the total of monthly average by 12. etc. we shall first obtain total for each month for five years and divide each total by 5 to obtain an average. MAY JUNE 2003 318 281 278 250 231 216 2004 342 309 299 268 249 236 2005 367 328 320 287 269 251 2006 392 349 342 311 290 273 2007 420 378 370 334 314 296 Find out seasonal variation by the method of monthly averages .272 1. MAR.2 Seasonal variation index (9) 116.839 1. February.311 1.88 101. For example. 245 262 284 305 330 SEP. 002 Average 1.7 3.Assuming that trend is absent .7 4.0 345. 3.1 3. Column no 7 gives the total for each month for five years.0 14.0 4.30 Page 24 .974 3.6 369.5 3.4 4.1 3.7 3.55 Seasonal Index 98.8 1.74 95.7 16.30 Notes fro calculating seasonal index 4th Quarter 3.1 3.9 3.52 124.6 4.7/316.2 Average 3.800. Thus. we shall compute weekly or quarterly average by following the same procedure as explained above.1 3.September October November December 269 302 325 347 288 321 342 364 309 245 367 394 328 364 389 417 356 396 422 452 Total 19.6 2006 4. Question15:.675 4. the percentage for January = 36.3 2005 3.55 95.0/316.we are given weekly or quarterly data.66 The above calculation are explained below: 1.3 4. The average of monthly averages is obtained by dividing the total of monthly averages by 12. 4.0 394. 2.0 Total 14.3 3.7 4.14 percentage for February = 329.5 14.843 1.4 316.COMPUTATION OF SEASONAL INDICES YEar 1st Quarter 2nd Quarter 3rd Quarter 2004 3.1 4.1 3. in column no 8 each total of column no 7 has been divided by 5 to obtain an average for each month.5 1.555 1.2 3.6 3.66 110.7 310.0 4.3 2007 3.7 X 100 =103.5 3.3 4.9 3.3 3. determined if there is any seasonality in the data given below : Year 2004 2005 2006 2007 Ist Quarter 2ndQuarter 3rdQuarter 4thQuarter 3.200 100 97.instead of monthly data .0 What are the seasonal indices for various quarters? ANSWER:.583.88 109. In column no 9 each monthly average has been expressed as a percentage of the average of monthly averages.0 4. if .125 3.88.4 4.13 116.728 1.6 3.7 X 100= 116. it is often desirable to use the median which is generally not affected by very high or very low values. The next step is to divide the original data month by month by the corresponding trend values and multiply these ratios by 100. for instance . Trend values are obtained by applying the method of least squares.30 ---------------------------------------------------------------------------------------------------------------------------------------------------------------- Ratio-to-Trend Method This method of calculating a seasonal index (also known as the percent-age—to trend method )is relatively simply and yet an improvement over the method of simply averages explained in the preceding section .725 X 100 Seasonal Index for the first quarter = Seasonal Index for the second quarter = X 100 = 98. 2.in effect T x S x C x I /T = S x C x I.66 X100 = 110.this method assumes that seasonal variation for a given method is constant fraction of trend .an earthquake. Since such scrutiny of the data requires considerable knowledge of prevailing conditions and is to a large extent subjective . 3. The seasonal index for each month is expressed as a percentage of the average month .200/ the sum of the 12 value). ---------------------------------------------------------------------------------------------------------------------------------------------- Page 25 . and adjustment is made by multiplying each index by a suitable factor (1.the ratio-to-trend method presumably isolates the seasonal factor in the following miner . February . it is sometimes possible to ascribe a definite cause to usually high or low values. the some of 12 values must equal 1. are averaged with any one of the usual measures of central value. If it is not.200 or 100 per cent. For series that are not subject to pronounced cyclical or random influences and for which trend can be computed accurately. the values so obtained are now free from trend and the problem that remains is to free them also of irregular and cyclical movements. 4. Trend is eliminated when the ratios are computed . The steps in the computation of seasonal index by this method are. if the data are examined month by month . A careful selection of the period of years used in the computation is expected to cause the influences of prosperity or depression to offset each other and thus remove the cycle. this method may suffice.The average of averages = Seasonal Index = = = 3. This gives the final seasonal index. 1.74 Seasonal index for the third and fourth quarters = X 100 = 95. famine and the like) they may be cast out and the mean of the remaining items is referred to as a modified mean. etc. the median or the mean . When such causes are found to be associated with irregular variations (extremely bad whether . In order to free the values from irregular and cyclical movements the figures given for the various years for January. Random elements are supposed to disappear when the ratios are averaged. CALCULATING TREND BY METHOD OF LEAST SQUARES Year Yearly Yearly Deviations XY X2 Trend totals average Y from midvalues year 2003 140 35 -2 -70 4 32 2004 180 45 -1 -45 1 44 2005 200 50 0 0 0 56 2006 260 65 +1 +65 1 68 2007 340 85 +2 +170 4 80 N=5 ∑Y = 280 ∑XY = ∑X2 = 10 120 The equation of the straight line trend is Y=a+bx A= = = 56 b= = = 12. half of 2nd and half of 3rd is 32.Find seasonal various by the ratio-to-trend method from the data given blow: Year Ist Quarter 2ndQuarter 3rdQuarter 4thQuarter 2003 2004 2005 2006 2007 30 34 40 54 80 40 52 58 76 92 36 50 54 68 86 34 44 48 62 82 Answer: .Question16:. Calculation of Quarterly Trend Values. Consider 2003. first we will determine the trend for yearly data and then concert it to quarterly data.e. So Page 26 . Quarterly increment is 3.For determining seasonal variation by ratio-to-trend method. Quarterly increment = = 3. trend value for the middle quarter i. 5 2006 63.5) X 100 = 109.92 + 89.e.36 102.28 + 102. i.the trend value of 2nd quarter is 32.67 106.41 514. Trend value for the 1st quarter is 30.72 2005 77.5 45.e.46 Total of average = 92. Thus for 1st Qtr.20 105.5 48. i. Adjusted 92.05 117.5 84.5 2004 39.04 114. ie 30.09 for 2nd Qtr.12 88. Of 2003.84 591.15 107.5 66.5 30.5 60.52 97.46 93.15 2004 86.52 2007 105.15 = 403. 36.5 72.91 79. GIVEN QUARTERLY values as % of TREND VALUES 2nd Quarter 3rd Quarter 4th Quarter Year 1st Quarter 2003 109. 33.35 109.12 Page 27 .I.5 and of 4th quarter is 33.5 42.08 122. (40/30.89 90. 27.5 78. We thus get quarterly trend values as shown below: TREND VALUES YEar 1st quarter 2nd quarter 3rd quarter 4th quarter 2003 27.77 + 118.15 S.04 Total 463.5 The given values are expresses as percentage of the corresponding trend values.5 33.5 54.92 89.e.5 81.5.77 118.77 Average 92.29 97.5 57.62 445.5.5 36.28 102.42 93.5) X100 = 131.5 + 3..5 2005 51. the percentage shall be (30/27.5 69.5 and for 3rd quarter is 32+ i.5 -3.15 etc.5 2007 75.96 117.09 131.34 2006 85.84 85. The elimination of extremes may be achieved while we are averaging all Januarys. If it totals less than 1. Since seasonal variations recur every year. the centered 12-month moving average approximates T.Since the total is more than 400 an adjustment is made by multiplying each average by and final indices are obtained. an adjustment is made to eliminate the discrepancy. Thus. It remains to rid the data of irregular variations.Thus by using the median as an average we can obtain the typical seasonal relative for each month which will not be affected by irregular factors. The steps necessary for determining seasonal pattern by this method are . 1:.C. In any array of seasonal relatives for each month.200 or 100 per cent.200 for 12 month by definition. Page 28 . By averaging these percentages for a given month (step 4) the irregular factors tend to cancel out and and the average itself reflects the seasonal influence alone. but also because when we come to eliminate sweaonality from the original data we do not wish to arise or lower the level of the data unduly. a centered 4 quarter moving average must be used.a centered 12 month moving average tends to eliminate these fluctuations(in case of quarterly data.200 ( or averages more than 100) then the original data adjusted in terms of it will total less than the unadjusted original data. A separate table is prepared in which the calculations involved in this step are shown.and – in the process of averaging—to eliminate the irregular factors. 2. These means are preliminary seasonal indexes. Februarys and the like We do this by using an appropriate type of average. We have now succeeded in eliminating from the original data to a considerable extent the disturbing influences of trend and cycles. We assume that the relatively high or extremely low values of seasonal relatives for any month are caused by irregular Factors. The adjustment consist of multiplying average of each month obtained in step 4 by 1200 the total of the modified mean for 12 months This adjustment is made not only to achieve accuracy. The median is appropriate since it is not affected by extremes .month moving average. 3. 4:. since the fluctuations have a time span of 12 months. Express the original data for each month as a percentage of the centered 12 month moving average corresponding to it . Thus if a seasonal index affregates more than 1. a value or several values on one end or both ends may be dropped and then the arithmetic mean of the remaining seasonal relatives is taken.) The centered 12 month moving average which aims to eliminate seasonal and irregular fluctuations ( S and I) represents the remaining elements of the origin al data . Divide each monthly item of the original data by the corresponding 12. and list the quotients as percentage of Moving average.the purpose of this step is to average . 5:-Sometimes a so-called modified mean is used as an average for ach month.that is. extreme values are omitted before the arithmetic mean is taken. trend and cycles.200 the opposite would be true. ---------------------------------------------------------------------------------------------------------------------------------------------------------------- Ratio-to-moving average method* The ratio-to-moving average method also known as the percentages of moving average method is the most widely used method of measuring seasonal variations. Here. They should average 100 per cent or total 1. namely.Eliminate seasonality from the data by ironing it out of the original data. If the total is not equal to 1. Calculate seasonal indices by the ratio to moving average method. represents irregular and seasonal influences. If the actual value for any month is divided by the 12 month moving average centered to that month. I ---------------------------------------------------------------------------------------------------------------------------------------------------------------- Question17:. from the following data : Year 2005 2006 2007 Answer:Year 2005 Ist Quarter 68 65 68 2ndQuarter 62 58 63 Quarter Given figures I II 68 62 3rdQuarter 61 66 63 4 figure moving totals 4thQuarter 63 61 67 2 figure moving totals 4 figure moving average Given figure as % of moving average 505 6.260 101. Hence in effect S x I = S.375 104.186 96. from which this method gets its name. Presumably cycle and trend are removed.The logical reasoning behind this method follows from the fact that 12-month moving average can be considered to represent the influence of cycle and trend C x T.19 499 62. This may be represented by the following expression: Tx SxCxI = SxI TXC Thus the ratio to the moving average.21 254 III 61 251 IV 63 247 2006 I 65 252 Page 29 . If the ratio for each worked over a period of years are then averaged most random influences will usually be eliminated.54 498 62. 80 100.05 Seasonal Index 105.67 Total 210.35 98.II 58 502 62.83 By expressing each quarterly average as percentage of 99.30 X 100 = 95.97 4th Quarter 101.04 64.83 we will obtain seasonal indices.25 190.20 95.43 2007 106.52 Arithmetic average of averages = = 99. Seasonal index of 1st Quarter = Seasonal index of 2nd Quarter = X 100 = 105.875 95.10 Average 105.30 95.67 250 66 253 IV 61 258 2007 I 68 513 255 II 63 516 261 III 63 IV 67 CALCULATION OF SEASONAL INDEX Percentage to Moving Average 1st Quarter 2nd Quarter 2005 2006 104.50 -196.97 -201.60 100.70 98.50 511 64.21 Page 30 .21 3rd Quarter 96.500 97.21 92.125 95.875 104.04 97.125 106.97 503 63.63 104. March April May Jun July August September October November December Sales (in thousand unit) 11 11 12 13 14 14 15 15 15 16 18 20 12 13 13 15 16 18 20 20 21 22 24 25 Solution COMPUTATION OF 12-MONTH MOVING AVERAGES Year & Month Sales (in thousand units) 12-month moving total 12-month moving average 2-monht moving total of col. March April May Jun July August September October November December Sales (in thousand unit) 10 12 13 15 16 16 17 18 18 19 22 22 10 12 11 12 13 15 15 17 18 20 22 24 Year and month 2006 Jan.6) X100 Page 31 . March April May Jun July August September October November December 2005 Jan.Seasonal index of 3rd Quarter = Seasonal index of 4th Quarter = X 100 = 100. Apply ratio to moving average method to calculate seasonal indices from the following data: Year and month 2004 Jan. 5 ÷ 2) Percentage of centred 12month moving average (col. March April May Jun July August September October November December 2007 Jan. Feb. 4 Centered 12month moving average (Col. Feb. 2 ÷ col.52 ---------------------------------------------------------------------------------------------------------------------------------------------------------------- Question18:-. Fab. Fab.97 X 100 = 98. 00 137.25 15. 16.5 31.8 33. 189 Feb.1 30.08 16.83 94.1 2004 Jan Feb.42 19 195 Nov.8 32.67 14.58 14.25 15.25 22 193 Dec.00 16.54 102.3 32.54 108.6) X100 7 10 12 13 15 16 16 17 199 Aug 18 198 Sept.46 109.75 15.7 Percentage of centred 12month moving average (col.4 32. 4 1 2 3 4 5 Page 32 .50 July Year & Month 15 Sales (in thousand units) 12-month moving total 12-month moving average 28.5 11 31.83 69.75 12 183 April 16.92 2-monht moving total of col.9 14.62 70.25 15. 16. 2 ÷ col.10 22 32.32 16.4 30.16 136.4 29.00 14 176 June 15.37 78.17 14.67 16.0 29. 16.08 11 186 March 16.46 Centered 12month moving average (Col.67 14 174 14. 5 ÷ 2) 6 103.50 33.92 16.67 15.12 86.50 13 180 May 15. MArch April May June July 2 3 4 5 6 7 198 16.50 18 197 Oct.33 116. 2005 Jan.08 16.59 95. 15. 80 33.9 34.50 10 172 Feb.66 14.0 28.34 15.9 32.25 12 181 103.08 18 194 14.17 16.59 16.3 31.5 30.58 15.25 14.42 17.7 14.42 20 197 111.173 Aug. Nov Dec.1 2-monht Centered 12- Percentage of 16.67 16.7 14.17 15.42 16.75 22 200 28.33 11 177 14.00 16.92 14.75 17 192 28.92 15.75 14.29 122.83 14.50 15 191 103.33 13 185 28.96 106. 15 174 173 172 171 172 2006 Jan March April May June July August Sept.59 16.75 31.46 14.25 85.0 33.29 140.58 96.0 28.5 32.92 12 208 12-month 126.46 14. 2007 Jan.09 16.25 17.41 83.92 24 203 14.42 12 174 28. Year & Month Sales (in 28.58 14.79 142.12 70. 14.33 15 189 14.91 80.67 15.08 111.33 12-month Page 33 .37 14.2 29.0 15.3 29.83 14.2 31.83 94.50 15.92 14. Oct.33 70.62 75.54 133.29 15.17 16. 2 90.195.28 74.08 88.83 18.6) X100 6 7 17.9 96.00 16 218 June 17.20 108.17 18.5 70.3 116.76 136.7 88. Median has been used to average the figure given for the individual Page 34 .2 80.7 103.4 35.9 70.91 month moving centred 12average (Col. 2 ÷ col.0 142.3 116.8 103.7 36.2 75. Feb March April May June July August Sept Oct.28 102.58 35.6 95.55 It should be noted that there are only three values for each month since the moving average failed to provide average for the first half of 2004 and the last half of 2007.70 73. Dec. Nov.5 98.1 Feb.85 95.4 106.84 116.5 111.92 109.5 85.21 98. Dec.17 18 219 July August Sept.5 73. 5 month moving ÷ 2) average (col.5 36.8 102.5 88.42 18.56 1.45 74. Nov. Oct.83 17.0 70.53 137.3 75.7 107.5 211 17.41 17.8 106. 4 2 13 3 4 5 34.9 103.0 133.4 109.4 95.0 140.8 1200. 69.50 83.8 2007 Median Seasonal Index 13 214 April 15 216 May 18.0 83.5 74.1 74.4 109.6 126.25 20 20 21 22 24 25 COMPUTATION OF SEASONAL INDICES 2004 2005 2006 Jan.1 86.0 94. March thousand units) moving total moving average moving total of col.1 133. 18.5 106.0 70.80 75.8 70.0 140.91 83.9 122.03 88.7 103.4 83.5 140.8 133.4 78.7 84. 8 8.28-1.1 Answer:CALCULATION OF SEASONAL INDICES BY THE METHOD OF LINK RELATIVES Quarter YEAR I II III IV 2003 2004 2005 2006 2007 Arithmetic average Chain relatives Corrected chain relatives Seasonal indices 62.5 9.86 =108.5 77.615 X100 = X100 = 100 100 = 88.3.1 129.28 =121.200.64 – 5.5 86.6 80.9 8.4 2006 7. The sum of 12 values obtained is 1.5 8.28 108.195.38 =123.3 95.0 6.3 109.21 94.3 146.0 7. It is necessary therefore to make an adjustment so that the total is 1.35 = 128.18 X100 = 113.6 108.8 113.8.3 8.8 =108.60 Page 35 .605 =131.01 The calculations in the above table are explained below: Chain relative of the first quarter ( on the basis of first quarter) = 100 Chain relative of the first quarter ( on the basis of the last quarter) 104.8 6.3 6.66 =93.4 7.03 per cent of htose of the average month.The final result thus obtained gives us the seasonal indices.The adjustment is done by multiplying the average (median) values by = 1.53 per cent of those of average month. and so on.73 131. typical November sales are 133.4 7.0 106.025 = 118.9 68.64 123.3 88.5 2007 6.3 QUARTERLY FIGURES 2005 6.73 .004.6 110.5 7.675 = 106.6 =86.8 7. ---------------------------------------------------------------------------------------------------------------------------------------------- Question19:. The interpretation of this index is very simple.months.6 7.3 143.Apply the method of link relatives to the following data and calculate seasonal indices: Quarter I II III IV 2003 6.7 2004 5.2 112.2 5. Typical April sales are 84.1 93.35 120.6 111. Plot the original and trend values on the same graph paper. 50 1.Using three-yearly moving averages.= = 106. 60 1. determine the trend and short-term flections.500 December 250 3.7 Difference per quarter = = 1.100 June 150 2.600 March 70 1.The following are the monthly figures of advertising expenditure and sales of a firm .900 ---------------------------------------------------------------------------------------------------------------------------------------------- Question21:.675 Adjusted chain relatives are obtained by subtracting 1 X 1.7 The difference between these chain relatives = 106.4 Seasonal variation index = X 100 ---------------------------------------------------------------------------------------------------------------------------------------------- Question20:.200 November 200 3.600 September 170 2. expenditure Rs. 2 X1.Plot the following data on a graph paper and ascertain trend by the method of semi-average: Year Production (in million tonnes) 2201 10 2002 120 2003 95 2004 105 2005 108 2006 102 2007 112 Question22:.500 August 160 2.it is generally found that advertising expenditure has its impact on sales generally after 2 month .675. 3 X 1.800 April 90 2. Year 1998 1999 2000 2001 2002 Production (in 000 tonnes) 21 22 23 25 24 year 2003 2004 2005 2006 2007 Production (in 000 tonnes) 22 25 26 27 26 Solution:.7 – 100 = 6.675 from the Average of corrected chain relatives = = = 113. Jan.Calculation of trend and Short-term Fluctuations Year Production 3-yeraly Totals 3-yearly average (Trend values) Short-term Fluctuations Page 36 .400 Feb.675.900 May 120 2.000 October 190 2.200 July 140 2.allowing for the time lag calculate coefficient of correlation: Month Advertising Sales (Rs) month Advertising Sales (Rs) expenditure Rs. 50 542.75 503.63 511.33 +1. 2003 83 2004 94 2005 99 2006 92 2007 104 Solution:.00 +0.63 529.67 24.00 23.00 26.67 - ---------------------------------------------------------------------------------------------------------------------------------------------------------------- Question23:.33 24.33 - 0 -0.50 ---------------------------------------------------------------------------------------------------------------------------------------------- Question24:.Calculate trend values by the method of least-squares from the data given below: Year 2000 2001 2002 sales 80 90 92 Plot the data showing also the trend line.5 506.50 553.1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 21 22 23 25 24 22 25 26 27 26 66 70 72 71 71 73 78 79 - Yc (y-yc) 22.0 500.50 - 4-yearly moving Average centered 495.67 +0.00 23.50 581.Calculation of trend Values By 4 –yearly Moving Averages Method Year Production 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 464 515 518 467 502 540 557 571 586 612 4-yearly moving Totals 1964 2002 2027 2066 2170 2254 2326 - 4-yearly moving Average 491.33 26.00 572.Fitting Straight Line Trend By Method of Least Squares Year Sales Deviations Deviations XY X2 Yc Page 37 .75 516.33 -1.67 23.50 563.Calculate the trend values by the method of 4-yearly moving average: Year Production 1998 464 1999 515 2000 518 2001 467 2002 502 2003 540 2004 557 2005 571 2006 586 2007 612 Solution:.67 0 +0. 0 95.25(-7) = 91.0 85.5 98.25 X Y 2000 = 91. b = ∑Xy /∑X2 = 210/168 = 1.75 = 83 For finding these values .5 2001 90 -2.0 90.5 N=8 ∑Y = 734 The equation of the straight line trend is Y = a +b X Since ∑X = 0.5 ∑Yc = 734 110 Trend by the Method of Least Squares 105 100 Trend Line 95 90 85 80 2000 Actual Data 2001 2002 2003 2004 2005 2006 2007 Year Y= 91.5 ---------------------------------------------------------------------------------------------------------------------------------------------- Question25:.Calculate the treed values by the method of least squares.75 + 1.75 + 1. a = ∑y/n= 734/8.0 100. 1.5 2007 104 +3.Y From 2003.25X2 = 2. Year 2001 2002 Sales (Rs 125 128 2003 133 2004 135 2005 140 2006 141 2007 143 Page 38 . 2005 99 +1.e. i.5 85.25 -560 -450 -276 -83 +94 +297 +460 +728 ∑XY = 210 49 25 9 1 1 9 25 49 ∑X2 = 168 83. Also calculate the monthly increase in sales and trend value for 2012.5 2006 92 +2.5 Multiplied By 2 X -7 -5 -3 -1 +1 +3 +5 +7 2000 80 -3.5 2004 94 +. double the value of B.5 93.5 – 8.5 2002 92 -1.5 and add to the preceding value: Y2001 = 83 + 2.5 88.5 2003 83 -. 3 4.1 2004 5. X shall be +8 Y2012 = 135 + 3.9 -12.7 5.3 6.7 2001 5.8 6.856 lakhs.679 2002 128 -2 -256 4 128.Computation of trend Values Year Y Deviation from 2004 XY X2 Yc X 2001 125 -3 -375 9 125.000 2005 140 +1 +140 1 138.856 The estimated sales for the year 2012 is Rs.786 2003 133 -1 -133 1 131.107 Hence Y = 135 +3.3 2003 6.5 5.321 = 125.5 -21.1 year 2004 2005 2006 2007 Number 5.893 2004 135 0 0 0 135.The following data relate to the number of passenger cars (in million) sold from 200 to 2007 Year 2000 2001 2002 2003 Number 6. a = ∑Y/N = 945/7 = 135 B = ∑XY/∑X2 = 87/28=3.2 -12. of passenger Cars(in Millions 2000 6.Y2001 = 135 Y2001 = 135 +3.107(8) = 135 + 24.1 (a) fit a straight line trend to the data thought 2005 only (b) Use your result in (a) to estimate production in 2007 and compare with the actual production.3 2002 4.6 7.6 Deviations From 2005 X -5 -4 -3 -2 -1 XY X2 -33.107X. 159.107(-3) =135 -9.2 -5.856 = 159. ------------------------------------------------------------------------------------------------------------------------------------------------------ Question26:.107 2006 141 +2 +282 4 141.lakhs) Solution:.321 2 N=7 ∑Y = 945 EX=0 ∑XY=87 ∑X =28 ∑Yc=945 The equation of the straight line trend is: Y=a +b X Since ∑x =0.214 2007 143 +3 +429 9 144.6 25 16 9 4 1 Page 39 . Solution:Fitting Straight line Trend Year No.679 Trend Value for 2012 For 2012. one year .8 = .4 ∑x2=60 The equation of the straight line trend is y =a+bX ∑Y=Na+b∑X ∑XY = a ∑X +b∑X2 47.the sales of a company for the last eight years are given blow : Year 2000 2001 2002 2003 2004 2005 2006 2007 Sales 52 45 98 92 110 185 175 220 Rs.4 = -12a +60b Multiplying Eqn.9 0 0 0 2006 5. Some difference is likely to be there between the actual and estimated figures because estimates are based on certain assumption-it may be a rare chance when actual and estimated figures may completely coincide. (ii) by 2 143.102 Putting the value of (b) in eqn (i) 47. production in million tonnes shift the origin to January 1.4 = 24a -36b -134. Estimate for 2007 For 2007 X is +2 Y = 6.8 +1 +5.2002.8 1 2007 6.Y units.8 or 8a = 47.the following table relates to the tourist arrivals (in million )during 2001to 2007 in India: Years: 2001 2002 2003 2004 2005 2006 2007 Tourist 18 20 23 25 24 28 30 arrivals (in millions): Fit a straight line trend by the method of last squares and estimate the number of tourists that would arrive in the year 20011: ---------------------------------------------------------------------------------------------------------------------------------------------------------------- Question28:.102(2) = 6.1 +2 +12.2005 7.6 = 84b 84b = 8.You are given the following trend equation: Yc =20+0.8 X Origin . ---------------------------------------------------------------------------------------------------------------------------------------------------------------- Question29:.322 Thus the estimated sale for 2007 is 6.102 X.8 =8a -12 X . Page 40 . --------------------------------------------------------------------------------------------------------------------------------------------------------------- Question27:.322 million passenger cars. There is some difference in the actual sales figure which is 6.2 4 N=8 ∑y=47.6/84 = .024 a = 6.8 +1.8 = 8a – 12b -67.224 = 47.128 +.128 +.2003.24a +120b 8.224 8a =49.102 8a -1.(000) Estimate sales figure for 2008 using an equation of the Y= abx where X= years and y= sales.8 ∑X=-12 ∑XY=-67.128 +. (i) by 3 and Eqn.1 million passenger cars and the estimated figure.128 Thus the required equation is Y =6.6: b =8.204 = 6. X units . I.06 These figures are calculated as follows: S.---------------------------------------------------------------------------------------------------------------------------------------------- Question30:. XS.673.000. 130 1.653. 82 83.I.00 April –June 90 91.the seasonal indices of the sales of readymade garment in a store are given below: Quarter January to march April to June July to September October to December Seasonal Index 98 90 82 130 If the total sales of garments in the first quarter is worth Rs. seasonal variations and seasonal indices for the following time series: Jan.I. for 3rd Qtr. S.) Jan-March 98 1.32. for 3 Qtr. S. for 2nd Qtr. For 2nd Qtr.Compute the seasonal average. for 1st qtrr.I.836. X S. For Ist Qtr.Calculate seasonal indices by the ratio-to-moving average method from the following data: Years quarters 2004 2005 2006 2007 Q1 75 86 90 100 Q2 60 65 72 78 Q3 54 63 66 72 Q4 59 80 85 93 ---------------------------------------------------------------------------------------------------------------------------------------------------------------- Question31:. = Figure for 1 qtr. Solution:- Calculation of Estimated Stock Quarter Seasonal Index Estimated Stock (Rs.73 July –Sept. 1. Feb. determine how much worth of garments of this type should be kept in stock to meet the demand in each of the remaining quarters. March April May June July August 2005 15 16 18 18 23 23 20 28 2006 23 22 28 27 31 28 22 28 2007 25 25 35 36 36 30 30 34 Page 41 . = Figure for 1qtr.I.000.I. ---------------------------------------------------------------------------------------------------------------------------------------------------------------- Question32:. rd S.00.00.45 Oct-Dec. The average of monthly averages is obtained by dividing the total of monthly averages by 12. 16 22 25 63 21 70 March 18 28 35 81 27 90 April 18 27 36 81 27 90 May 23 31 36 90 30 100 June 23 28 30 81 27 90 July 20 22 30 72 24 80 August 28 28 34 90 30 100 Sept. 15 23 25 63 21 70 Frb.5 gives the total for each month for 3 years in col. Thus the percentage for January is : = 21/30 X 100 =70 Similarly the percentage for February is: = 21/30X 100 = 70 etc.080 360 1. ---------------------------------------------------------------------------------------------------------------------------------------------------------------- Question33:. In col.September October November December 29 33 33 38 32 37 34 44 38 47 41 53 Solution:. 29 32 38 99 33 110 Oct 33 37 47 117 39 130 Nov 33 34 41 108 36 120 Dec 38 44 53 135 45 150 Total 1.6 each total of col.CALCULATION OF SEASONAL INDICES BY The Method of Monthly Averages Month 2005 2006 2007 Total Average % of average of monthly averages 1 2 3 4 5 6 7 Jan.200 Average 90 30 100 Col.Calculate the seasonal variation indices by the method of link relatives for the following figures: Quarter I II III IV 2003 45 54 72 60 QUARTERLY FIGURES FOR FIVE YEARS Year 2004 2005 2006 2007 48 49 52 60 56 63 65 70 63 70 75 84 96 65 72 66 Page 42 .5 has been divided by 3 to obtain an average for each month.7 each monthly averages has been expressed as a percentage of the average of monthly averages. 2nd Qtr.2 68 70 66 74 74 352 70. March April May June July August Sept.33 80. Nov.67 -- ---------------------------------------------------------------------------------------------------------------------------------------------------------------- Question35:.67 92.00 73. 57 Feb.Calculation of 3 monthly moving Average Month Values Jan Feb.33 87. 2003 2004 2005 2006 2007 Total Average 72 76 74 76 78 376 75. 4th Qtr. 65 March 63 April 72 May 69 June 78 July 82 Aug Sep 81 90 Oct 92 Nov 95 Dec 97 Solution:.8 Page 43 .Calculated the three-monthly moving averages from the following data: Jan.Calculate the seasonal index from the following data using the average method: Year 2003 2004 2005 2006 2007 Its Quarter 72 76 74 76 78 2nd Quarter 68 70 66 74 74 3rd Quarter 80 82 84 84 86 4th Quarter 70 74 80 78 82 Solution:. 3rd Qtr. Dec.98. 116.Computation of Seasonal Indices Year 1st qtr.Answer:.67 66. 57 65 63 72 69 78 82 81 90 92 95 97 3 monthly Totals 185 200 204 219 229 241 253 263 277 284 - 3 monthly moving average 61. 102.67 68. Oct.00 76.2 70 74 80 78 82 384 76.4 80 82 84 84 86 416 83.62 ---------------------------------------------------------------------------------------------------------------------------------------------------------------- Question34:.86.33 94.33 84.48 . 52 Grand Average = 75. ---------------------------------------------------------------------------------------------------------------------------------------------------------------- Question36:.4/76.b and c.4X100 = 108.8 = 305.8/76.52.2/76.We have a fit the equation Y= a +b x cx2 To determine the values of a.43 Seasonal Index for 2nd Qtr = 70.43 92.2/76.2 + 76.2 +70.9 Seasonal Index for 4th Qtr. = 83. = Average of 1 Qtr. X 100 Grand Average =75.4 4 4 st st Seasonal Index for 1 Qtr.9 100. these equations are reduced to ∑Y = Na + c∑X2 ∑XY=b∑X2 ∑X2Y=a∑X2 + ∑X4 X4 +81 +16 +1 0 +1 +16 +81 XY -126 -98 -62 0 +92 +244 +474 X2Y 378 196 62 0 92 488 1422 ∑X4=196 ∑XY=524 ∑X2Y=2638 Substituting the values 600= 7a+28c 524 = 28b 2.4 X 100 = 100. we solve the following equations: ∑Y=Na+b∑X + c∑X2 ∑XY =a∑X +b∑X2+ C∑X3 ∑X2Y =a ∑X2+ b∑X3+ C∑X4 Production Year 2001 2002 2003 2004 2005 2006 2007 N=7 Y 42 49 62 75 92 122 158 ∑y=600 X -3 -2 -1 0 +1 +2 +3 ∑X=0 X2 9 4 1 0 1 4 9 X3 -27 -8 -1 0 +1 +8 +27 ∑X2=28 ∑X3=0 Since ∑X=0 and ∑X3 =0.4X100 = 91.6 = 76.15 Seasonal Index for 3rd Qtr.15 108.Seasonal Index 98. = 76.Fit a parabolic trend to the following time-series data and estimate the production in 2012: Year Production (in ‘000 units) 2001 42 2002 49 2003 62 2004 75 2005 92 2006 122 2007 158 Answer:.4 X 100 = 98.4 +83.638 = 28a+196c Page 44 . 2 102. II.83 Fitting the value of c in Eqn.From Eqn.8 96. 40 42 41 45 44 212 42.Computation of Seasonal Indices Year 2003 2004 2005 2006 2007 Total Average Seasonal index 1st Qtr.43 4th Qtr.39 + 149.24= 520.4 108.83x2 For 2010.16 2nd Qtr.83(8)2 = 74.55 Page 45 .12 = 405. X will be +8 Putting X = 8 in the above equation Y2012 = 74. 35 37 35 36 38 181 36.39 +18. 38 39 38 36 38 189 37.39 Hence the required equation would be Y= 74.2 92.39 +18.71(8) -2.83) 7a = 600 -79.71X +2. 40 38 40 41 42 201 40.19 thousand units.------------------------238 = -84 c C = 2.(i) 600 =7a +28(2. ---------------------------------------------------------------------------------------------------------------------------------------------------------------- Question37:. (ii) b= 524/28 = 18.68 + 181.76 A = 74.From the data given below.35 3rd Qtr. III and IV quarters assuming the trend is absent: I II III IV 2003 40 35 38 40 2004 42 37 39 38 Year 2005 41 35 38 40 2006 45 36 36 41 2007 44 38 38 42 Solution:.71 Multiplying Eq(i) by 4 2400 = 28a +112c 2638 = 28a +196C .19 The estimate value for 2012 is 405. calculate seasonal indices for I. 4325 -5.2/39.5X2 X -5 -3 LogY X2 X Log Y YC 1. = 42.1399 0 2006 40.0 +2 2.8865 1.2 X100 = 108.6042 1 2007 125.2 +1 1.8 (in Rs.6 4.The average of averages = S.2/39.8/392 X100 = 96. for 2nd Qtr.0 x SOLUTION:. ---------------------------------------------------------------------------------------------------------------------------------------------------------------- Question39:. Taking logarithms.0969 4 N =5 ∑Y = 185. = 37.22 Page 46 .1397 +.Fitting Equation of The Type Y = ab Year Profits X Log Y X2 Y 2003 1.70 87. = 36. Log Y = Log a + X Log b.8335 78.35 S.4/39.16 S.6983 ∑x2 = 10 We have to fit an equation of the type Yc = abx.2X100 = 92. = X.6532 0 +1. = 40.1938 ∑X.43 S.1397 = =0.474 X.6 -2 0.4082 -0.Below are Given figures of production (lakh tones) of a sugar factory: Year 2002 2003 2004 2005 Production 77 88 95 114 x Fit a trend Y = ab to this data and tabulate the trend values.5 13.Log Y -0.I for 1st Qtr.2X100= 102. 2006 119 2007 127 Solution:.7366 =1.2 2007 125.The following table gives the profits of a concern for 5 years ending 2007: Year 2003 2004 2005 Profit 1.474 Log Y =1.log Y = 4.I.5 -1 0.9445 25 9 -9.Fitting Trend Line of the Form Y = abx Year Production 2002 2003 77 88 Deviation From 2004.6042 +4.8 0 1.55 ---------------------------------------------------------------------------------------------------------------------------------------------------------------- Question38:.6532 1 2005 13.2041 4 2004 4.1 ∑X=0 ∑log Y = 5.I for 4th Qtr.I for 3rd Qtr. Thousands) Fit an equation of the type Yc = abX 2006 40. 65 Log Y2005= 2.0015=1.0744 = 118.0223(-5)=2.0569 2.0569 +6.0223 = Hence Log Y2002 = 2.0075 +.0075 .1 Log Y 2006= 2.2004 2005 2006 2007 N=6 95 114 119 127 ∑Y = 620 -1 +1 +3 +5 ∑X =0 1. and (ii) Linear trend.10 118.896=78.0298 = 107.log Y = 1.0075 + .5587 96.1038 ∑log Y = 12.9852 = 96.00 Page 47 .519 ∑x.0298 Y = AL 9.50 =2.7 Log Y 2007 = 2.67 5.22 Log Y2004= 2.0669 = 1.0223(1) = 2. 7 9 8 8 9 10 SOLUTION:.The production of cement by a firm in years 1 to 9 is given below: Year 1 2 3 4 5 6 5 5 6 7 8 Production 4 (in tones) Calculate the trend values for the above series by the following two methods: (i) 3 yearly moving average.896 Y=AL 1.119 = 131.119 Y=AL 2.0223(-3) = 2.7 Log Y 2003= 2.5 ---------------------------------------------------------------------------------------------------------------------------------------------------------------- Question40:.9852 Y= AL 1.9406 = 87.0075 +.0223(-1) = 1..65 107.0075 +.70 131.0744 Y= AL 2.2265 +10.9406 Y = AL 1.33 6.9777 +2.0223(=3) =2.0075-.0449 1 1 9 25 ∑x2 =70 -1.0755 2.Calculation of 3 yearly Moving Average Year Production 3 yearly Totals 1 2 3 4 5 6 4 5 5 6 7 8 14 16 18 21 24 3-yearly moving average 4.0075 +.0223(+5) = 2.00 8.00 7.0075 +.9777 2.0075 = =0. 33 9. Value Y 75 67 68 65 50 54 41 ∑Y = 420 2004 68 2005 65 2006 50 2007 54 2008 41 of Trend by the Method of Least Squares Deviation from 2005 X -6 -3 -1 0 +2 +3 +5 ∑X =0 XY X2 Yc -450 -201 -68 0 +100 +162 +205 ∑XY = -252 36 9 1 0 4 9 25 ∑X2 = 84 78 69 63 60 54 51 45 ∑Yc=420 Page 48 .89 + 0.00 - Calculation of Linear Trend Year 1 2 3 4 5 6 7 8 9 N=9 Production Y 4 5 5 6 7 8 9 8 10 ∑Y=62 X -4 -3 -2 -1 0 +1 +2 +3 +4 ∑X =0 XY -16 -15 -10 -6 0 +8 +18 +24 +40 ∑XY=43 X2 16 9 4 1 0 1 4 9 16 ∑X2=60 Yc = a+ b X Hence Y = 6.Calculate trend values by the method of least squares from the data given below: Year Value 2002 75 2003 67 SOLUTION: Calculation Year 2002 2003 2004 2005 2006 2007 2008 N=7 Yc = a+ b X Since ∑X 0.7 8 9 9 8 10 25 27 - 8.71 X. ---------------------------------------------------------------------------------------------------------------------------------------------- Question41:. Bx Log Y = log A + X log B Normal Equation Page 49 .Additive Model (b) Y = T X S X C X1 – Multiplicative Model (ii) Isolation of Trend (a) Moving Average Method (b) Method of Least Squares (i) LINEAR TREND Y = a + bx ∑(Y) = Na + b∑(X) ∑(Xy) = a ∑ (X) + b∑ (X2) (ii) Non-Linear Trend Y = a + bX + cX2 ∑(y) = Na + b (X) + c∑(X2) ∑(XY) = a∑(X) + b∑(X2)+ c∑(X2) ∑(X2Y) = a∑(X2) + b∑(X3) X c∑(X4) (C) Exponential Curve Y = A.Fit a straight line trend by least squares method and figures of production of a sugar factory: Year 2001 2002 Production 7 88 (‘000 tonnes) 2003 94 2004 85 2005 91 tabulate the trend values from the following 2006 98 2007 90 ANSWER:.Y =89+2X.b= Hence Y = 60 -3 X Y2002 = 60 -3(-6) = 60 + 18 =78 Y2003 = 60 -3 (-3) = 60 + 9 = 69 Y2004 = 60 -3(-1)=60+3 =63 Y2005 = 60 Y2006 = 60-3(2) =60 – 6 = 54 Y2007 = 60 -3(3) = 60-9=51 Y2008 = 60-3(5) = 60-15=45 ---------------------------------------------------------------------------------------------------------------------------------------------------------------- Question42:. FORMULAE USED IN TIME SERIES (i)Components of a Time Series (a) Y = T +S+C +I . 1026 0.0793 0.2190 0.05 0.3106 0.0517 0.2580 0.0 0.3051 0.01 0.1331 0.06 0.1951 0.2881 0.1915 0.2054 0.1628 0.07 0.2157 0.8 0.02 0.2291 0.4 0.1406 0.6 0.2910 0.0199 0.1664 0.1179 0.1064 0.0871 0. AREA OF A STANDARD NORMAL DISTRIBUTION Table for Areas under the Standard Normal Curve from 0 to Z (Type II) (p (0< X < x) = n ( o < Z < Z ) Z 0.0040 0.0636 0.2995 0.1141 0.2704 0.1368 0.1700 0.0160 0.2642 0.1443 0.2612 0.1293 0.5 0.1217 0.1808 0.0239 0.2257 0.3133 Page 50 .3 0.1517 0.0080 0.078 0.1 0.0279 0.2088 0.1879 0.2389 0.1985 0.0948 0.0910 0.2939 0.0714 0.1554 0.2518 0.1591 0.0596 0.2422 0.2454 0.0120 0.0359 0.3023 0.0478 0.0675 0.00 0.08 0.2794 0.2549 0.0557 0.2357 0.2823 0.0753 0.1480 0. FICWA.1844 0.2486 0.2967 0.0438 0.1772 0.03 0.2019 0.2224 0.0987 0.∑log (Y) = N logA + logB∑(X) ∑(log Y) = logA∑(X) + log B∑(X2) ---------------------------------------------------------------------------------------------------------------------------------------------By:-SANJAY AGGARWAL FCA.2673 0.7 0.1255 0.2123 0.2324 0.04 0.0832 0.1736 0.2734 0.09 0.0398 0.2825 0.1103 0.0319 0.0000 0.2 0.2764 0. 3686 0.2 2.4834 0.4192 0.4906 0.4616 0.4983 0.4980 0.4974 0.4608 0.3315 0.4495 0.4251 0.4406 0.4951 0.4994 0.4998 0.3849 0.4332 0.4 2.4985 0.4999 0.4999 Chi-Square PDF(TWO SIDED Test at Alpha = 0.0.4913 0.4970 0.4999 0.4995 0.4972 0.4803 0.4974 0.4015 0.4732 0.4875 0.4936 0.4999 0.4207 0.4955 0.4999 0.05 Page 51 .4997 0.4582 0.3708 0.4854 0.4394 0.4 3.4846 0.4484 0.4830 0.4812 0.4988 0.4991 0.9 2.4772 0.4896 0.4999 0.4998 0.4306 0.4998 0.3289 0.4441 0.4940 0.4997 0.4938 0.4744 0.4998 0.4996 0.4999 0.4909 0.4922 0.7 1.4999 0.4948 0.4931 0.4999 0.4999 0.4984 0.4292 0.4082 0.4999 0.4925 0.4625 0.3531 0.4599 0.4981 0.4995 0.4992 0.4999 0.3810 0.3461 0.7 2.4997 0.4778 0.4927 0.8 3.4997 0.3365 0.4981 0.4969 0.4990 0.4975 0.4998 0.6 2.4998 0.4986 0.4429 0.4864 0.3438 0.4999 0.4884 0.4999 0.4918 0.3599 0.4998 0.1 3.4236 0.4986 0.4985 0.4999 0.4988 0.0 3.4591 0.4977 0.4901 0.4999 0.4162 0.1 2.4979 0.4418 0.4573 0.4999 0.4999 0.3 1.4826 0.4998 0.4989 0.4946 0.4932 0.3212 0.4989 0.4890 0.4996 0.4904 0.4649 0.4999 0.4756 0.4999 0.4965 0.9 0.3508 0.4994 0.4999 0.4783 0.4993 0.4992 0.3790 0.4963 0.4495 0.3621 0.4999 0.3907 0.4997 0.4989 0.4984 0.4990 0.4990 0.4999 0.4993 0.4463 0.3 3.4995 0.3997 0.3888 0.4066 0.4793 0.3554 0.4999 0.4957 0.4949 0.4977 0.4978 0.3962 0.4967 0.3643 0.4452 0.4671 0.15 0.5 1.4966 0.4319 0.4995 0.8 1.4964 0.4998 0.4656 0.4999 0.4911 0.4726 0.4767 0.4999 0.4961 0.4999 0.3665 0.4032 0.05) 0.4861 0.4941 0.4994 0.4999 0.4525 0.4887 0.4999 0.4956 0.4279 0.4999 0.6 1.3159 0.6 3.4868 0.4788 0.4535 0.4881 0.3577 0.4998 0.4750 0.4920 0.4987 0.2 3.4713 0.4515 0.4996 0.4678 0.4798 0.4177 0.4995 0.4345 0.4943 0.4959 0.4998 0.4999 0.4998 0.4962 0.3980 0.4686 0.4222 0.8 2.4999 0.4996 0.4997 0. 0.4997 0.4998 0.4049 0.4953 0.4871 0.4987 0.3749 0.4971 0.4929 0.4997 0.4994 0.4976 0.4817 0.3186 0.4842 0.3264 0.9 3.4808 0.3830 0.4857 0.3 0.4693 0.3413 0.4699 0.4998 0.4545 0.4991 0.5 3.4738 0.1 1.4982 0.4945 0.4505 0.4916 0.3 2.25 0.4131 0.4633 0.3869 0.4 1.4994 0.3770 0.4265 0.4878 0.4893 0.0 1.9 1.0 2.4968 0.4641 0.4821 0.4992 0.4992 0.4999 0.4996 0.4564 0.4719 0.4986 0.4982 0.5 2.7 3.4999 0.3389 0.4952 0.4993 0.4838 0.3340 0.3944 0.4664 0.4850 0.4999 0.4474 0.4761 0.4382 0.4973 0.4997 0.4934 0.4991 0.4996 0.4999 0.4147 0.4993 0.4115 0.4554 0.4357 0.4370 0.4999 0.4979 0.3238 0.3729 0.3925 0.4706 0.2 1.4898 0.4999 0.4960 0.4099 0.3485 0. 520 11.838 14.455 4.245 19.085 10.307 19.571 4.828 24.209 24.773 0.620 33.141 28.963 48.103 0.666 23.181 45.337 24.087 40.571 7. df2) Page 52 .219 11.336 29.660 5.067 15.633 30.340 12.659 38.790 8.143 12.090 21.064 1.718 37.319 32.107 4.114 18.024 7.115 0.9 0.191 37.443 13.01 6.526 32.191 21.757 28.879 10.404 5.554 0.357 4.711 34.657 23.920 23.023 20.390 10.980 44.815 9.873 28.611 15.578 32.005 7.736 26.170 35.336 0.955 23.196 34.026 22.004 0.290 49.140 45.493 0.844 7.923 2.461 45.260 8.408 7.346 33.338 15.10.369 20.196 10.962 0.852 34.240 18.337 24.684 15.344 17.121 13.567 24.336 53.984 17.297 0.886 10.844 14.856 44.296 27.072 0.088 2.645 50.566 42.591 12.651 12.557 43.479 36.635 2.953 0.586 11.141 30.599 0.255 5.300 29.338 13.323 2.542 10.289 41.145 1.037 19.137 19.064 22.241 29.524 12.df1.204 2.346 7.592 14.010 0.165 11.260 9.908 7.635 9.0 5 10 15 20 Upper-tail area α df 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0.848 15.800 0.076 39.924 35.278 21.831 1.962 8.618 24.000 0.916 39.559 46.549 19.816 4.256 14.488 11.865 11.389 12.352 0.605 22.75 0.041 14.549 13.362 23.000 0.211 0.575 5.95 0.805 36.741 37.007 33.935 26.993 52.05 3.845 15.605 6.160 11.558 3.025 5.216 0.528 32.750 18.119 16.267 35.343 37.247 3.615 30.749 22.848 14.338 19.584 8.277 15.671 33.899 6.113 41.348 6.490 4.144 31.588 50.892 0.701 14.675 21.622 18.968 40.725 26.711 1.054 25.382 35.051 0.718 23.151 16.652 38.366 3.659 16.02 5.364 40646 41.603 3.668 13.336 27.452 16.791 0.009 5.312 10.074 3.207 0.337 22.813 32.236 10.338 17.489 21.633 8.885 40.685 24.940 4.401 42.675 14.879 13.706 4.020 0.412 0.042 7.837 11.989 1.675 3.307 23.017 13.679 21.091 13.647 2.473 17.340 13.016 0.229 5.339 30.251 7.645 12.733 3.687 35.843 21.787 0.869 30.037 10.351 5.412 29.535 19.722 46.259 29.841 9.860 16.575 1.812 21.548 20.239 1.975 0.939 20.672 F(.117 18.589 25.108 5.565 14.419 46.819 31.337 20.119 27.299 10.278 49.845 30.399 15.939 19.378 9.053 3.379 16.102 0.808 12.226 5.172 36.865 5.261 7.198 12.337 21.872 1.643 9.478 0.989 27.231 8.995 32.584 1.314 45.979 0.217 27.275 18.601 5.488 28.304 7.587 28.996 26.779 9.020 36.676 0.262 6.341 11.410 32.483 21.982 11.564 8.928 17.697 6.337 25.689 12.339 14.573 15.579 12.642 46.769 25.386 2.120 13.773 4.337 42.566 38.475 20.071 5.812 6.461 13.824 9.013 17.897 9.582 39.075 4.991 7.336 26.638 42.213 1.923 43.455 1.039 27.168 19.833 3.438 9.167 2.086 16.265 6.629 6.45 13.928 48.562 15.565 4.033 16.919 18.578 6.626 7.409 34.337 23.693 47.034 8.99 0.812 18.308 16.195 44.796 44.768 20.71 12.591 10.987 17.484 0.542 24.507 16.348 11.188 26.892 6.037 11.344 8.708 18.5 0.841 5.256 0.690 2.781 38.672 9.401 13.270 41.700 3.117 10.142 5.833 14.344 1.1 2.292 18.339 16.168 4.25 1.792 13.907 9.000 33.161 22.325 3.240 14.237 1.210 11.737 7.610 2.343 9.563 36.156 2.472 26.801 34.283 10.156 38.047 16.435 31.336 28.547 9.434 8.412 7.001 0.362 14.735 2.015 7.912 12.995 0.342 10.385 6.688 29.997 41.204 28.338 18.932 40.180 2.415 37.851 11. 099 3.354 3.411 3.838 3.028 3.378 2.634 3.600 4.510 2.290 233.768 19.342 2.922 2.682 3.422 3.960 2.396 8.818 4.413 8.175 236.171 4.917 1.604 2.445 2.106 3.474 2.494 4.344 3.009 2.366 2.591 2.573 2.965 2.073 3.225 4.223 2.849 2.726 3.388 2.410 4.944 5.587 3.085 4.249 2.147 3.326 3.118 2.467 3.773 4.124 2.300 2.165 2.787 2.513 10.293 3.148 2.543 4.247 9.685 2.459 2.353 8.095 2.911 243.555 3.425 2.016 240.866 2.885 3.759 2.661 2.901 2.204 3.049 3.004 1.776 2.342 2.707 19.094 4.522 3.F TABLE FOR ALPHA=.041 4.097 2.260 4.232 3.587 2.490 2.845 6.558 2.975 2.739 3.128 7.357 3.385 8.887 6.608 5.385 3.325 4.167 2.641 2.359 2.525 2.204 2.514 2.743 2.291 2.050 4.001 3.211 2.423 2.740 2.279 4.544 2.217 3.745 5.863 3.747 4.355 2.092 2.687 2.606 2.934 2.707 2.791 2.549 2.661 2.000 9.136 3.374 3.972 3.958 2.711 2.368 2.657 2.000 3.369 3.320 2.226 2.737 4.277 6.162 19.688 3.882 19.987 5.236 2.375 2.397 2.678 4.708 3.948 2.494 2.633 3.421 2.423 2.183 4.387 3.978 2.602 2.220 2.637 3.482 3.883 19.912 4.628 2.336 2.508 2.478 3.180 2.552 6.592 3.316 3.340 3.143 4.699 2.337 2.347 4.786 5.621 2.796 2.072 2.381 2.284 3.494 2.df1.297 2.250 2.735 4.581 3.832 2.05 DF FOR DENOMINATOR(V2) 1 2 3 4 5 6 7 8 9 10 12 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 161.257 4.183 2.127 3.728 2.414 4.538 2.321 2.284 3.447 2.459 4.087 238.456 2.405 2.464 2.278 2.701 2.757 4.854 2.120 3.348 2.10.528 2.179 3.351 4.591 5.346 2.667 4.834 F(.328 3.282 2. df2) Page 53 .986 19.103 3.572 2.388 3.117 4.321 2.699 2.025 2.242 4.534 2.105 2.920 199.447 230.999 4.014 6.866 3.160 3.179 3.207 3.993 1.347 3.318 5.741 2.928 2.844 4.915 2.296 9.438 3.190 2.588 2.583 19.640 2.545 2.117 6.230 3.073 2.810 2.421 2.477 2.164 9.614 2.237 2.373 2.287 3.758 2.450 2.848 2.177 2.906 19.266 2.137 2.599 2.773 2.132 2.714 2.677 3.098 3.876 4.493 3.764 2.941 6.543 19.500 19.996 2.150 3.112 3.403 3.197 3.947 2.796 2.991 2.913 2.965 4.255 2.066 3.577 2.475 2.266 2.895 2.534 2.709 6.432 2.196 4.767 2.275 2.072 215.204 2.450 2.007 2.964 4.388 5.950 4.239 3.381 4.806 3.259 3.591 5.788 2.393 2.603 2.056 3.714 2.308 2.371 8.959 241.192 4.896 2.443 3.786 5.060 3.852 2.501 3.690 2.451 4.278 2.334 2.488 2.330 8.840 2.680 224.163 4.301 4.040 1.077 1.812 5.165 2.671 2.982 3.575 3.442 2.250 2.534 4.254 2.412 2.913 2.839 2.448 18.544 2.548 2.210 4.490 3.012 2.646 2.020 2.256 5.305 2.817 2.753 2. 327 2.054 2.041 1.820 1.505 2.233 2.865 1.361 2.007 2.082 2.005 1.877 1.900 1.416 2.316 2.589 2.975 2.040 2.249 2.827 1.705 9.113 3.936 3.845 1.645 2.894 2.102 2.055 2.536 2.952 1.924 2.195 9.219 2.561 2.307 2.005 1.158 2.990 2.781 1.875 1.440 2.668 2.490 2.283 2.291 2.193 2.405 3.207 2.074 2.234 2.391 4.924 2.014 2.395 2.887 2.252 3.257 3.979 3.128 2.343 4.560 2.873 1.245 2.059 2.538 2.624 2.935 1.162 5.941 1.726 2.995 1.462 2.953 1.722 59.152 2.240 9.927 2.218 2.511 2.196 2.188 2.985 1.892 1.075 2.214 2.309 4.304 2.920 3.849 1.948 1.526 5.138 2.216 3.360 3.937 2.551 2.408 5.195 2.522 2.416 2.961 2.451 2.892 1.191 3.463 3.858 9.906 1.956 1.028 2.108 2.035 2.520 3.905 2.983 1.462 4.091 2.164 2.785 2.147 2.073 2.997 1.285 4.367 5.283 2.297 2.611 2.119 2.394 2.244 2.008 1.339 2.243 5.347 2.109 2.538 4.996 1.927 1.958 2.909 2.495 2.605 2.461 2.748 49.177 3.060 3.967 1.095 2.961 2.875 1.693 2.057 2.606 2.248 2.883 2.807 2.884 1.984 1.000 5.333 2.545 4.010 3.128 2.919 1.176 2.142 2.073 3.397 2.824 58.832 1.347 53593 9.086 2.225 3.977 1.174 2.949 2.088 2.971 1.342 2.414 2.266 2.528 2.389 2.047 2.006 2.130 55.286 2.226 2.026 3.715 1.440 2.F TABLE FOR ALPHA=.859 1.048 3.285 3.325 3.273 2.855 1.028 2.307 2.992 57.896 58.136 3.728 2.912 1.023 2.791 2.857 1.933 1.268 2.695 2.02 2.965 1.024 2.890 1.230 3.284 2.776 3.489 2.909 1.001 1.130 2.860 2.809 1.283 2.184 2.331 2.933 1.919 1.703 2.178 2.895 1.980 1.937 1.884 1.929 1.276 2.943 1.469 2.619 3.317 2.347 2.480 2.955 3.502 2.097 2.293 5.896 3.668 2.274 2.181 2.107 3.767 59.601 LOGORITHMA TABLES COMMON LOGORITHAMS 10 0 0000 0043 1 2 0086 3 0128 4 0170 11 0414 0453 0492 0531 0569 5 0212 6 0253 7 0294 8 0334 9 0374 0607 0645 0682 0719 0755 1 5 4 4 4 2 9 8 8 7 3 13 12 12 11 4 17 16 16 15 5 21 20 20 18 6 26 24 23 22 7 30 28 27 26 8 34 32 31 29 9 38 36 35 33 Page 54 .763 1.752 2.149 2.881 2.323 2.308 2.918 2.793 1.999 1.349 5.195 2.982 1.726 2.381 5.158 2.351 2.961 1.874 1.589 2.064 2.208 2.904 1.561 2.589 3.017 1.657 1.813 2.289 3.845 1.958 1.299 2.835 2.866 1.836 1.240 3.393 2.055 2.154 2.519 2.243 2.061 2.575 2.906 9.549 2.819 1.983 2.624 2.500 9.453 3.937 2.377 2.079 2.049 1.901 2.102 3.819 1.10 DF FOR DENOMINATOR(V2) 1 2 3 4 5 6 7 8 9 10 12 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 39.927 1.775 1.165 2.606 2.103 2.458 3.538 2.204 9.339 2.707 1.773 1.051 3.503 2.061 2.522 2.091 2.038 2.058 2.157 2.920 1.780 3.660 2.437 2.177 2.439 9.668 2.368 3.652 60.829 1.142 2.365 2.266 3.326 5.833 9.738 1.684 60.014 2.946 1.379 2.434 2.806 2.392 5.017 1.122 2.380 2.827 2.863 8.725 2.988 1.209 2.763 2.115 2.799 1.790 1. 12 0792 0828 0864 0899 0934 0969 1004 1038 1072 1106 1303 1335 1367 1399 1430 1614 1644 1673 1703 1732 1903 1931 1959 1987 2014 2175 2201 2227 2253 2279 2430 2455 2480 2504 2529 2672 2695 2718 2742 2765 13 1139 1173 1206 1239 1271 14 1461 1492 1523 1553 1584 15 1761 1790 1818 1547 1875 16 2041 2068 2095 2122 2148 17 2304 2330 2355 2380 2405 18 2553 2577 2601 2625 2648 19 2788 2810 2833 2856 2878 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 3010 3222 3424 3617 3802 3979 4150 4314 4472 4624 4771 4914 5051 5185 5315 5441 5563 5682 5798 5911 6021 6128 6232 6335 6435 6535 6628 6721 3032 3054 3263 6464 3655 3838 4014 4183 4346 4502 4654 4800 4942 5079 5211 5340 5464 5587 5705 5821 5933 6042 6149 6253 6355 6454 6551 6646 6739 3075 3284 3483 3674 3856 4031 4200 4362 4518 4669 4814 4955 5092 5224 5353 5478 5599 5717 5832 5944 6053 6160 6263 6365 6464 6561 6656 6749 3096 3304 3502 3692 9874 4048 4216 4378 4533 4683 4829 4969 5105 5237 5366 5490 5611 5729 5843 5955 6064 6170 5274 6375 6474 6571 6665 6758 2900 3118 3324 3522 3711 3892 4065 4232 4393 4548 4698 4843 4983 5119 5250 5378 5502 5623 5740 5855 5966 6075 6180 6284 6385 6484 6580 6675 6767 2923 3139 3345 3541 3729 3909 4082 4249 4409 4564 4713 4857 4997 5132 5263 5391 5514 5635 5752 5866 5977 6085 6191 6294 6395 6493 6590 6684 6776 2945 3160 3365 3560 3747 3927 4099 4265 4425 4579 4728 4871 5011 5145 5276 5403 5527 5647 5763 5877 5988 6096 6201 6304 6405 6503 6609 6702 6794 2967 3181 3385 3579 3766 3945 4116 4281 4440 4594 4742 4886 5024 5159 5289 5416 5539 5658 5775 5888 5999 6107 6212 6314 6415 6513 6609 6702 6794 48 49 6812 6902 6821 6830 6920 6839 6928 6848 6937 6857 6946 6866 6955 6884 6964 6884 6972 2989 3201 3404 3598 3784 3962 4133 4298 4456 4609 4757 4900 5038 5172 5302 5428 5551 5670 5786 5899 6010 6117 6222 6325 6425 6522 6618 6712 68 03 6893 6981 3 4 5 6 7 8 9 3243 3444 3636 3820 3997 4166 4330 4487 4639 4786 4928 5065 5198 5328 5453 5575 5694 5809 5922 6031 6138 6243 6345 6444 6542 6637 6730 6911 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7 7 6 7 6 6 6 6 6 5 5 5 5 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 11 10 10 10 9 9 9 8 8 8 8 8 7 7 7 6 6 6 6 6 5 5 5 5 5 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 14 14 13 13 12 12 11 11 11 10 10 10 9 9 9 8 8 8 8 7 7 7 7 6 6 6 6 6 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 4 18 17 17 16 15 14 14 14 14 13 13 12 12 11 11 11 11 10 10 9 9 9 8 8 8 7 7 7 7 6 6 6 6 6 6 5 5 5 5 5 5 5 5 5 21 20 20 19 19 17 17 17 16 16 15 15 14 14 13 13 13 12 12 11 11 10 10 9 9 9 9 8 8 8 8 7 7 7 7 7 6 6 6 6 6 6 6 5 25 24 24 23 22 20 20 19 19 18 18 17 17 16 16 15 15 14 14 13 12 12 11 11 11 10 10 10 9 9 9 9 8 8 8 8 8 7 7 7 7 7 7 6 28 27 27 26 25 25 23 22 22 21 20 20 19 18 18 17 17 16 15 15 14 14 13 13 12 12 11 11 11 10 10 10 10 9 9 9 9 8 8 8 8 8 7 7 32 31 31 29 29 26 26 25 24 23 23 22 21 21 20 19 19 18 17 17 16 15 15 14 14 13 13 12 12 12 11 11 11 10 10 10 10 9 9 9 9 9 8 8 1 1 2 2 3 3 4 4 4 4 5 4 6 6 7 7 8 8 1 2 3 4 5 LOGORITHMS TABLES COMMON LOGORITHAMS 0 1 2 6 7 8 9 Page 55 . 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 6990 7076 7160 7243 7324 7404 7482 7559 7634 7709 7782 7853 7924 7993 8062 8129 8195 8261 8325 8388 8451 8513 8573 8633 8692 8751 8808 8865 8921 8976 9031 9085 9138 9191 9243 9294 9345 9395 9445 9494 9542 9590 9638 9685 9731 9777 9823 9868 9912 9956 6998 7084 7168 7251 7332 7412 7490 7566 7642 7716 7789 7860 7931 8000 8069 8136 8202 8267 8331 8395 8457 8519 8579 8639 8698 8756 8814 8871 8927 8982 9036 9090 9143 9196 9248 9299 9350 9400 9450 9499 9547 9595 9643 9689 9736 9782 9827 9872 9917 9961 7007 7093 7177 7259 7340 7419 7497 7574 7649 7723 7796 7868 7938 8007 8075 8142 8209 8274 8338 8401 8463 8525 8585 8645 8704 8762 8820 8876 8932 8987 9042 9096 9149 9201 9253 9304 9355 9405 9455 9504 9552 9600 9647 9694 9741 9786 9832 9877 9921 9965 7016 7101 7185 7267 7348 7427 7505 7582 7657 7731 7803 7875 7945 8014 8082 8149 8215 8280 8344 8407 8470 8531 8591 8651 8710 8768 8825 8882 8938 8993 9047 9101 9154 9206 9258 9309 9360 9410 9460 9509 9557 9605 9652 9699 9745 9791 9836 9881 9926 9969 7024 7110 7193 7275 7356 7435 7513 7589 7664 7738 7810 7882 7952 8021 8089 8156 8222 8287 8351 8414 8476 8537 8597 8657 8716 8774 8831 8887 8943 8998 9253 9106 9159 9212 9263 9315 9365 9415 9465 9513 9562 9609 9657 9703 9750 9795 9841 9886 9930 9974 7033 7118 7202 7284 7364 7443 7520 7597 7672 7745 7818 7889 7959 8028 8096 8162 8228 8293 8357 8420 8482 8543 8603 8663 8722 8779 8837 8893 8949 9004 9058 9112 9165 9217 9269 9320 9370 9420 9469 9518 9566 9614 9661 9708 9754 9800 9845 9890 9934 9978 7042 7126 7210 7292 7372 7451 7528 7604 7679 7752 7825 7896 7966 8035 8102 8169 8235 8299 8363 8426 8488 8549 8609 8669 8727 8785 8842 8899 8954 9009 9063 9117 9170 9222 9274 9325 9375 9425 9474 9523 9571 9619 9666 9713 9759 9805 9850 9894 9939 9983 7050 7135 7218 7300 7380 7459 7536 7612 7686 7760 7832 7903 7973 8041 9109 8176 8241 8306 8370 8432 8494 8555 8615 8675 8733 8791 8848 8904 8960 9015 9069 9122 9175 9227 9279 9330 9380 9430 9479 9528 9576 9624 9671 9717 9763 9809 9854 9899 9943 9987 7059 7143 7226 7308 7388 7466 7543 7619 7694 7767 7839 7910 7980 8048 8116 8182 8248 8312 8376 8439 8500 8561 8621 8681 8739 8797 8854 8910 8965 9020 9074 9128 9180 9232 9284 9335 9385 9435 9484 9533 9581 9628 9675 9722 9768 9814 9859 9903 9948 9991 7067 7152 7235 7316 7396 7474 7551 7627 7701 7774 7846 7917 7987 8055 8122 8189 8254 8319 8382 8445 8506 8567 8627 8686 8745 8802 8859 8915 8971 9025 9079 9133 9186 9238 9289 9340 9390 9440 9489 9538 9586 9633 9680 9727 9773 9818 9863 9908 9952 9996 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 6 6 6 6 6 5 5 5 5 5 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 7 7 7 6 6 6 6 6 6 6 6 6 6 5 5 5 5 5 5 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 8 8 7 7 7 7 7 7 7 7 6 6 6 6 6 6 6 6 6 6 6 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 4 4 4 4 4 ANTILOGORITHMS TABLES Page 56 . 09 0.03 0.36 0.12 0.02 0.42 0.04 0.18 0.15 0.29 0.27 0.33 0.39 0.41 0.10 0.22 0.00 0.17 0.30 0.26 0.11 0.31 0.20 0.49 0 1000 1023 1047 1072 1096 1122 1148 1175 1202 1230 1259 1288 1318 1349 1380 1413 1445 1479 1514 1549 1585 1622 1660 1698 1738 1778 1820 1862 1905 1950 1995 2042 2089 2138 2188 2239 2291 2344 2399 2455 2512 2570 2630 2692 2754 2818 2884 2951 3020 3090 1 1002 1026 1050 1074 1099 1125 1151 1178 1205 1233 1262 1291 1321 1352 1384 1416 1449 1483 1517 1552 1589 1626 1663 1702 1742 1782 1824 1866 1910 1954 2000 2046 2094 2143 2193 2244 2296 2350 2404 2460 2518 2576 2636 2698 2761 2825 2891 2958 3027 3097 2 1005 1028 1052 1076 1102 1127 1153 1180 1208 1236 1265 1294 1324 1355 1387 1419 1452 1486 1521 1556 1592 1629 1667 1706 1746 1786 1828 1871 1914 1959 2004 2051 2099 2148 2198 2249 2301 2355 2410 2466 2523 2582 2642 2704 2767 2831 2897 2965 3034 3105 3 4 5 6 7 8 9 1007 1009 1012 1014 1016 1019 1021 1030 1033 1054 111057 1079 1081 1104 1107 1130 1132 1156 1159 1183 1186 1211 1213 1239 1242 1268 1271 1297 1300 1327 1330 1358 1361 1390 1393 1422 1426 1455 1459 1489 1493 1524 1528 1560 1563 1596 1600 1633 1637 1671 1675 1710 1714 1750 1754 1791 1795 1832 1837 1875 1879 1919 1923 1963 1968 2009 2014 2056 2061 2104 2109 2153 2158 2203 2208 2254 2259 2307 2312 2360 2366 2415 2421 2472 2477 2529 2535 2588 2594 2649 2655 2710 2716 2773 2780 2838 2844 2904 2911 2972 2979 3041 3048 3112 3119 1035 1059 1084 1109 1135 1161 1189 1216 1245 1274 1303 1334 1365 1396 1429 1462 1496 1531 1567 1603 1641 1679 1718 1758 1799 1841 1884 1928 1972 2018 2065 2113 2163 2213 2265 2317 2371 2427 2483 2541 2600 2661 2723 2786 2851 2917 2985 3055 3126 1038 1062 1086 1112 1138 1164 1191 1219 1247 1276 1306 1337 1368 1400 1432 1466 1500 1535 1570 1607 1644 1683 1722 1762 1803 1845 1888 1932 1977 2023 2070 2118 2168 2218 2270 2323 2377 2432 2489 2547 2606 2667 2729 2793 2858 2924 2992 3062 3133 1040 1064 1089 1114 1140 1167 1194 1222 1250 1279 1309 1340 1371 1403 1435 1469 1503 1538 1574 1611 1648 1687 1726 1766 1807 1849 1892 1936 1982 2028 2075 2123 2173 2223 2275 2328 2382 2438 2495 2553 2612 2673 2735 2799 2864 2931 2999 3069 3141 1042 1067 1091 1117 1143 1169 1197 1225 1253 1282 1312 1343 1374 1406 1439 1472 1507 1542 1578 1614 1652 1690 1730 1770 1811 1854 1897 1941 1986 2032 2080 2128 2178 2228 2280 2333 2388 2443 2500 2559 2618 2679 2742 2805 2871 2938 3006 3076 3148 1045 1069 1094 1119 1146 1172 1199 1227 1256 1285 1315 1346 1377 1409 1442 1476 1510 1545 1581 1618 1656 1694 1734 1774 1816 1858 1901 1945 1991 2037 2084 2133 2183 2234 2286 2339 2393 2449 2506 2564 2624 2685 2748 2812 2877 2944 3013 3083 3155 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 5 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 6 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 7 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 8 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 6 6 9 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 Page 57 .40 0.34 0.38 0.47 0.13 0.08 0.45 0.43 0.32 0.05 0.48 0.06 0.25 0.21 0.23 0.01 0.0.16 0.24 0.28 0.44 0.37 0.46 0.07 0.14 0.19 0.35 0. 65 0.74 0.89 0.85 0.50 0.95 0.52 0.57 0.60 0.94 0.96 0.51 0.53 0.54 0.78 0.97 0.98 0.69 0.99 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 3162 3236 3311 3388 3467 3548 3631 3715 3802 3890 3981 4074 4169 4266 4365 4467 4571 4677 4786 4898 5012 5129 5248 5370 5495 5623 5754 5888 6026 6166 6310 6457 6607 6761 6918 7079 7244 7413 7586 7762 7943 8128 8318 8511 8710 8913 9120 9333 9550 9772 3170 3243 3319 3396 3475 3556 3639 3724 3811 3899 3990 4083 4178 4276 4375 4477 4581 4688 4797 4909 5023 5140 5260 5383 5508 5636 5768 5902 6039 6180 6324 6471 6622 6776 6934 7096 7261 7430 7603 7780 7962 8147 8337 8531 8730 8933 9141 9354 9572 9795 3177 3251 3327 3404 3483 3565 3648 3733 3819 3908 3999 4093 4188 4285 4385 4487 4592 4699 4808 4920 5035 5152 5272 5395 5521 5649 5781 5916 6053 6194 6339 6486 6637 6792 6950 7112 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0.91 0.83 0.73 0.90 0.