IMS 502 Information Analysis for Decision Making Descriptive StatisticsContent • Measure of Location • Measure of Spread • Measure of Shape Learning Outcomes • After completing this chapter, you should be able to: – Able to conduct appropriate measure of location, measure of spread & measure of shape Concept of Descriptive Statistics Measure of Shape Measure of Spread (Variability) Describing Data Numerically Measure of Location (Central Tendency) • Three common measures of central tendency are: Mode Mean Median .Concept of Descriptive Statistics Measure of Location (Central Tendency) • Descriptive information about the single numerical value that is considered to be the most typical of the values of a quantitative variable. • A measure of central tendency is a measure which indicates where the middle of the data is. Measures of Central Tendency • Statistics to represent the ‘centre’ of a distribution – Mode (most frequent) – Median (50th percentile) – Mean (average) • Choice of measure dependent on – Type of data – Shape of distribution (esp. skewness) . Measures of Central Tendency Measure of Central Tendency Level of Measurement Nominal Ordinal Interval Ratio Mode X X X X? X X X X X X Median Mean . 3. The median always lies at the exact center of a distribution. You want a quick and easy measure for ordinal and interval-ratio variables. You anticipate additional statistical analysis. 2.Measures of Central Tendency • Choosing a measure of central tendency Use the Mode When : 1. The variable is measured at the ordinal level. You want to report the central score. The mean is “the fulcrum that exactly balances all of the scores. The variable is measured at the nominal level.” 3. 2. You want to report the typical score. : 1. A variables measured at the interval-ratio level has a highly skewed distribution. : 1. Use the Median When Use the Mean When . 2. The variable is measured at the interval-ratio level (except when the variable is highly skewed). You want to report the most common score. 3. • The Sample Mean: x = n ∑ x i =1 i n .Measures of Central Tendency • Mean – Is the average of the data • The Population Mean: N µ= ∑X i =1 i which is usually unknown. then we use the N sample mean to estimate or approximate it. where χ 1 = 42. χ 5 = 31.6 . χ 8 = 34. χ 6 = 23. x = (42 + 28 + … + 37) / 10 = 36. χ 3 = 28.Measures of Central Tendency Example: Here is a random sample of size 10 of ages. χ 9 = 32. χ 7 = 50. χ 2 = 28. χ 4 = 61. χ 10 = 37. . – Simplicity: It is easy to understand and to Simplicity compute.Measures of Central Tendency • Properties of the Mean – Uniqueness: For a given set of data there is Uniqueness one and only one mean. – Affected by extreme values: Since all values values enter into the computation. 80 and 280.Measures of Central Tendency Example: Assume the values are 115. 119. 75. 110. a value that is not representative of the set of data as a whole. 117. But assume that the values are 75. . The mean = 118. The mean = 118. 80. 121 and 126. Concept of Descriptive Statistics • Median – When ordering the data. 50% below) . – The median is the center point in a set of numbers (50% above. it is the observation that divide the set of observations into two equal parts such that half of the data are before it and the other are after it. 4. 9 .Concept of Descriptive Statistics • Median – Check to see which of the following two rules applies: • Rule One If n is odd. It will be the (n+1)/2 th ordered observation. then the median is the 6th observation. 3. 1. Example: Three is the median for the numbers 1. When n = 11. the median will be the middle of observations. 34. = (32+34)/2 = 33 . 32. there are two middle observations. 31. Since n = 10.5th observation. The median will be the mean of these two middle observations.e. 61. then the median is the 6. When n = 12. 50. 42.Concept of Descriptive Statistics • Median – Check to see which of the following two rules applies: • Rule Two If n is even. 37. i. which is an observation halfway between the 6th and 7th ordered observation. 28. It will be the (n+1)/2 th ordered observation. then the median is the 5. Example: 23.5th observation. 28. Simplicity – It is not affected by extreme values as is the mean.Measures of Central Tendency • Properties of the Median – Uniqueness: For a given set of data there is Uniqueness one and only one median. . – Simplicity: It is easy to calculate. – If all values are different there is no mode. there are more than one mode. mode . mode – Sometimes.Concept of Descriptive Statistics • Mode – The mode is simply the most frequently occurring number. 31. 37. 28. 34.Measures of Central Tendency Example: Assume the values are 23. 28. The median = 28 (repeated two times). 42. 32. . 50. 61. it is not unique. – It may be used for describing qualitative data. .Measures of Central Tendency • Properties of the Mode – Sometimes. • To begin: .Measures of Central Tendency • The only procedure in SPSS that will produce all three commonly used measure of location is Frequency. Gender and number of aces were recorded for each player. • Follow steps 1. 8 & 11. 2. 3. 4. The data can be found in Work4. .Working Example(Pg. 6. 59) • One hundred tennis players participated in a serving competition.sav on the iLearn web site that accompanies this title. Exercises • Use the Frequencies command to get all three (3) measures of location. You may choose three(3) variables to report on. . • Write a sentence or two reporting each measure. • Measures of deviation from the central tendency. • Non-parametric/non-normal: • range. percentiles.Concept of Descriptive Statistics Measure of Spread (Variability/Dispersion) • Give information on the spread or variability of the data values. max • Parametric: • standard deviation (SD) & properties of the normal distribution . min. Concept of Descriptive Statistics Measure of Spread (Variability/Dispersion) • Four common measures of spread are: Range Quartiles Variance Standard Deviation . Measures of Spread Measure of Spread Level of Measurement Nominal Ordinal Interval Ratio X X X X X X? X Range. Min/Max Percentile Standard Deviation (SD) . Therefore. – The range is limited as a means of telling about the general spread of a group of data. .Measures of Spread • Range (R) – The range is the difference between the highest and lowest scores in a distribution. – If we examine the marks of the 100 students above. the range is 50 (85 – 35 = 50). it does set the boundaries of the scores. then we can see that the highest score was 85 and the lowest was 35. or 25th percentile -.0% of the observations – The difference between Q3 . or 75th percentile -. or IQR. is the median -.0% of the observations • Q3. 25% Q1 25% Q2 25% Q3 25% .25.75.0% of the observations • Q2.50.Q1 is called the inter-quartile range.Measures of Spread • Quartiles (Q) – The quartiles split the ordered data into four quarters: • Q1. Measures of Spread • Variance – The variance is the square of the standard deviation. . the more accurately the mean represents the scores of all cases in a distribution of data. – The lower the variance. . – The higher the standard deviation.Measures of Spread • Standard Deviation – The standard deviation provides the researcher with an indicator of how scores for variables are spread around the mean average. the more scores around the mean are spread out. • To begin: .Measures of Spread • Using SPSS for measure of spread. 5. 7. 8 & 11. 3. The data can be found in Work4. Gender and number of aces were recorded for each player. • Follow steps 1. 2. 4. .Working Example(Pg.sav on the iLearn web site that accompanies this title. 59) • One hundred tennis players participated in a serving competition. • Write a few sentences summarizing these tables for each under measurement of spread. .Exercises • Choose three(3) variables to work on. • Describe the difference (if any). Concept of Descriptive Statistics Measure of Shape • To describes how data are distributed. • Two common measures of shape are: Kurtosis is a measure of whether the data are peaked or flat relative to a normal distribution. Skewness is a measure of symmetry. or more precisely. the lack of symmetry. • Use the normal curve (combination of mean & standard deviation) to construct precise descriptive statements. Platykurtic Leptokurtic Normal Positive Skewness Negative Skewness . Measure of Shape .Skewness Mean < Median < Mode Coefficient = Negative Mean = Median = Mode Coefficient = 0 Mode < Median < Mean Coefficient = Positive . • The data will cluster around or close to the Mean.Measure of Shape . • Kurtosis. • Data sets with low kurtosis tend to have a flat top near the mean rather than a sharp peak. • Data sets with high kurtosis tend to have a distinct peak near the mean. • Kurtosis. γ2 > 0 (Leptokurtic) • Data distribution with large standard deviation. γ2 < 0 (Platykurtic) . • The data will be far away from the mean.Kurtosis • Data distribution with small standard deviation. and have heavy tails. decline rather rapidly. • To begin: .Measure of Shape • Using SPSS for measure of shape. 2. 59) • One hundred tennis players participated in a serving competition. The data can be found in Work4. 9.Working Example(Pg. 8. . 10 & 11. Gender and number of aces were recorded for each player. Click on Skewness & Kurtosis.sav on the iLearn web site that accompanies this title. • Follow steps 1. 3. 4. • Describe the difference (if any).Exercise • Choose three(3) variables to work on. • Write a few sentences summarizing these tables for each under measurement of shape. .