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Measures of Variability Professor David A Walsh Lecture 5 1 What is Variability Descriptive statistics that summarize the amount of the differences between scores Variability is a different dimension than central tendency How much dispersion of the scores is there How much individual difference exists between the participants on the measured variable Sample Distributions N 9 A 41 41 41 41 41 41 41 41 41 41 XA 369 B 38 39 40 41 41 41 42 43 44 XB 369 41 C 21 26 33 41 41 41 50 51 65 XC 369 41 Comparing Distribution A B C The means of all three distributions are the same BUT the scores in C have greater differences among them than B or A Sample Distributions N 9 A 41 41 41 41 41 41 41 41 41 41 XA 369 B 38 39 40 41 41 41 42 43 44 XB 369 41 C 21 26 33 41 41 41 50 51 65 XC 369 41 9 5 3 1 0 F r e q F r e q F r e q x x x x x X x x x x X xxxxxxx x x Distribution A Distribution B Distribution C 10 20 30 40 50 60 70 10 20 30 40 50 60 70 x x x x x x x 10 20 30 40 50 60 70 Range Describes the distance in scale units between the largest and smallest scores in a distribution Range XHighest XLowest 100 of the scores fall within the number of scales units indicated by the range Sample Distributions N 9 A 41 41 41 41 41 41 41 41 41 41 XA 369 B 38 39 40 41 41 41 42 43 44 XB 369 41 C 21 26 33 41 41 41 50 51 65 XC 369 41 In our sample distributions Range of A 41 41 0 100 of the scores fall at the same scale value Range of B 44 38 6 100 of the scores fall within 6 scale units of each Range of C 65 21 44 100 of the scores fall within 44 scale units of one other another Problems with the Range Insensitive Based on only 2 scores Subject to bias of outlying or extreme scores Changing only 2 of 9 values DRAMATICAL Changes the Range A1 19 X 41 X X X X X X X 63 X 10 20 30 40 50 60 70 XA1 41 XA1 41 has same range as C but it certainly has fewer differences among allof the scores Range 63 19 44 Sample Distribution A1 has a Range of 41 63 19 which equals the Range of Distribution C 65 21 A1 19 41 41 41 41 41 41 41 63 41 XA 369 B 38 39 40 41 41 41 42 43 44 XB 369 41 C 21 26 33 41 41 41 50 51 65 XC 369 41 Measures of Variability Professor David A Walsh Lecture 5 2 Interquartile Range 25 50 75 100 10 20 30 40 50 60 70 X25 30 X75 51 IQR 51 30 21 Interquartile Range IQR The range of values for the middle 50 of the scores in a distribution IQR X75 X25 IQR Q3 Q1 IQR XQ3 XQ1 More stable less subject to bias than range because does not use extreme outlying scores Still insensitive uses only 2 scores But more Stable ones Interquartile Range 25 50 75 100 10 20 30 40 50 60 70 X25 30 X75 51 IQR 51 30 21 How do you identify the X corresponding to Q1 and Q3 Must compute a cumulative percentile distribution for the scores Start from the bottom of a set of scores ordered from smallest to largest Accumulate upward to identify the X values at the 25 and 75 boundaries In my previous example 25 30 75 51 Semi interquartile Range SIQR Equal to of the IQR Indicates the number of scale units around the median that include 25 of the scores 12 5 below the vertical axis at 50 or Q2 12 5 above the vertical axis at 50 or Q2 SIQR IQR 2 X75 X25 2 Comparison of IQR and SIQR 25 50 75 100 10 20 30 40 50 60 70 IQR SIQR IQR Measures of Variability Professor David A Walsh Lecture 5 3 Variance Most widely used and useful measure of variability Four important properties 1 Sensitive every score is used in its computation 2 Logical value of variance increases as the difference between scores increases 3 Independent of the size of the scores or the mean of the 4 Independent of the number of terms or scores in the distribution distribution Variance deviation scores Is an average or mean of the squared is the Greek symbol for mean Computed on a population Where N is the total number of observations in the population Population Variance Formula Parts Sum of squared deviations SS Number of scores 9 5 3 1 0 F r e q F r e q F r e q x x x x x X x x x x X xxxxxxx x x Distribution A Distribution B Distribution C 10 20 30 40 50 60 70 10 20 30 40 50 60 70 x x x x x x x 10 20 30 40 50 60 70 Distribution B 2 41 41 41 41 41 41 41 41 41 9 3 4 2 1 1 0 0 0 0 0 0 1 1 4 2 9 3 2 28 X 38 39 40 41 41 41 42 43 44 X 369 3699 41 289 3 11 9 5 3 1 0 F r e q F r e q F r e q x x x x x X x x x x X xxxxxxx x x Distribution A 3 11 Distribution B Distribution C 10 20 30 40 50 60 70 10 20 30 40 50 60 70 x x x x x x x 10 20 30 40 50 60 70 Changing only 2 of 9 values DRAMATICAL Changes the Range A1 19 X 41 X X X X X X X 63 X 10 20 30 40 50 60 70 XA1 41 XA1 41 has same range as C but it certainly has fewer differences among allof the scores Range 63 19 44 The Variance is sensitive to changes in every score 2 Dist A1 Two scores moved to extremes X 19 41 41 41 41 41 41 41 63 369 22 484 0 0 0 0 0 0 0 0 0 0 0 0 0 0 22 484 2 968 41 41 41 41 41 41 41 41 41 X 3699 41 9689 107 55 9 5 3 1 0 F r e q F r e q F r e q x x x X x x x x X xxxxxxx x x 19 X 63 X 10 20 30 40 50 60 70 110 120 130 140 150 160 170 x x x x x x x 10 20 30 40 50 60 70 107 55 Distribution A1 3 11 Distribution B100 Distribution …


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USC PSYC 274 - Measures of Variability

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