Variability
Variability Central tendency locates middle of distribution How are scores distributed around that point? Low variability vs. high variability Ways to measure variability Range Interquartile range Sum of squares Variance Standard deviation
Why Variability is Important Inference Reliability of estimators For its own sake Consistency (manufacturing, sports, etc.) Diversity (attitudes, strategies) 49 58 100 61 97 55 31 52 13 43 178 154 136 103 94 181 91 109 46 28 34 175 37 139 106 19 184 88 112 64 76 172 157 22 16 85 142 130 151 67 121 124 160 40 145 127 82 133 169 166 163 118 148 79 25 115 73 187 70 Average = 111.63 49 58 100 61 97 55 31 52 13 43 178 154 136 103 94 181 91 109 46 28 34 175 37 139 106 19 184 88 112 64 76 172 157 22 16 85 142 130 151 67 121 124 160 40 145 127 82 133 169 166 163 118 148 79 25 115 73 187 70 Average = 87.25 49 58 100 61 97 55 31 52 13 43 178 154 136 103 94 181 91 109 46 28 34 175 37 139 106 19 184 88 112 64 76 172 157 22 16 85 142 130 151 67 121 124 160 40 145 127 82 133 169 166 163 118 148 79 25 115 73 187 70 Average = 121 m = 100 100 105 108 99 97 106 104 93 96 109 95 92 103 102 94 98 101 107 Average = 99.5 100 105 108 99 97 106 104 93 96 109 95 92 103 102 94 98 101 107 Average = 99.13 100 105 108 99 97 106 104 93 96 109 95 92 103 102 94 98 101 107 Average = 100.5 M 87.3 121 111.6 M 99.1 100.5 99.5
Range Distance from minimum to maximum Sample range depends on n More useful as population parameter Theoretical property of measurement variable E.g. memory test: min and max possible Rough guidelines, e.g. height Measurement unit or precision X = [66.2, 78.6, 69.6, 65.3, 62.7] 78.6 – 62.7 + .1 = 16.0 78.65 – 62.65 = 16.0 (11 – 1) + 1 = 11
Interquartile range Quartiles Interquartile range Values of X based on dividing data into quarters 1st quartile: greater than 1/4 of data 3rd quartile: greater than 3/4 of data 2nd quartile = median Interquartile range Difference between 1st and 3rd quartiles Like range, but for middle half of distribution Not sensitive to n more stable 6 – 3 = 3 X = [1,1,2,2,2,3,3,4,4,4,4,5,5,5,5,6,6,6,6,6,7,7,7,8] 1st quartile = 3 3rd quartile = 6
Sum of Squares Based on deviation of each datum from the mean: (X – m) Square each deviation and add them up m 115 88 94 108 122 133 145 729 = 272 441 = 212 49 = 72 27 21 7 7 18 30 2 = 49 2 = 324 2 = 900
Variance Most sophisticated statistic for variability Sum of squares divided by N Mean Square: average squared deviation 729 = 272 441 = 212 49 = 72 m 115 88 94 108 122 133 145 27 21 7 18 30 2 = 49 2 = 324 2 = 900
Standard deviation Typical difference between X and m Again, based on (X – m)2 Variance is average squared deviation, so sqrt(variance) is standard deviation X = [5, 3, 7, 6, 4, 6, 8, 7, 4, 2, 3, 5] m = 5 Square-root Average X – m = [0, -2, 2, 1, -1, 1, 3, 2, -1, -3, -2, 0] (X – m)2 = [0, 4, 4, 1, 1, 1, 9, 4, 1, 9, 4, 0] Square Average
Why squared difference? Could use absolute distances, |X - µ| Would be more intuitive: average distance from the mean Squares have special mathematical properties Can be broken into different parts Central Tendency: Common to all scores Sum of Squares: Differences among scores