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1 Measures of variation

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Variability measures In addition to locating the center of the observed values of the variable in the data, another important aspect of a descriptive study of the variable is numerically measuring the extent of variation around the center. Two data sets of the same variable may exhibit similar positions of center but may be remarkably different with respect to variability. The variability measures should have the following characteristics: - be minimum if all the value of the distribution are the same -increase as increase the difference among the values of the distribution 2

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Shop s RevenuesCostsemploy ee placeDirector gender Shop On-line R.O 13502055citymaleyes145 22001003suburbsmaleyes100 360035010Near the city femaleno250 450027010suburbsfemaleno230 52702006citymaleno70 61801203citymaleno60 72051053suburbsmaleno100 83402105Near the city femaleno120 92801404cityfemaleyes140 3

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Variability revenu e 350 200 600 500 270 180 205 340 280 revenu e (A) revenu e (B) revenu e (C) 325 300140 325 350270 325 400830 325 200605 325 300120 325 200 325 300190 325 400200 325 350370 Observed distribution Possible distribution All the 3 possible distribution have the same mean of the observed one BUT the distribution are very different!!! 4

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Some measures of variability Range It is the width of the interval that contain all the values of the distribution. Interquartile range It is the width of the interval that contain 50% the values of the distribution. (central ones). 5

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Example Revenue 350 200 600 500 270 180 205 340 280 Revenue (A) Revenue (B) Revenue (C) 325 300140 325 350270 325 400830 325 200605 325 300120 325 200 325 300190 325 400200 325 350370 x min 180325200120 x max 600325400830 Range=x max -x min 4200200710 A No Variability All values are the same From A to B and from B to C, the variability increasaes, the range is higher. 6

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Deviation from the mean The variance σ 2 is function of the differences among each value x i and the mean The sum of squared deviation is 7

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The standard is the squared root of the variance The coefficient of variation CV is the ratio between the standard dev. and the mean, multiplied 100 8

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9 Example Revenue x j Differences from mean (x j -μ) Squared differences (x j -μ) 2 350 25625 200 -12515625 600 27575625 500 17530625 270 -553025 180 -14521025 205 -12014400 340 15225 280 -452025 Mean property s.s.dev.=163200 Variance=18133,3 Std.Dev.=134,7 9

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Variabilità dei ricavi dei punti vendita Un basso grado di variabilità indica che i punti vendita realizzano performance simili (i ricavi si discostano poco tra di loro) Viceversa un alto grado di variabilità fa capire che c’è una certa eterogeneità nei risultati delle vendite ottenuti nei diversi negozi 10

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Variance from a frequency distribution Employee (x j ) Shops (n j ) 32 41 63 71 102 (x j -μ) 2 *n j 19,34 4,45 0,04 0,79 30,26 11

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Standardised values If a quantitative variable X as mean and standard deviation σ, it is possible to obtain its standardised values The distribution of Y has zero mean and standard deviation equal to 1

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Comparison among two founds (equal mean) In last 5 years F 1 and F 2 had the same performance in mean, but variances are different Var(F 1 )>Var(F 2 ) F1F1 F2F2 20037,76,4 20046,15,9 20050,43,2 20069,87,1 20073,54,9 mean5,5 var10,71,8 Higher variability means that performance very different from the mean are more frequent. Higher volatility Higher risk 13

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Comparison among the performance of two founds (different mean) F 1 has a mean and a variance higher than F 2. Can we say that F 1 is an higher risk found than F 2 ? F1F1 F2F2 20039,71,4 20047,11,9 20050,92,2 20069,92,1 20077,54,9 media7,02,5 var10,61,5 CV46,549,3 We have to compare the CV F 1 has less variability 14

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