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1 Measures of variation. Variability measures In addition to locating the center of the observed values of the variable in the data, another important.

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Presentation on theme: "1 Measures of variation. Variability measures In addition to locating the center of the observed values of the variable in the data, another important."— Presentation transcript:

1 1 Measures of variation

2 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

3 Shop s RevenuesCostsemploy ee placeDirector gender Shop On-line R.O citymaleyes suburbsmaleyes Near the city femaleno suburbsfemaleno citymaleno citymaleno suburbsmaleno Near the city femaleno cityfemaleyes140 3

4 Variability revenu e revenu e (A) revenu e (B) revenu e (C) Observed distribution Possible distribution All the 3 possible distribution have the same mean of the observed one BUT the distribution are very different!!! 4

5 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

6 Example Revenue Revenue (A) Revenue (B) Revenue (C) x min x max Range=x max -x min 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

7 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

8 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

9 9 Example Revenue x j Differences from mean (x j -μ) Squared differences (x j -μ) Mean property s.s.dev.= Variance=18133,3 Std.Dev.=134,7 9

10 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

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

12 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

13 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 F2F ,76, ,15, ,43, ,87, ,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

14 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 F2F ,71, ,11, ,92, ,92, ,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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