Measures of Variability

Transcription

1 Sample I: 30, 35, 40, 45, 50, 55, 60, 65, 70 Sample II: 30, 41, 48, 49, 50, 51, 52, 59, 70 Sample III: 41, 45, 48, 49, 50, 51, 52, 55, 59

2 Sample I: 30, 35, 40, 45, 50, 55, 60, 65, 70 Sample II: 30, 41, 48, 49, 50, 51, 52, 59, 70 Sample III: 41, 45, 48, 49, 50, 51, 52, 55, 59

3 Sample Range: the difference between the largest and the smallest sample values.

4 Sample Range: the difference between the largest and the smallest sample values. e.g. for Sample I: 30, 35, 40, 45, 50, 55, 60, 65, 70 the sample range is 40(= 70 30).

5 Sample Range: the difference between the largest and the smallest sample values. e.g. for Sample I: 30, 35, 40, 45, 50, 55, 60, 65, 70 the sample range is 40(= 70 30). Deviation from the Sample Mean: the diffenence between the individual sample value and the sample mean.

6 Sample Range: the difference between the largest and the smallest sample values. e.g. for Sample I: 30, 35, 40, 45, 50, 55, 60, 65, 70 the sample range is 40(= 70 30). Deviation from the Sample Mean: the diffenence between the individual sample value and the sample mean. e.g. for Sample I: 30, 35, 40, 45, 50, 55, 60, 65, 70 the sample mean is 50 and thus the deviation from the sample mean for each data is -20, -15, -10, -5, 0, 5, 10, 15, 20.

7 Sample Variance: the mean (or average) of the sum of squares of the deviations from the sample mean for each individual data.

8 Sample Variance: the mean (or average) of the sum of squares of the deviations from the sample mean for each individual data. If our sample size is n, and we use x to denote the sample mean, then the sample variance s 2 is given by: s 2 = n i=1 (x i x) 2 n 1 = S xx n 1

9 Sample Variance: the mean (or average) of the sum of squares of the deviations from the sample mean for each individual data. If our sample size is n, and we use x to denote the sample mean, then the sample variance s 2 is given by: s 2 = n i=1 (x i x) 2 n 1 = S xx n 1 Sample Standard Deviation: the square root of the sample variance s = s 2

10 e.g. for Sample I: 30, 35, 40, 45, 50, 55, 60, 65, 70, the mean is 50 and we have x i x i x (x i x) Therefore the sample variance is ( )/(9 1) = and the standard deviation is = 13.7.

11 e.g. for Sample II: 30, 41, 48, 49, 50, 51, 52, 59, 70, the mean is also 50 and we have x i x i x (x i x) Therefore the sample variance is ( )/(9 1) = and the standard deviation is = 11.0.

12 e.g. for Sample III: 41, 45, 48, 49, 50, 51, 52, 55, 59, the mean is also 50 and we have x i x i x (x i x) Therefore the sample variance is ( )/(9 1) = and the standard deviation is = 4.9.

13 sample variance for Sample I is 187.5, for Sample II is and for Sample III is

14 Remark: 1. Why use the sum of squares of the deviations? Why not sum the deviations?

15 Remark: 1. Why use the sum of squares of the deviations? Why not sum the deviations? Because the sum of the deviations from the sample mean EQUAL TO 0!

16 Remark: 1. Why use the sum of squares of the deviations? Why not sum the deviations? Because the sum of the deviations from the sample mean EQUAL TO 0! n n n (x i x) = x i x i=1 = = = 0 i=1 i=1 n x i n x i=1 n x i n( 1 n i=1 n x i ) i=1

17 Remark: 2. Why do we use divisor n 1 in the calculation of sample variance while we use use divisor N in the calculation of the population variance?

18 Remark: 2. Why do we use divisor n 1 in the calculation of sample variance while we use use divisor N in the calculation of the population variance? The variance is a measure about the deviation from the center. However, the center for sample and population are different, namely sample mean and population mean.

19 Remark: 2. Why do we use divisor n 1 in the calculation of sample variance while we use use divisor N in the calculation of the population variance? The variance is a measure about the deviation from the center. However, the center for sample and population are different, namely sample mean and population mean. If we use µ instead of x in the definition of s 2, then s 2 = (x i µ)/n.

20 Remark: 2. Why do we use divisor n 1 in the calculation of sample variance while we use use divisor N in the calculation of the population variance? The variance is a measure about the deviation from the center. However, the center for sample and population are different, namely sample mean and population mean. If we use µ instead of x in the definition of s 2, then s 2 = (x i µ)/n. But generally, population mean is unavailable to us. So our choice is the sample mean. In that case, the observations x i s tend to be closer to their average x then to the population average µ. So to compensate, we use divisor n 1.

21 Remark: 3. It customary to refer to s 2 as being based on n 1 degrees of freedom (df).

22 Remark: 3. It customary to refer to s 2 as being based on n 1 degrees of freedom (df). s 2 is the average of n quantities: (x 1 x) 2, (x 2 x) 2,..., (x n x) 2. However, the sum of x 1 x, x 2 x,..., x n x is 0. Therefore if we know any n 1 of them, we know all of them.

23 Remark: 3. It customary to refer to s 2 as being based on n 1 degrees of freedom (df). s 2 is the average of n quantities: (x 1 x) 2, (x 2 x) 2,..., (x n x) 2. However, the sum of x 1 x, x 2 x,..., x n x is 0. Therefore if we know any n 1 of them, we know all of them. e.g. {x 1 = 4, x 2 = 7, x 3 = 1, and x 4 = 10}.

24 Remark: 3. It customary to refer to s 2 as being based on n 1 degrees of freedom (df). s 2 is the average of n quantities: (x 1 x) 2, (x 2 x) 2,..., (x n x) 2. However, the sum of x 1 x, x 2 x,..., x n x is 0. Therefore if we know any n 1 of them, we know all of them. e.g. {x 1 = 4, x 2 = 7, x 3 = 1, and x 4 = 10}. Then the mean is x = 5.5 and x 1 x = 1.5, x 2 x = 1.5 and x 3 x = 4.5. From that, we know directly that x 4 x = 4.5 since their sum is 0.

25 Some mathematical results for s 2 :

26 Some mathematical results for s 2 : s 2 = Sxx n 1 where S xx = (x i x) 2 = x 2 i ( x i ) 2 n ;

27 Some mathematical results for s 2 : s 2 = Sxx n 1 where S xx = (x i x) 2 = x 2 i ( x i ) 2 n ; If y 1 = x 1 + c, y 2 = x 2 + c,..., y n = x n + c, then s 2 y = s 2 x ;

28 Some mathematical results for s 2 : s 2 = Sxx n 1 where S xx = (x i x) 2 = x 2 i ( x i ) 2 n ; If y 1 = x 1 + c, y 2 = x 2 + c,..., y n = x n + c, then s 2 y = s 2 x ; If y 1 = cx 1, y 2 = cx 2,..., y n = cx n, then s y = c s x. Here s 2 x is the sample variance of the x s and s 2 y is the sample variance of the y s. c is any nonzero constant.

29 e.g. in the previous example, Sample III is {41, 45, 48, 49, 50, 51, 52, 55, 59} then we can calculate the sample variance as following x i x 2 i xi 450 x 2 i Therefore the sample variance is ( )/(9 1) =

30 Boxplots

31 Boxplots e.g. A recent article ( Indoor Radon and Childhood Cancer ) presented the accompanying data on radon concentration (Bq/m 2 ) in two different samples of houses. The first sample consisted of houses in which a child diagnosed with cancer had been residing. Houses in the second sample had no recorded cases of childhood cancer. The following graph presents a stem-and-leaf display of the data. 1. Cancer 2. No cancer Stem: Tens digit 8 5 Leaf: Ones digit

32 The boxplot for the 1st data set is:

33 The boxplot for the 2nd data set is:

34 We can also make the boxplot for both data sets:

35 Some terminology: Lower Fourth: the median of the smallest half

36 Some terminology: Lower Fourth: the median of the smallest half Upper Fourth: the median of the largest half

37 Some terminology: Lower Fourth: the median of the smallest half Upper Fourth: the median of the largest half Fourth spread: the difference between lower fourth and upper fourth f s = upper fourth lower fourth

38 Some terminology: Lower Fourth: the median of the smallest half Upper Fourth: the median of the largest half Fourth spread: the difference between lower fourth and upper fourth f s = upper fourth lower fourth Outlier: any observation farther than 1.5f s from the closest fourth

39 Some terminology: Lower Fourth: the median of the smallest half Upper Fourth: the median of the largest half Fourth spread: the difference between lower fourth and upper fourth f s = upper fourth lower fourth Outlier: any observation farther than 1.5f s from the closest fourth An outlier is extreme if it is more than 3f s from the nearest fourth, and it is mild otherwise.

40 The boxplot for the 2nd data set is: