1. Number of observations in the population The population mean of a data set is the average of all the data values. Sum of the values of the N observations Measures of Location

2. The population mean of a data set is the average of all the data values. Sum of the values of the n observations The sample mean is the point estimator of the population mean . Number of observations in the sample Measures of Location

3. Example: Recall the Hudson Auto Repair example The manager of Hudson Auto would like to have better understanding of the cost of parts used in the engine tune-ups performed in the shop. She examines 50 customer invoices for tune-ups. The costs of parts, rounded to the nearest dollar, are listed below. 91789357755299809762 71697289667579757276 10474626897105776580109 85978868836871696774 6282981017910579696273 3949 50 78.98 Measures of Location

4. For an odd number of observations: in ascending order 2618271427 19 7 observations the median is the middle value. 12 Measures of Location

5. in ascending order the median is the average of the middle two values. Median = (19 + 26)/2 = 22.5 For an even number of observations: 2618271427 19 8 observations 12 30 Measures of Location

6. Averaging the 25th and 26th data values: = (75 + 76)/2 = 75.5 Note: Data is in ascending order. 525762 65666768 69 71 72 73 74 75 76777879 80 82838588899193 97 9899101104105 109 Example: Hudson Auto Repair Measures of Location Median

7. = 62 Note: Data is in ascending order. 525762 65666768 69 71 72 73 74 75 76777879 80 82838588899193 97 9899101104105 109 Example: Hudson Auto Repair Measures of Location Mode

8. First quartile = 25 th percentile = 13 th First quartile = 69 525762 65666768 69 71 72 73 74 75 76777879 80 82838588899193 97 9899101104105 109 Example: Hudson Auto Repair i th = ( p /100) n = ( 25 /100)50= 12.5 Note: Data is in ascending order. Measures of Location

9. i th = ( p /100) n = Average the 40 th and 41 st data values 80 th Percentile = 525762 65666768 69 71 72 73 74 75 76777879 80 82838588899193 97 9899101104105 109 Note: Data is in ascending order. Example: Hudson Auto Repair ( 80 /100)50= 40 th (93 + 97)/2 = 95 Measures of Location

10. 525762 65666768 69 71 72 73 74 75 76777879 80 82838588899193 97 9899101104105 109 Example: Hudson Auto Repair: 80 th Percentile 95 Note: Data is in ascending order. Measures of Location

11. data_pelican.xls Pelican Stores -- continued Pelican Stores is chain of women’s apparel stores. It recently ran a promotion in which discount coupons were set to customers of other National Clothing stores. Data collected for a sample of 100 in-store credit card transactions at Pelican Stores during one day while the promotion was running are shown in Table 2.18. Customers who made a purchase using a discount coupon are referred to as promotional customers and customers who made a purchase but did not use a discount coupon are referred to as regular customers. Because the promotional coupons were not set to regular Pelican Stores customers, management considers the sales made to people presenting the promotional coupons as sales it would not otherwise make. Pelican’s management would like to use this sample data to learn about its customer base and to evaluate the promotion involving discounts. Managerial Report 1.Using graphs and tables, summarize the qualitative variables. 2.Using graphs and tables, summarize the quantitative variables. 3.Using pivot tables and scatter plots, summarize the variables. 4.Compute the mean, mode, median, and the 25 th and 75 th percentiles.

12. Range = maximum – minimum Range = 109 – 52 = 57 Note: Data is in ascending order. 525762 65666768 69 71 72 73 74 75 76777879 80 82838588899193 97 9899101104105 109 Example: Hudson Auto Repair Measures of Variability

13. Note: Data is in ascending order. 525762 65666768 69 71 72 73 74 75 76777879 80 82838588899193 97 9899101104105 109 Example: Hudson Auto Repair Measures of Variability 3rd Quartile ( Q 3) = 89 1st Quartile ( Q 1) = 69 = Q 3 – Q 1 = 89 – 69 = 20 Interquartile Range

14. The population mean The population variance is the average variation Measures of Variability

15. i th deviation from the population mean The population variance is the average variation Measures of Variability

16. i th squared deviation from the population mean The population variance is the average variation Measures of Variability

17. Sum of squared deviations from the population mean The population variance is the average variation Measures of Variability

18. Total variation of x The population variance is the average variation Measures of Variability

19. Number of observations in the population The population variance is the average variation Measures of Variability

20. The population variance is the average variation Measures of Variability The sample variance is an unbiased estimator of   Number of observations in the sample

21. The population variance is the average variation Measures of Variability The sample variance is an unbiased estimator of  

22. The population variance is the average variation Measures of Variability The sample variance is an unbiased estimator of   Degrees of freedom

23. Measures of Variability

24. Sorted invoices Observed value Sqrd Dev from the mean 152727.92 257483.12 362288.32 462288.32 562288.32 662288.32 765195.44 49105677.04 50109901.20 Sum3949 9592.98 x = 78.98 Measures of Variability

25. Variance Standard Deviation Example: Hudson Auto Repair Coefficient of variation Measures of Variability

26. Pelican Stores -- continued Pelican Stores is chain of women’s apparel stores. It recently ran a promotion in which discount coupons were set to customers of other National Clothing stores. Data collected for a sample of 100 in-store credit card transactions at Pelican Stores during one day while the promotion was running are shown in Table 2.18. Customers who made a purchase using a discount coupon are referred to as promotional customers and customers who made a purchase but did not use a discount coupon are referred to as regular customers. Because the promotional coupons were not set to regular Pelican Stores customers, management considers the sales made to people presenting the promotional coupons as sales it would not otherwise make. Pelican’s management would like to use this sample data to learn about its customer base and to evaluate the promotion involving discounts. Managerial Report 1.Using graphs and tables, summarize the qualitative variables. 2.Using graphs and tables, summarize the quantitative variables. 3.Using pivot tables and scatter plots, summarize the variables. 4.Compute the mean, mode, median, and the 25 th and 75 th percentiles. 5.Compute the range, IQR, variance, and standard deviations. data_pelican.xls

27. 525762 65666768 69 71 72 73 74 75 76777879 80 82838588899193 97 9899101104105 109 Note: Data is in ascending order. Example: Hudson Auto Repair z-Score of Smallest Value Measures of Shape

28. Observed value Dev from the meanz-score 52-26.98-1.93 57-21.98-1.57 62-16.98-1.21 62-16.98-1.21 62-16.98-1.21 62-16.98-1.21 65-13.98 10526.021.86 10930.022.15 39490 0 Measures of Shape x = 78.98 s = 13.992

29. An important measure of the shape of a distribution is called skewness. It is just the average of the n cubed z-scores when n is “large” Measures of Shape

30. Observed valuez-score cubed z-score 52-1.93-7.17 57-1.57-3.88 62-1.21-1.79 62-1.21-1.79 62-1.21-1.79 62-1.21-1.79 65 1051.866.43 1092.159.88 39490 22.567 Measures of Shape

31. 2 4 6 8 10 12 14 16 18 Parts Cost ($) Frequency 50 60 70 80 90 100 110 Tune-up Parts Cost $78.98 $75.50 $62 Measures of Shape

32. Moderately Skewed Left Symmetric Highly Skewed Right skew = 0 skew = .31 skew = 1.25 Measures of Shape

33. Chebyshev's Theorem: At least (1 - 1/z 2 ) of the data values are within z standard deviations of the mean. At least 75% of the data values are within 2 standard deviations of the mean At least 89% of the data values are within 3 standard deviations of the mean At least 94% of the data values are within 4 standard deviations of the mean Measures of Shape At least 0% of the data values are within 1 standard deviation of the mean

34. Empirical Rule: 95.44% of the data values are within 2 standard deviations of the mean 99.74% of the data values are within 3 standard deviations of the mean 99.99% of the data values are within 4 standard deviations of the mean Measures of Shape 68.26% of the data values are within 1 standard deviation of the mean

35. z-score Is the observation within 2 std dev? -1.93Yes -1.57Yes -1.21Yes -1.21Yes -1.21Yes -1.21Yes Yes 1.86Yes 2.15No 49 of the 50 data values are within 2 s of the mean = 98% 50 of the 50 data values are within 3 s of the mean = 100% None of the values are outliers Measures of Shape

36. data_pelican.xls Pelican Stores -- continued Pelican Stores is chain of women’s apparel stores. It recently ran a promotion in which discount coupons were set to customers of other National Clothing stores. Data collected for a sample of 100 in-store credit card transactions at Pelican Stores during one day while the promotion was running are shown in Table 2.18. Customers who made a purchase using a discount coupon are referred to as promotional customers and customers who made a purchase but did not use a discount coupon are referred to as regular customers. Because the promotional coupons were not set to regular Pelican Stores customers, management considers the sales made to people presenting the promotional coupons as sales it would not otherwise make. Pelican’s management would like to use this sample data to learn about its customer base and to evaluate the promotion involving discounts. Managerial Report 1.Using graphs and tables, summarize the qualitative variables. 2.Using graphs and tables, summarize the quantitative variables. 3.Using pivot tables and scatter plots, summarize the variables. 4.Compute the mean, mode, median, and the 25 th and 75 th percentiles. 5.Compute the range, IQR, variance, and standard deviations. 6.Compute the z-scores and skew, find the outliers, and count the observations that are within 1, 2, & 3 standard deviations of the mean.

37. The covariance is computed as follows: (for samples) (for populations) Measures of the relationship between 2 variables

38. i th deviation from x’s means The covariance is computed as follows: (for samples) (for populations) Measures of the relationship between 2 variables

39. i th deviation from y’s means The covariance is computed as follows: (for samples) (for populations) Measures of the relationship between 2 variables

40. The sizes of the sample and population The covariance is computed as follows: (for samples) (for populations) Measures of the relationship between 2 variables

41. Degrees of freedom The covariance is computed as follows: (for samples) (for populations) Measures of the relationship between 2 variables

42. The covariance is computed as follows: Measures of the relationship between 2 variables

43. Reed Auto periodically has a special week-long sale. As part of the advertising campaign Reed runs one or more television commercials during the weekend preceding the sale. Data from a sample of 5 previous sales are shown below. Example: Reed Auto Sales Number of TV Ads (x) Number of Cars Sold (y) 1321313213 14 24 18 17 27 Measures of the relationship between 2 variables

44. TV Ads Cars sold 5 10 15 20 25 30 0 35 1 2304 Example: Reed Auto Sales Measures of the relationship between 2 variables

45. x y 1321313213 14 24 18 17 27 1 3 2 1 3 2 2 2 2 2      x – x (x – x) 1 1 0 1 1 20      y – y (y – y) 2 36 16 4 9 49 (y – y) 6 4 0 3 7 (x – x) 2 14 24 18 17 27 10. 100. 114. 4. 4. 20. 5 5 4 4 4 = 2 = 20 = 28.5 = 1 = 5 x y s xx s yy s xy Example: Reed Auto Sales (ads) (cars) (ads squared) (cars squared) (ads-cars) = 5.34 = 1 sxsx sysy (ads) (cars) Measures of the relationship between 2 variables

46. = 5 s xy Example: Reed Auto Sales (ads-cars) = 5.34 = 1 sxsx sysy (ads) (cars) (ads-cars) (ads) (cars) Measures of the relationship between 2 variables

47. data_pelican.xls Pelican Stores -- continued Pelican Stores is chain of women’s apparel stores. It recently ran a promotion in which discount coupons were set to customers of other National Clothing stores. Data collected for a sample of 100 in-store credit card transactions at Pelican Stores during one day while the promotion was running are shown in Table 2.18. Customers who made a purchase using a discount coupon are referred to as promotional customers and customers who made a purchase but did not use a discount coupon are referred to as regular customers. Because the promotional coupons were not set to regular Pelican Stores customers, management considers the sales made to people presenting the promotional coupons as sales it would not otherwise make. Pelican’s management would like to use this sample data to learn about its customer base and to evaluate the promotion involving discounts. Managerial Report 1.Using graphs and tables, summarize the qualitative variables. 2.Using graphs and tables, summarize the quantitative variables. 3.Using pivot tables and scatter plots, summarize the variables. 4.Compute the mean, mode, median, and the 25 th and 75 th percentiles. 5.Compute the range, IQR, variance, and standard deviations. 6.Compute the z-scores and skew, find the outliers, and count the observations that are within 1, 2, & 3 standard deviations of the mean. 7.Compute the covariances and correlations.