1. 1 1 Slide © 2001 South-Western/Thomson Learning  Anderson  Sweeney  Williams Anderson  Sweeney  Williams  Slides Prepared by JOHN LOUCKS  CONTEMPORARYBUSINESSSTATISTICS WITH MICROSOFT  EXCEL CONTEMPORARYBUSINESSSTATISTICS

2. 2 2 Slide Chapter 3 Descriptive Statistics II: Numerical Methods - Part A n Measures of Location n Measures of Variability

3. 3 3 Slide Measures of Location n Mean n Median n Mode n Percentiles n Quartiles x x     % %

4. 4 4 Slide Example: Apartment Rents Given below is a sample of monthly rent values ($) for one-bedroom apartments. The data is a sample of 70 apartments in a particular city. The data are presented in ascending order.

5. 5 5 Slide Mean n The mean of a data set is the average of all the data values. n If the data are from a sample, the mean is denoted by. n If the data are from a population, the mean is denoted by (mu).

6. 6 6 Slide Example: Apartment Rents n Mean

7. 7 7 Slide Median n The median of a data set is the value in the middle when the data items are arranged in ascending order. n If there is an odd number of items, the median is the value of the middle item. n If there is an even number of items, the median is the average of the values for the middle two items.

8. 8 8 Slide Example: Apartment Rents n Median Median = 50th percentile Median = 50th percentile i = ( p /100) n = (50/100)70 = 35.5 Averaging the 35th and 36th data values: Median = (475 + 475)/2 = 475

9. 9 9 Slide Mode n The mode of a data set is the value that occurs with greatest frequency.

10. 10 Slide Example: Apartment Rents n Mode 450 occurred most frequently (7 times) 450 occurred most frequently (7 times) Mode = 450 Mode = 450

11. 11 Slide Using Excel to Compute the Mean, Median, and Mode n Formula Worksheet Note: Rows 7-71 are not shown.

12. 12 Slide Using Excel to Compute the Mean, Median, and Mode n Value Worksheet Note: Rows 7-71 are not shown.

13. 13 Slide Percentiles n The p th percentile of a data set is a value such that at least p percent of the items take on this value or less and at least (100 - p ) percent of the items take on this value or more. Arrange the data in ascending order.Arrange the data in ascending order. Compute index i, the position of the p th percentile.Compute index i, the position of the p th percentile. i = ( p /100) n i = ( p /100) n If i is not an integer, round up. The p th percentile is the value in the i th position.If i is not an integer, round up. The p th percentile is the value in the i th position. If i is an integer, the p th percentile is the average of the values in positions i and i +1.If i is an integer, the p th percentile is the average of the values in positions i and i +1.

14. 14 Slide Example: Apartment Rents n 90th Percentile i = ( p /100) n = (90/100)70 = 63 Averaging the 63rd and 64th data values: 90th Percentile = (580 + 590)/2 = 585 90th Percentile = (580 + 590)/2 = 585

15. 15 Slide Quartiles n Quartiles are specific percentiles n First Quartile = 25th Percentile n Second Quartile = 50th Percentile = Median n Third Quartile = 75th Percentile

16. 16 Slide Example: Apartment Rents n Third Quartile Third quartile = 75th percentile Third quartile = 75th percentile i = ( p /100) n = (75/100)70 = 52.5 = 53 i = ( p /100) n = (75/100)70 = 52.5 = 53 Third quartile = 525 Third quartile = 525

17. 17 Slide Using Excel to Compute Percentiles and Quartiles n Unsorted Monthly Rent ($) Note: Rows 7-71 are not shown.

18. 18 Slide Using Excel to Compute Percentiles and Quartiles n Sorting Data Step 1 Select any cell containing data in column B Step 2 Select the Data pull-down menu Step 3 Choose the Sort option Step 4 When the Sort dialog box appears: In the Sort by box, make sure that Monthly Rent ($) appears and that Ascending is selected In the My list has box, make sure that Header row is selected Click OK

19. 19 Slide Using Excel to Compute Percentiles and Quartiles n Sorted Monthly Rent ($) Note: Rows 7-71 are not shown.

20. 20 Slide Using Excel to Compute Percentiles and Quartiles n Formula Worksheet for 90 th Percentile’s Index Note: Rows 7-71 are not shown.

21. 21 Slide Using Excel to Compute Percentiles and Quartiles n Value Worksheet for 90 th Percentile’s Index Note: Rows 7-71 are not shown.

22. 22 Slide Using Excel to Compute Percentiles and Quartiles n Value Worksheet for 3 rd Quartile’s Index Note: Rows 7-71 are not shown.

23. 23 Slide Measures of Variability n Range n Interquartile Range n Variance n Standard Deviation n Coefficient of Variation

24. 24 Slide Range n The range of a data set is the difference between the largest and smallest data values. n It is the simplest measure of variability. n It is very sensitive to the smallest and largest data values.

25. 25 Slide Example: Apartment Rents n Range Range = largest value - smallest value Range = largest value - smallest value Range = 615 - 425 = 190 Range = 615 - 425 = 190

26. 26 Slide Interquartile Range n The interquartile range of a data set is the difference between the third quartile and the first quartile. n It is the range for the middle 50% of the data. n It overcomes the sensitivity to extreme data values.

27. 27 Slide Example: Apartment Rents n Interquartile Range 3rd Quartile ( Q 3) = 525 3rd Quartile ( Q 3) = 525 1st Quartile ( Q 1) = 445 1st Quartile ( Q 1) = 445 Interquartile Range = Q 3 - Q 1 = 525 - 445 = 80 Interquartile Range = Q 3 - Q 1 = 525 - 445 = 80

28. 28 Slide Variance n The variance is the average of the squared differences between each data value and the mean. n If the data set is a sample, the variance is denoted by s 2. If the data set is a population, the variance is denoted by  2. If the data set is a population, the variance is denoted by  2.

29. 29 Slide Standard Deviation n The standard deviation of a data set is the positive square root of the variance. n It is measured in the same units as the data, making it more easily comparable, than the variance, to the mean. n If the data set is a sample, the standard deviation is denoted s. If the data set is a population, the standard deviation is denoted  (sigma). If the data set is a population, the standard deviation is denoted  (sigma).

30. 30 Slide Coefficient of Variation n The coefficient of variation indicates how large the standard deviation is in relation to the mean. n If the data set is a sample, the coefficient of variation is computed as follows: n If the data set is a population, the coefficient of variation is computed as follows:

31. 31 Slide Example: Apartment Rents n Variance n Standard Deviation n Coefficient of Variation

32. 32 Slide Using Excel to Compute the Sample Variance and Standard Deviation n Formula Worksheet Note: Rows 8-71 are not shown.

33. 33 Slide Using Excel to Compute the Sample Variance and Standard Deviation n Value Worksheet Note: Rows 8-71 are not shown.

34. 34 Slide Using Excel’s Descriptive Statistics Tool Step 1 Select the Tools pull-down menu Step 2 Choose the Data Analysis option Step 3 Choose Descriptive Statistics from the list of Analysis Tools Analysis Tools … continued

35. 35 Slide Using Excel’s Descriptive Statistics Tool Step 4 When the Descriptive Statistics dialog box appears: Enter B1:B71 in the Input Range box Select Grouped By Columns Select Labels in First Row Select Output Range Enter D1 in the Output Range box Select Summary Statistics Select OK

36. 36 Slide Using Excel’s Descriptive Statistics Tool n Value Worksheet (Partial)

37. 37 Slide Using Excel’s Descriptive Statistics Tool n Value Worksheet (Partial)

38. 38 Slide End of Chapter 3, Part A