1 1 Slide © 2003 South-Western/Thomson Learning TM Slides Prepared by JOHN S. LOUCKS St. Edwards University

2 2 Slide © 2003 South-Western/Thomson Learning TM Chapter 3 Descriptive Statistics: Numerical Methods, Part A n Measures of Location n Measures of Variability x x % %

3 3 Slide © 2003 South-Western/Thomson Learning TM Measures of Location n Mean n Median n Mode n Percentiles n Quartiles

4 4 Slide © 2003 South-Western/Thomson Learning TM 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 Slide © 2003 South-Western/Thomson Learning TM 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. If the data are from a population, the mean is denoted by (mu). If the data are from a population, the mean is denoted by (mu).

6 6 Slide © 2003 South-Western/Thomson Learning TM Example: Apartment Rents n Mean

7 7 Slide © 2003 South-Western/Thomson Learning TM Median n The median is the measure of location most often reported for annual income and property value data. n A few extremely large incomes or property values can inflate the mean.

8 8 Slide © 2003 South-Western/Thomson Learning TM Median n The median of a data set is the value in the middle when the data items are arranged in ascending order. n For an odd number of observations, the median is the middle value. n For an even number of observations, the median is the average of the two middle values.

9 9 Slide © 2003 South-Western/Thomson Learning TM 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 = ( )/2 = 475

10 Slide © 2003 South-Western/Thomson Learning TM Mode n The mode of a data set is the value that occurs with greatest frequency. n The greatest frequency can occur at two or more different values. n If the data have exactly two modes, the data are bimodal. n If the data have more than two modes, the data are multimodal.

11 Slide © 2003 South-Western/Thomson Learning TM Example: Apartment Rents n Mode 450 occurred most frequently (7 times) 450 occurred most frequently (7 times) Mode = 450 Mode = 450

12 Slide © 2003 South-Western/Thomson Learning TM Using Excel to Compute the Mean, Median, and Mode n Formula Worksheet Note: Rows 7-71 are not shown.

13 Slide © 2003 South-Western/Thomson Learning TM n Value Worksheet Using Excel to Compute the Mean, Median, and Mode Note: Rows 7-71 are not shown.

14 Slide © 2003 South-Western/Thomson Learning TM Percentiles n A percentile provides information about how the data are spread over the interval from the smallest value to the largest value. n Admission test scores for colleges and universities are frequently reported in terms of percentiles.

15 Slide © 2003 South-Western/Thomson Learning TM 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. Percentiles

16 Slide © 2003 South-Western/Thomson Learning TM Example: Apartment Rents n 90th Percentile i = ( p /100) n = (90/100)70 = 63 Averaging the 63rd and 64th data values: 90th Percentile = ( )/2 = th Percentile = ( )/2 = 585

17 Slide © 2003 South-Western/Thomson Learning TM Quartiles n Quartiles are specific percentiles n First Quartile = 25th Percentile n Second Quartile = 50th Percentile = Median n Third Quartile = 75th Percentile

18 Slide © 2003 South-Western/Thomson Learning TM 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

19 Slide © 2003 South-Western/Thomson Learning TM Using Excel to Compute Percentiles and Quartiles n Unsorted Monthly Rent ($) Note: Rows 7-71 are not shown.

20 Slide © 2003 South-Western/Thomson Learning TM 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 Using Excel to Compute Percentiles and Quartiles

21 Slide © 2003 South-Western/Thomson Learning TM n Sorted Monthly Rent ($) Using Excel to Compute Percentiles and Quartiles Note: Rows 7-71 are not shown.

22 Slide © 2003 South-Western/Thomson Learning TM n Formula Worksheet for 90 th Percentiles Index Using Excel to Compute Percentiles and Quartiles Note: Rows 7-71 are not shown.

23 Slide © 2003 South-Western/Thomson Learning TM n Value Worksheet for 90 th Percentiles Index Using Excel to Compute Percentiles and Quartiles Note: Rows 7-71 are not shown.

24 Slide © 2003 South-Western/Thomson Learning TM n Value Worksheet for 3 rd Quartiles Index Using Excel to Compute Percentiles and Quartiles Note: Rows 7-71 are not shown.

25 Slide © 2003 South-Western/Thomson Learning TM Measures of Variability n It is often desirable to consider measures of variability (dispersion), as well as measures of location. n For example, in choosing supplier A or supplier B we might consider not only the average delivery time for each, but also the variability in delivery time for each.

26 Slide © 2003 South-Western/Thomson Learning TM Measures of Variability n Range n Interquartile Range n Variance n Standard Deviation n Coefficient of Variation

27 Slide © 2003 South-Western/Thomson Learning TM 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.

28 Slide © 2003 South-Western/Thomson Learning TM Example: Apartment Rents n Range Range = largest value - smallest value Range = largest value - smallest value Range = = 190 Range = = 190

29 Slide © 2003 South-Western/Thomson Learning TM 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.

30 Slide © 2003 South-Western/Thomson Learning TM 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 = = 80 Interquartile Range = Q 3 - Q 1 = = 80

31 Slide © 2003 South-Western/Thomson Learning TM Variance n The variance is a measure of variability that utilizes all the data. It is based on the difference between the value of each observation ( x i ) and the mean ( x for a sample, for a population). It is based on the difference between the value of each observation ( x i ) and the mean ( x for a sample, for a population).

32 Slide © 2003 South-Western/Thomson Learning TM 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.

33 Slide © 2003 South-Western/Thomson Learning TM 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).

34 Slide © 2003 South-Western/Thomson Learning TM 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:

35 Slide © 2003 South-Western/Thomson Learning TM Example: Apartment Rents n Variance n Standard Deviation n Coefficient of Variation

36 Slide © 2003 South-Western/Thomson Learning TM Using Excel to Compute the Sample Variance, Standard Deviation, and Coefficient of Variation n Formula Worksheet Note: Rows 8-71 are not shown.

37 Slide © 2003 South-Western/Thomson Learning TM n Value Worksheet Using Excel to Compute the Sample Variance, Standard Deviation, and Coefficient of Variation Note: Rows 8-71 are not shown.

38 Slide © 2003 South-Western/Thomson Learning TM Using Excels 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

39 Slide © 2003 South-Western/Thomson Learning TM 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 Click OK Using Excels Descriptive Statistics Tool

40 Slide © 2003 South-Western/Thomson Learning TM n Value Worksheet (Partial) Using Excels Descriptive Statistics Tool Note: Rows 9-71 are not shown.

41 Slide © 2003 South-Western/Thomson Learning TM n Value Worksheet (Partial) Using Excels Descriptive Statistics Tool Note: Rows 1-8 and are not shown.

42 Slide © 2003 South-Western/Thomson Learning TM End of Chapter 3, Part A