1. BIA 2610 – Statistical Methods Chapter 3 – Descriptive Statistics: Numerical Measures

2. Chapter 3 Descriptive Statistics: Numerical Measures n Measures of Variability n Measures of Location

3. Measures of Location If the measures are computed for data from a sample, they are called sample statistics. If the measures are computed for data from a population, they are called population parameters. A sample statistic is referred to as the point estimator of the corresponding population parameter. n Mean n Median n Mode n Percentiles n Quartiles n Weighted Mean n Geometric Mean

4. Mean n Perhaps the most important measure of location is the mean. n The mean provides a measure of central location. n The mean of a data set is the average of all the data values.

5. Number of observations in the sample Sum of the values of the n observations

6. Population Mean  Number of observations in the population Sum of the values of the N observations

7. Seventy efficiency apartments were randomly sampled in a college town. The monthly rents for these apartments are listed below. Example: Apartment Rents

8. Central Location Measures Averaging the 35th and 36th data values: Median = (575 + 575)/2 = 575 Example: Apartment Rents Note: Data is in ascending order. Mode = 550 550 occurred most frequently (7 times)

9. Excel’s Mean, Median, and Mode Functions Excel’s Mean function =AVERAGE(data cell range) Excel’s Median function =MEDIAN(data cell range) Excel’s Mode function =MODE.SNGL(data cell range)

10. Percentiles Arrange the data in ascending order. Compute L p, the location of the pth percentile. L p = (p/100)(n + 1) For example, in a sample of n = 70 values, the location of the 80 th percentile (p = 80) would be calculated as: L p = (p/100)(n + 1) = (80/100)(70 + 1) = 56.8

11. 80 th Percentile L p = (p/100)(n + 1) = (80/100)(70 + 1) = 56.8 (the 56 th value plus.8 times the difference between the 57 th and 56 th values) 80 th Percentile = 635 +.8(649 – 635) = 646.2 Example: Apartment Rents Note: Data is in ascending order.

12. 80 th Percentile “At least 80% of the items take on a value of 646.2 or less.” “At least 20% of the items take on a value of 646.2 or more.” 56/70 =.8 or 80% 14/70 =.2 or 20% Example: Apartment Rents

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

14. Third Quartile (75 th Percentile) L p = (p/100)(n + 1) = (75/100)(70 + 1) = 53.25 Third quartile = 625 +.25(625 – 625) = 625 Example: Apartment Rents Note: Data is in ascending order. (the 53 rd value plus.25 times the difference between the 54 th and 53 rd values)

15. Measures of Variability Range Variance Standard Deviation Coefficient of Variation

16. Range Range = largest value - smallest value Range = 715 - 525 = 190 Note: Data is in ascending order. Example: Apartment Rents

17. Variance The variance is a measure of variability that utilizes all the data. The variance is useful in comparing the variability of two or more variables.

18. Variance The variance is computed as follows: The variance is the average of the squared differences between each data value and the mean. for a sample for a population

19. Standard Deviation The standard deviation is computed as follows: for a sample for a population

20. Excel’s Variance and Standard Deviation Functions Excel’s Sample Variance function =VAR.S(data cell range) Excel’s Sample Standard Deviation function =STDEV.S(data cell range)

21. Coefficient of Variation The coefficient of variation is computed as follows: The coefficient of variation indicates how large the standard deviation is in relation to the mean. for a sample for a population

22. Sample Variance, Standard Deviation, and Coefficient of Variation Standard deviation is about 9% of the mean Standard deviation is about 9% of the mean Variance Standard Deviation Coefficient of Variation Example: Apartment Rents

23. Using Excel to Compute the Sample Variance, Standard Deviation, and Coefficient of Variation Formula Worksheet Note: Rows 8-71 are not shown. ABCDE 1 Apart- ment Monthly Rent ($) 21545Mean=AVERAGE(B2:B71) 32715Median=MEDIAN(B2:B71) 43530Mode=MODE.SNGL(B2:B71) 54690Variance=VAR.S(B2:B71) 65535Std. Dev.=STDEV.S(B2:B71) 76700C.V.=E6/E2*100

24. Using Excel to Compute the Sample Variance, Standard Deviation, and Coefficient of Variation Value Worksheet Note: Rows 8-71 are not shown. ABCDE 1 Apart- ment Monthly Rent ($) 21545Mean590.80 32715Median575.00 43530Mode550.00 54690Variance2996.16 65535Std. Dev.54.74 76700C.V.9.27

25. End of Chapter 3