1. 1 1 Slide STATISTICS FOR BUSINESS AND ECONOMICS Seventh Edition AndersonSweeneyWilliams Slides Prepared by John Loucks © 1999 ITP/South-Western College Publishing

2. 2 2 Slide Chapter 3, Part 1 Descriptive Statistics II: Numerical Methods n Measures of Location n Measures of Variability   x s

3. 3 3 Slide Measures of Location n Mean n Median n Mode n Percentiles n Quartiles and Hinges

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 Trimmed Mean n The trimmed mean is the mean of the data after percent of the smallest and percent of the largest values are removed from the data set. n It is a good measure of central location when extremely small and/or large data values are present in the data set.

8. 8 8 Slide Example: Apartment Rents n Trimmed Mean With n = 70, a 5% trimmed mean removes.05(70) = 3.5 = 4 values from each end of the set..05(70) = 3.5 = 4 values from each end of the set. 5% trimmed mean = 5% trimmed mean =

9. 9 9 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.

10. 10 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

11. 11 Slide Mode n The mode of a data set is the value that occurs with greatest frequency.

12. 12 Slide Example: Apartment Rents n Mode 450 occurred most frequently (7 times) 450 occurred most frequently (7 times) Mode = 450 Mode = 450

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 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: 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 Hinges n Hinges are similar to quartiles. n The lower hinge is approximately the first quartile. n The upper hinge is approximately the third quartile. n Hinge and quartile values might differ slightly due to differing computational conventions.

18. 18 Slide Measures of Variability n Range n Inter-quartile Range n Variance n Standard Deviation n Coefficient of Variation

19. 19 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 dispersion. n It is very sensitive to the smallest and largest data values.

20. 20 Slide Example: Apartment Rents n Range Range = largest value - smallest value Range = largest value - smallest value Range = 615 - 425 = 190 Range = 615 - 425 = 190

21. 21 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.

22. 22 Slide Example: Apartment Rents n Interquartile Range 3rd Quartile ( Q 3) = 525 3rd Quartile ( Q 3) = 525 1st Quartile ( Q 1) = 450 1st Quartile ( Q 1) = 450 Interquartile Range = Q 3 - Q 1 = 525 - 450 = 75 Interquartile Range = Q 3 - Q 1 = 525 - 450 = 75

23. 23 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.

24. 24 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 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).

25. 25 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:

26. 26 Slide Example: Apartment Rents n Variance n Standard Deviation n Coefficient of Variation

27. 27 Slide The End of Chapter 3, Part 1