Descriptive Statistics: Numerical Measures Location and Variability

1. Descriptive Statistics: Numerical MeasuresLocation and Variability Chapter 3 BA 201

2. Sample Statistics, Population Parameters,and Point Estimators 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.

3. Location

4. Measures of Location • Mean • Median • Mode • Percentiles • Quartiles

5. The sample mean is the point estimator of the population mean m. Mean • The mean of a data set is the average of all the data values. • The mean provides a measure of central location.

6. Mean Sample Population

7. Sample Mean Apartment Rents

8. Trimmed Mean • Another measure, sometimes used when extreme values are present, is the trimmed mean. • It is obtained by deleting a percentage of the smallest and largest values from a data set and then computing the mean of the remaining values. • For example, the 5% trimmed mean is obtained by removing the smallest 5% and the largest 5% of the data values and then computing the mean of the remaining values.

9. Median • The median of a data set is the value in the middle • when the data items are arranged in ascending order. • Whenever a data set has extreme values, the median • is the preferred measure of central location.

10. Median • For an odd number of observations: 26 18 27 12 14 27 19 7 observations 27 12 14 18 19 26 27 in ascending order the median is the middle value. Median = 19

11. Median • For an even number of observations: 26 18 27 12 14 27 30 19 8 observations 27 30 12 14 18 19 26 27 in ascending order the median is the average of the middle two values. Median = (19 + 26)/2 = 22.5

12. Median • Example: Apartment Rents Averaging the 35th and 36th data values: Median = (475 + 475)/2 = 475

13. Mode • The mode of a data set is the value that occurs with • greatest frequency. • The greatest frequency can occur at two or more • different values. • If the data have exactly two modes, the data are • bimodal. • If the data have more than two modes, the data are • multimodal.

14. Mode Apartment Rents 450 occurred most frequently (7 times) Mode = 450

15. Practice

16. Practice #1 • Compute the … • Mean • Median • Mode

17. Practice #1 - Median

18. Practice #1 - Mode

19. Percentiles • A percentile provides information about how the • data are spread over the interval from the smallest • value to the largest value. • The pth 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.

20. Percentiles Arrange the data in ascending order. Compute index i, the position of the pth percentile. i = (p/100)n If i is not an integer, round up. The pth percentile is the value in the ith position. If i is an integer, the pth percentile is the average of the values in positions i and i+1.

21. 80th Percentile Apartment Rents i = (p/100)n = (80/100)70 = 56 Averaging the 56th and 57th data values: 80th Percentile = (535 + 549)/2 = 542

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

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

24. Practice

25. Practice #2 - Percentiles 80th Percentile

26. Variability

27. Measures of Variability • Range • Interquartile Range • Variance • Standard Deviation • Coefficient of Variation

28. Range • The range of a data set is the difference between the largest and smallest data values.

29. Range Apartment Rents Range = largest value - smallest value Range = 615 - 425 = 190

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

31. Interquartile Range Apartment Rents 3rd Quartile (Q3) = 525 1st Quartile (Q1) = 445 Interquartile Range = Q3 - Q1 = 525 - 445 = 80

32. PracticeRange

33. Practice #3 - Range Range

34. Variance

35. It is based on the difference between the value of each observation (xi) and the mean ( for a sample, m for a population). 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.

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

37. Sample Variance Apartment Rents • Variance

38. Detailed Example - Variance b d c e s2 = 4/(5-1) = 1 g f a

39. Standard Deviation The standard deviation of a data set is the positive square root of the variance. It is measured in the same units as the data, making it more easily interpreted than the variance.

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

41. Standard Deviation Apartment Rents • Standard Deviation

42. Detailed Example – Standard Deviation s2 = 4/(5-1) = 1 a

43. PracticeVariance and Standard Deviation

44. Practice #4 – Variance

45. Practice #4 – Standard Deviation

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

47. Sample Variance, Standard Deviation, And Coefficient of Variation Apartment Rents • Variance • Standard Deviation the standard deviation is about 11% of the mean • Coefficient of Variation

48. PracticeCoefficient of variation

49. Practice #5 – Coefficient of Variation s=