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

2. 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 3 Slide © 2003 South-Western/Thomson Learning TM Measures of Location n Mean n Median n Mode n Percentiles n Quartiles

4. 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 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 6 Slide © 2003 South-Western/Thomson Learning TM Example: Apartment Rents n Mean

7. 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 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 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 = (475 + 475)/2 = 475

10. 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. 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. 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. 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. 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. 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. 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 = (580 + 590)/2 = 585 90th Percentile = (580 + 590)/2 = 585

17. 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. 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. 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. 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. 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. 22 Slide © 2003 South-Western/Thomson Learning TM n Formula Worksheet for 90 th Percentile’s Index Using Excel to Compute Percentiles and Quartiles Note: Rows 7-71 are not shown.

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

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

25. 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. 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. 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. 28 Slide © 2003 South-Western/Thomson Learning TM Example: Apartment Rents n Range Range = largest value - smallest value Range = largest value - smallest value Range = 615 - 425 = 190 Range = 615 - 425 = 190

29. 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. 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 = 525 - 445 = 80 Interquartile Range = Q 3 - Q 1 = 525 - 445 = 80

31. 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. 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. 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. 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. 35 Slide © 2003 South-Western/Thomson Learning TM Example: Apartment Rents n Variance n Standard Deviation n Coefficient of Variation

36. 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. 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. 38 Slide © 2003 South-Western/Thomson Learning TM 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

39. 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 Excel’s Descriptive Statistics Tool

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

41. 41 Slide © 2003 South-Western/Thomson Learning TM n Value Worksheet (Partial) Using Excel’s Descriptive Statistics Tool Note: Rows 1-8 and 17-71 are not shown.

42. 42 Slide © 2003 South-Western/Thomson Learning TM Descriptive Statistics: Numerical Methods, Part B Descriptive Statistics: Numerical Methods, Part B n Measures of Relative Location and Detecting Outliers n Exploratory Data Analysis n Measures of Association Between Two Variables n The Weighted Mean and Working with Grouped Data Working with Grouped Data     % % x x

43. 43 Slide © 2003 South-Western/Thomson Learning TM Measures of Relative Location and Detecting Outliers n z-Scores n Chebyshev’s Theorem n Empirical Rule n Detecting Outliers

44. 44 Slide © 2003 South-Western/Thomson Learning TM z -Scores n The z -score is often called the standardized value. n It denotes the number of standard deviations a data value x i is from the mean. n A data value less than the sample mean will have a z -score less than zero. n A data value greater than the sample mean will have a z -score greater than zero. n A data value equal to the sample mean will have a z -score of zero.

45. 45 Slide © 2003 South-Western/Thomson Learning TM n z -Score of Smallest Value (425) Standardized Values for Apartment Rents Example: Apartment Rents

46. 46 Slide © 2003 South-Western/Thomson Learning TM Chebyshev’s Theorem At least (1 - 1/ k 2 ) of the items in any data set will be At least (1 - 1/ k 2 ) of the items in any data set will be within k standard deviations of the mean, where k is any value greater than 1. At least 75% of the items must be within At least 75% of the items must be within k = 2 standard deviations of the mean. At least 89% of the items must be within At least 89% of the items must be within k = 3 standard deviations of the mean. At least 94% of the items must be within At least 94% of the items must be within k = 4 standard deviations of the mean. At least (1 - 1/ k 2 ) of the items in any data set will be At least (1 - 1/ k 2 ) of the items in any data set will be within k standard deviations of the mean, where k is any value greater than 1. At least 75% of the items must be within At least 75% of the items must be within k = 2 standard deviations of the mean. At least 89% of the items must be within At least 89% of the items must be within k = 3 standard deviations of the mean. At least 94% of the items must be within At least 94% of the items must be within k = 4 standard deviations of the mean.

47. 47 Slide © 2003 South-Western/Thomson Learning TM Example: Apartment Rents n Chebyshev’s Theorem Let k = 1.5 with = 490.80 and s = 54.74 Let k = 1.5 with = 490.80 and s = 54.74 At least (1 - 1/(1.5) 2 ) = 1 - 0.44 = 0.56 or 56% of the rent values must be between of the rent values must be between - k ( s ) = 490.80 - 1.5(54.74) = 409 - k ( s ) = 490.80 - 1.5(54.74) = 409 and and + k ( s ) = 490.80 + 1.5(54.74) = 573 + k ( s ) = 490.80 + 1.5(54.74) = 573

48. 48 Slide © 2003 South-Western/Thomson Learning TM n Chebyshev’s Theorem (continued) Actually, 86% of the rent values Actually, 86% of the rent values are between 409 and 573. are between 409 and 573. Example: Apartment Rents

49. 49 Slide © 2003 South-Western/Thomson Learning TM Empirical Rule For data having a bell-shaped distribution: For data having a bell-shaped distribution: Approximately 68% of the data values will be within one standard deviation of the mean. Approximately 68% of the data values will be within one standard deviation of the mean.

50. 50 Slide © 2003 South-Western/Thomson Learning TM Empirical Rule For data having a bell-shaped distribution: Approximately 95% of the data values will be within two standard deviations of the mean. Approximately 95% of the data values will be within two standard deviations of the mean.

51. 51 Slide © 2003 South-Western/Thomson Learning TM Empirical Rule For data having a bell-shaped distribution: Almost all (99.7%) of the items will be within three standard deviations of the mean. Almost all (99.7%) of the items will be within three standard deviations of the mean.

52. 52 Slide © 2003 South-Western/Thomson Learning TM Example: Apartment Rents n Empirical Rule Interval % in Interval Interval % in Interval Within +/- 1 s 436.06 to 545.5448/70 = 69% Within +/- 2 s 381.32 to 600.2868/70 = 97% Within +/- 3 s 326.58 to 655.0270/70 = 100%

53. 53 Slide © 2003 South-Western/Thomson Learning TM Detecting Outliers n An outlier is an unusually small or unusually large value in a data set. n A data value with a z-score less than -3 or greater than +3 might be considered an outlier. n It might be an incorrectly recorded data value. n It might be a data value that was incorrectly included in the data set. n It might be a correctly recorded data value that belongs in the data set !

54. 54 Slide © 2003 South-Western/Thomson Learning TM Example: Apartment Rents n Detecting Outliers The most extreme z-scores are -1.20 and 2.27. Using | z | > 3 as the criterion for an outlier, there are no outliers in this data set. Standardized Values for Apartment Rents

55. 55 Slide © 2003 South-Western/Thomson Learning TM Exploratory Data Analysis n Five-Number Summary n Box Plot

56. 56 Slide © 2003 South-Western/Thomson Learning TM Five-Number Summary n Smallest Value n First Quartile n Median n Third Quartile n Largest Value

57. 57 Slide © 2003 South-Western/Thomson Learning TM Example: Apartment Rents n Five-Number Summary Lowest Value = 425 First Quartile = 450 Median = 475 Median = 475 Third Quartile = 525 Largest Value = 615

58. 58 Slide © 2003 South-Western/Thomson Learning TM Box Plot n A box is drawn with its ends located at the first and third quartiles. n A vertical line is drawn in the box at the location of the median. n Limits are located (not drawn) using the interquartile range (IQR). The lower limit is located 1.5(IQR) below Q 1. The lower limit is located 1.5(IQR) below Q 1. The upper limit is located 1.5(IQR) above Q 3. The upper limit is located 1.5(IQR) above Q 3. Data outside these limits are considered outliers. Data outside these limits are considered outliers. … continued

59. 59 Slide © 2003 South-Western/Thomson Learning TM Box Plot (Continued) n Whiskers (dashed lines) are drawn from the ends of the box to the smallest and largest data values inside the limits. n The locations of each outlier is shown with the symbol *.

60. 60 Slide © 2003 South-Western/Thomson Learning TM Example: Apartment Rents n Box Plot Lower Limit: Q1 - 1.5(IQR) = 450 - 1.5(75) = 337.5 Lower Limit: Q1 - 1.5(IQR) = 450 - 1.5(75) = 337.5 Upper Limit: Q3 + 1.5(IQR) = 525 + 1.5(75) = 637.5 Upper Limit: Q3 + 1.5(IQR) = 525 + 1.5(75) = 637.5 There are no outliers. 37 5 40 0 42 5 45 0 47 5 50 0 52 5 550 575 600 625

61. 61 Slide © 2003 South-Western/Thomson Learning TM Measures of Association between Two Variables n Covariance n Correlation Coefficient

62. 62 Slide © 2003 South-Western/Thomson Learning TM Covariance n The covariance is a measure of the linear association between two variables. n Positive values indicate a positive relationship. n Negative values indicate a negative relationship.

63. 63 Slide © 2003 South-Western/Thomson Learning TM n If the data sets are samples, the covariance is denoted by s xy. n If the data sets are populations, the covariance is denoted by. Covariance

64. 64 Slide © 2003 South-Western/Thomson Learning TM Correlation Coefficient n The coefficient can take on values between -1 and +1. n Values near -1 indicate a strong negative linear relationship. n Values near +1 indicate a strong positive linear relationship. n If the data sets are samples, the coefficient is r xy. n If the data sets are populations, the coefficient is.

65. 65 Slide © 2003 South-Western/Thomson Learning TM Using Excel to Compute the Covariance and Correlation Coefficient n Formula Worksheet

66. 66 Slide © 2003 South-Western/Thomson Learning TM n Value Worksheet Using Excel to Compute the Covariance and Correlation Coefficient

67. 67 Slide © 2003 South-Western/Thomson Learning TM The Weighted Mean and Working with Grouped Data n Weighted Mean n Mean for Grouped Data n Variance for Grouped Data n Standard Deviation for Grouped Data

68. 68 Slide © 2003 South-Western/Thomson Learning TM Weighted Mean n When the mean is computed by giving each data value a weight that reflects its importance, it is referred to as a weighted mean. n In the computation of a grade point average (GPA), the weights are the number of credit hours earned for each grade. n When data values vary in importance, the analyst must choose the weight that best reflects the importance of each value.

69. 69 Slide © 2003 South-Western/Thomson Learning TM Weighted Mean x =  w i x i x =  w i x i  w i  w iwhere: x i = value of observation i x i = value of observation i w i = weight for observation i w i = weight for observation i

70. 70 Slide © 2003 South-Western/Thomson Learning TM Grouped Data n The weighted mean computation can be used to obtain approximations of the mean, variance, and standard deviation for the grouped data. n To compute the weighted mean, we treat the midpoint of each class as though it were the mean of all items in the class. n We compute a weighted mean of the class midpoints using the class frequencies as weights. n Similarly, in computing the variance and standard deviation, the class frequencies are used as weights.

71. 71 Slide © 2003 South-Western/Thomson Learning TM n Sample Data n Population Data where: f i = frequency of class i f i = frequency of class i M i = midpoint of class i M i = midpoint of class i Mean for Grouped Data

72. 72 Slide © 2003 South-Western/Thomson Learning TM Example: Apartment Rents Given below is the previous sample of monthly rents for one-bedroom apartments presented here as grouped data in the form of a frequency distribution.

73. 73 Slide © 2003 South-Western/Thomson Learning TM Example: Apartment Rents n Mean for Grouped Data This approximation differs by $2.41 from This approximation differs by $2.41 from the actual sample mean of $490.80. the actual sample mean of $490.80.

74. 74 Slide © 2003 South-Western/Thomson Learning TM Variance for Grouped Data n Sample Data n Population Data

75. 75 Slide © 2003 South-Western/Thomson Learning TM Example: Apartment Rents n Variance for Grouped Data n Standard Deviation for Grouped Data This approximation differs by only $.20 from the actual standard deviation of $54.74.

76. 76 Slide © 2003 South-Western/Thomson Learning TM End of Chapter 3