1. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Chapter Numerically Summarizing Data 3

2. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Section Measures of Central Tendency 3.1

3. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-3 Objectives 1.Determine the arithmetic mean of a variable from raw data 2.Determine the median of a variable from raw data 3.Explain what it means for a statistic to be resistant 4.Determine the mode of a variable from raw data

4. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-4 Objective 1 Determine the Arithmetic Mean of a Variable from Raw Data

5. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-5 The arithmetic mean of a variable is computed by adding all the values of the variable in the data set and dividing by the number of observations.

6. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-6 The population arithmetic mean, μ (pronounced “mew”), is computed using all the individuals in a population. The population mean is a parameter.

7. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-7 The sample arithmetic mean, (pronounced “x-bar”), is computed using sample data. The sample mean is a statistic.

8. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-8 If x 1, x 2, …, x N are the N observations of a variable from a population, then the population mean, µ, is

9. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-9 If x 1, x 2, …, x n are the n observations of a variable from a sample, then the sample mean,, is

10. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-10 EXAMPLEComputing a Population Mean and a Sample Mean The following data represent the travel times (in minutes) to work for all seven employees of a start-up web development company. 23, 36, 23, 18, 5, 26, 43 (a)Compute the population mean of this data. (b)Then take a simple random sample of n = 3 employees. Compute the sample mean. Obtain a second simple random sample of n = 3 employees. Again compute the sample mean.

11. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-11 EXAMPLEComputing a Population Mean and a Sample Mean (a)

12. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-12 EXAMPLEComputing a Population Mean and a Sample Mean (b) Obtain a simple random sample of size n = 3 from the population of seven employees. Use this simple random sample to determine a sample mean. Find a second simple random sample and determine the sample mean. 1 2 3 4 5 6 7 23, 36, 23, 18, 5, 26, 43

13. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-13

14. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-14 Objective 2 Determine the Median of a Variable from Raw Data

15. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-15 The median of a variable is the value that lies in the middle of the data when arranged in ascending order. We use M to represent the median.

16. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-16 Steps in Finding the Median of a Data Set Step 1Arrange the data in ascending order. Step 2Determine the number of observations, n. Step 3Determine the observation in the middle of the data set.

17. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-17 Steps in Finding the Median of a Data Set If the number of observations is odd, then the median is the data value exactly in the middle of the data set. That is, the median is the observation that lies in then (n + 1)/2 position. If the number of observations is even, then the median is the mean of the two middle observations in the data set. That is, the median is the mean of the observations that lie in the n/2 position and the n/2 + 1 position.

18. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-18 EXAMPLEComputing a Median of a Data Set with an Odd Number of Observations The following data represent the travel times (in minutes) to work for all seven employees of a start-up web development company. 23, 36, 23, 18, 5, 26, 43 Determine the median of this data. Step 1: 5, 18, 23, 23, 26, 36, 43 Step 2: There are n = 7 observations. Step 3: M = 23 5, 18, 23, 23, 26, 36, 43

19. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-19 EXAMPLEComputing a Median of a Data Set with an Even Number of Observations Suppose the start-up company hires a new employee. The travel time of the new employee is 70 minutes. Determine the median of the “new” data set. 23, 36, 23, 18, 5, 26, 43, 70 Step 1: 5, 18, 23, 23, 26, 36, 43, 70 Step 2: There are n = 8 observations. Step 3: 5, 18, 23, 23, 26, 36, 43, 70

20. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-20 Objective 3 Explain What it Means for a Statistic to Be Resistant

21. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-21 EXAMPLEComputing a Median of a Data Set with an Even Number of Observations The following data represent the travel times (in minutes) to work for all seven employees of a start-up web development company. 23, 36, 23, 18, 5, 26, 43 Suppose a new employee is hired who has a 130 minute commute. How does this impact the value of the mean and median? Mean before new hire: 24.9 minutes Median before new hire: 23 minutes Mean after new hire: 38 minutes Median after new hire: 24.5 minutes

22. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-22 A numerical summary of data is said to be resistant if extreme values (very large or small) relative to the data do not affect its value substantially.

23. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-23

24. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-24 EXAMPLE Describing the Shape of the Distribution The following data represent the asking price of homes for sale in Lincoln, NE. Source: http://www.homeseekers.com 79,995128,950149,900189,900 99,899130,950151,350203,950 105,200131,800154,900217,500 111,000132,300159,900260,000 120,000134,950163,300284,900 121,700135,500165,000299,900 125,950138,500174,850309,900 126,900147,500180,000349,900

25. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-25 Find the mean and median. Use the mean and median to identify the shape of the distribution. Verify your result by drawing a histogram of the data.

26. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-26 Find the mean and median. Use the mean and median to identify the shape of the distribution. Verify your result by drawing a histogram of the data. The mean asking price is $168,320 and the median asking price is $148,700. Therefore, we would conjecture that the distribution is skewed right.

27. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-27

28. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-28 Objective 4 Determine the Mode of a Variable from Raw Data

29. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-29 The mode of a variable is the most frequent observation of the variable that occurs in the data set. A set of data can have no mode, one mode, or more than one mode. If no observation occurs more than once, we say the data have no mode.

30. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-30 EXAMPLE Finding the Mode of a Data Set The data on the next slide represent the Vice Presidents of the United States and their state of birth. Find the mode.

31. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-31 Joe Biden Pennsylvani a

32. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-32

33. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-33 The mode is New York.

34. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-34 Tally data to determine most frequent observation

35. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Section Measures of Dispersion 3.2

36. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-36 Objectives 1.Determine the range of a variable from raw data 2.Determine the standard deviation of a variable from raw data 3.Determine the variance of a variable from raw data 4.Use the Empirical Rule to describe data that are bell shaped 5.Use Chebyshev’s Inequality to describe any data set

37. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-37 To order food at a McDonald’s restaurant, one must choose from multiple lines, while at Wendy’s Restaurant, one enters a single line. The following data represent the wait time (in minutes) in line for a simple random sample of 30 customers at each restaurant during the lunch hour. For each sample, answer the following: (a) What was the mean wait time? (b) Draw a histogram of each restaurant’s wait time. (c ) Which restaurant’s wait time appears more dispersed? Which line would you prefer to wait in? Why?

38. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-38 1.500.791.011.660.940.67 2.531.201.460.890.950.90 1.882.941.401.331.200.84 3.991.901.001.540.990.35 0.901.230.921.091.722.00 3.500.000.380.431.823.04 0.000.260.140.602.332.54 1.970.712.224.540.800.50 0.000.280.441.380.921.17 3.082.750.363.102.190.23 Wait Time at Wendy’s Wait Time at McDonald’s

39. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-39 (a) The mean wait time in each line is 1.39 minutes.

40. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-40 (b)

41. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-41 Objective 1 Determine the Range of a Variable from Raw Data

42. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-42 The range, R, of a variable is the difference between the largest data value and the smallest data values. That is, Range = R = Largest Data Value – Smallest Data Value

43. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-43 EXAMPLEFinding the Range of a Set of Data The following data represent the travel times (in minutes) to work for all seven employees of a start-up web development company. 23, 36, 23, 18, 5, 26, 43 Find the range. Range = 43 – 5 = 38 minutes

44. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-44 Objective 2 Determine the Standard Deviation of a Variable from Raw Data

45. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-45 The population standard deviation of a variable is the square root of the sum of squared deviations about the population mean divided by the number of observations in the population, N. That is, it is the square root of the mean of the squared deviations about the population mean. The population standard deviation is symbolically represented by σ (lowercase Greek sigma).

46. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-46 where x 1, x 2,..., x N are the N observations in the population and μ is the population mean.

47. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-47 A formula that is equivalent to the one on the previous slide, called the computational formula, for determining the population standard deviation is

48. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-48 EXAMPLE Computing a Population Standard Deviation The following data represent the travel times (in minutes) to work for all seven employees of a start- up web development company. 23, 36, 23, 18, 5, 26, 43 Compute the population standard deviation of this data.

49. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-49 xixi μ x i – μ(x i – μ) 2 2324.85714-1.857143.44898 3624.8571411.14286124.1633 2324.85714-1.857143.44898 1824.85714-6.8571447.02041 524.85714-19.8571394.3061 2624.857141.1428571.306122 4324.8571418.14286329.1633 902.8571

50. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-50 xixi (x i ) 2 23529 361296 23529 18324 525 26676 431849 Σ x i = 174Σ (x i ) 2 = 5228 Using the computational formula, yields the same result.

51. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-51 The sample standard deviation, s, of a variable is the square root of the sum of squared deviations about the sample mean divided by n – 1, where n is the sample size. where x 1, x 2,..., x n are the n observations in the sample and is the sample mean.

52. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-52 A formula that is equivalent to the one on the previous slide, called the computational formula, for determining the sample standard deviation is

53. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-53 We call n - 1 the degrees of freedom because the first n - 1 observations have freedom to be whatever value they wish, but the n th value has no freedom. It must be whatever value forces the sum of the deviations about the mean to equal zero.

54. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-54 EXAMPLEComputing a Sample Standard Deviation Here are the results of a random sample taken from the travel times (in minutes) to work for all seven employees of a start-up web development company: 5, 26, 36 Find the sample standard deviation.

55. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-55 xixi 522.33333-17.333300.432889 2622.333333.66713.446889 3622.3333313.667186.786889 500.66667

56. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-56 xixi (x i ) 2 525 26676 361296 Σ x i = 67Σ (x i ) 2 = 1997 Using the computational formula, yields the same result.

57. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-57

58. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-58 EXAMPLE Comparing Standard Deviations Determine the standard deviation waiting time for Wendy’s and McDonald’s. Which is larger? Why?

59. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-59 1.500.791.011.660.940.67 2.531.201.460.890.950.90 1.882.941.401.331.200.84 3.991.901.001.540.990.35 0.901.230.921.091.722.00 3.500.000.380.431.823.04 0.000.260.140.602.332.54 1.970.712.224.540.800.50 0.000.280.441.380.921.17 3.082.750.363.102.190.23 Wait Time at Wendy’s Wait Time at McDonald’s

60. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-60 EXAMPLE Comparing Standard Deviations Sample standard deviation for Wendy’s: 0.738 minutes Sample standard deviation for McDonald’s: 1.265 minutes Recall from earlier that the data is more dispersed for McDonald’s resulting in a larger standard deviation.

61. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-61 Objective 3 Determine the Variance of a Variable from Raw Data

62. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-62 The variance of a variable is the square of the standard deviation. The population variance is σ 2 and the sample variance is s 2.

63. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-63 EXAMPLE Computing a Population Variance The following data represent the travel times (in minutes) to work for all seven employees of a start-up web development company. 23, 36, 23, 18, 5, 26, 43 Compute the population and sample variance of this data.

64. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-64 EXAMPLE Computing a Population Variance Recall that the population standard deviation (from slide #49) is σ = 11.36 so the population variance is σ 2 = 129.05 minutes and that the sample standard deviation (from slide #55) is s = 15.82, so the sample variance is s 2 = 250.27 minutes

65. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-65 Objective 4 Use the Empirical Rule to Describe Data That Are Bell Shaped

66. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-66 The Empirical Rule If a distribution is roughly bell shaped, then Approximately 68% of the data will lie within 1 standard deviation of the mean. That is, approximately 68% of the data lie between μ – 1σ and μ + 1σ. Approximately 95% of the data will lie within 2 standard deviations of the mean. That is, approximately 95% of the data lie between μ – 2σ and μ + 2σ.

67. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-67 The Empirical Rule If a distribution is roughly bell shaped, then Approximately 99.7% of the data will lie within 3 standard deviations of the mean. That is, approximately 99.7% of the data lie between μ – 3σ and μ + 3σ. Note: We can also use the Empirical Rule based on sample data with used in place of μ and s used in place of σ.

68. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-68

69. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-69 EXAMPLE Using the Empirical Rule The following data represent the serum HDL cholesterol of the 54 female patients of a family doctor. 414843383537444444 627577588239855554 676969706572747474 606060616263646464 545455565656575859 454747484850525253

70. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-70 (a) Compute the population mean and standard deviation. (b) Draw a histogram to verify the data is bell-shaped. (c) Determine the percentage of all patients that have serum HDL within 3 standard deviations of the mean according to the Empirical Rule. (d) Determine the percentage of all patients that have serum HDL between 34 and 69.1 according to the Empirical Rule. (e) Determine the actual percentage of patients that have serum HDL between 34 and 69.1.

71. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-71 (a) Using a TI-83 plus graphing calculator, we find (b)

72. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-72 22.3 34.0 45.7 57.4 69.1 80.8 92.5 (e) 45 out of the 54 or 83.3% of the patients have a serum HDL between 34.0 and 69.1. (c) According to the Empirical Rule, 99.7% of the all patients that have serum HDL within 3 standard deviations of the mean. (d) 13.5% + 34% + 34% = 81.5% of all patients will have a serum HDL between 34.0 and 69.1 according to the Empirical Rule.

73. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-73 Objective 5 Use Chebyshev’s Inequality to Describe Any Set of Data

74. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-74 Chebyshev’s Inequality For any data set or distribution, at least of the observations lie within k Note: We can also use Chebyshev’s Inequality based on sample data. standard deviations of the mean, where k is any number greater than 1. That is, at least of the data lie between μ – kσ and μ + kσ for k > 1.

75. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-75 EXAMPLE Using Chebyshev’s Theorem Using the data from the previous example, use Chebyshev’s Theorem to (a) determine the percentage of patients that have serum HDL within 3 standard deviations of the mean. (b) determine the actual percentage of patients that have serum HDL between 34 and 80.8 (within 3 SD of mean). 52/54 ≈ 0.96 ≈ 96%

76. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Section Measures of Central Tendency and Dispersion from Grouped Data 3.3

77. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-77 Objectives 1.Approximate the mean of a variable from grouped data 2.Compute the weighted mean 3.Approximate the standard deviation of a variable from grouped data

78. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-78 Objective 1 Approximate the Mean of a Variable from Grouped Data

79. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-79 We have discussed how to compute descriptive statistics from raw data, but often the only available data have already been summarized in frequency distributions (grouped data). Although we cannot find exact values of the mean or standard deviation without raw data, we can approximate these measures using the techniques discussed in this section.

80. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-80 Approximate the Mean of a Variable from a Frequency Distribution Sample MeanPopulation Mean where x i is the midpoint or value of the i th class f i is the frequency of the i th class n is the number of classes

81. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-81 Hours01-56-1011-1516-2021-2526-3031-35 Frequency01302502301801006050 The National Survey of Student Engagement is a survey that (among other things) asked first year students at liberal arts colleges how much time they spend preparing for class each week. The results from the 2007 survey are summarized below. Approximate the mean number of hours spent preparing for class each week. Source:http://nsse.iub.edu/NSSE_2007_Annual_Report/docs/withhold/NSSE_2007_Annual_Report.pdf EXAMPLEApproximating the Mean from a Relative Frequency Distribution

82. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-82 TimeFrequencyxixi x i f i 0000 1 - 51303390 6 - 1025082000 11 - 15230132990 16 - 20180183240 21 - 25100232300 26 – 3060281680 31 – 3550331650

83. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-83 Objective 2 Compute the Weighted Mean

84. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-84 The weighted mean,, of a variable is found by multiplying each value of the variable by its corresponding weight, adding these products, and dividing this sum by the sum of the weights. It can be expressed using the formula where w is the weight of the i th observation x i is the value of the i th observation

85. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-85 EXAMPLE Computed a Weighted Mean Bob goes to the “Buy the Weigh” Nut store and creates his own bridge mix. He combines 1 pound of raisins, 2 pounds of chocolate covered peanuts, and 1.5 pounds of cashews. The raisins cost $1.25 per pound, the chocolate covered peanuts cost $3.25 per pound, and the cashews cost $5.40 per pound. What is the cost per pound of this mix.

86. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-86 Objective 3 Approximate the Standard Deviation of a Variable from Grouped Data

87. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-87 Approximate the Standard Deviation of a Variable from a Frequency Distribution Sample Standard Deviation Population Standard Deviation where x i is the midpoint or value of the i th class f i is the frequency of the i th class

88. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-88 An algebraically equivalent formula for the population standard deviation is

89. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-89 Hours01-56-1011-1516-2021-2526-3031-35 Frequency01302502301801006050 The National Survey of Student Engagement is a survey that (among other things) asked first year students at liberal arts colleges how much time they spend preparing for class each week. The results from the 2007 survey are summarized below. Approximate the standard deviation number of hours spent preparing for class each week. Source:http://nsse.iub.edu/NSSE_2007_Annual_Report/docs/withhold/NSSE_2007_Annual_Report.pdf EXAMPLEApproximating the Mean from a Relative Frequency Distribution

90. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-90 Time Frequ encyxixi 00000 1 - 51303–11.2516,453.125 6 - 102508–6.259765.625 11 - 1523013–1.25359.375 16 - 20180183.752531.25 21 - 25100238.757656.25 26 – 30602813.7511,343.75 31 – 35503318.7517,578.125

91. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Section Measures of Position and Outliers 3.4

92. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-92 Objectives 1.Determine and interpret z-scores 2.Interpret percentiles 3.Determine and interpret quartiles 4.Determine and interpret the interquartile range 5.Check a set of data for outliers

93. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-93 Objective 1 Determine and Interpret z-scores

94. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-94 The z-score represents the distance that a data value is from the mean in terms of the number of standard deviations. We find it by subtracting the mean from the data value and dividing this result by the standard deviation. There is both a population z-score and a sample z-score: Sample z-scorePopulation z-score The z-score is unitless. It has mean 0 and standard deviation 1.

95. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-95 EXAMPLE Using Z-Scores The mean height of males 20 years or older is 69.1 inches with a standard deviation of 2.8 inches. The mean height of females 20 years or older is 63.7 inches with a standard deviation of 2.7 inches. Data is based on information obtained from National Health and Examination Survey. Who is relatively taller? Kevin Garnett whose height is 83 inches or Candace Parker whose height is 76 inches

96. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-96 Kevin Garnett’s height is 4.96 standard deviations above the mean. Candace Parker’s height is 4.56 standard deviations above the mean. Kevin Garnett is relatively taller.

97. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-97 Objective 2 Interpret Percentiles

98. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-98 The kth percentile, denoted, P k, of a set of data is a value such that k percent of the observations are less than or equal to the value.

99. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-99 EXAMPLE Interpret a Percentile The Graduate Record Examination (GRE) is a test required for admission to many U.S. graduate schools. The University of Pittsburgh Graduate School of Public Health requires a GRE score no less than the 70th percentile for admission into their Human Genetics MPH or MS program. (Source: http://www.publichealth.pitt.edu/interior.php?pageID=1 01.) Interpret this admissions requirement.

100. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-100 EXAMPLE Interpret a Percentile In general, the 70 th percentile is the score such that 70% of the individuals who took the exam scored worse, and 30% of the individuals scores better. In order to be admitted to this program, an applicant must score as high or higher than 70% of the people who take the GRE. Put another way, the individual’s score must be in the top 30%.

101. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-101 Objective 3 Determine and Interpret Quartiles

102. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-102 Quartiles divide data sets into fourths, or four equal parts. The 1 st quartile, denoted Q 1, divides the bottom 25% the data from the top 75%. Therefore, the 1 st quartile is equivalent to the 25 th percentile. The 2 nd quartile divides the bottom 50% of the data from the top 50% of the data, so that the 2 nd quartile is equivalent to the 50 th percentile, which is equivalent to the median. The 3 rd quartile divides the bottom 75% of the data from the top 25% of the data, so that the 3 rd quartile is equivalent to the 75 th percentile.

103. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-103 Finding Quartiles Step 1Arrange the data in ascending order. Step 2Determine the median, M, or second quartile, Q 2. Step 3Divide the data set into halves: the observations below (to the left of) M and the observations above M. The first quartile, Q 1, is the median of the bottom half, and the third quartile, Q 3, is the median of the top half.

104. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-104 A group of Brigham Young University—Idaho students (Matthew Herring, Nathan Spencer, Mark Walker, and Mark Steiner) collected data on the speed of vehicles traveling through a construction zone on a state highway, where the posted speed was 25 mph. The recorded speed of 14 randomly selected vehicles is given below: 20, 24, 27, 28, 29, 30, 32, 33, 34, 36, 38, 39, 40, 40 Find and interpret the quartiles for speed in the construction zone. EXAMPLE Finding and Interpreting Quartiles

105. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-105 EXAMPLE Finding and Interpreting Quartiles Step 1: The data is already in ascending order. Step 2: There are n = 14 observations, so the median, or second quartile, Q 2, is the mean of the 7 th and 8 th observations. Therefore, M = 32.5. Step 3: The median of the bottom half of the data is the first quartile, Q 1. 20, 24, 27, 28, 29, 30, 32 The median of these seven observations is 28. Therefore, Q 1 = 28. The median of the top half of the data is the third quartile, Q 3. Therefore, Q 3 = 38.

106. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-106 Interpretation: 25% of the speeds are less than or equal to the first quartile, 28 miles per hour, and 75% of the speeds are greater than 28 miles per hour. 50% of the speeds are less than or equal to the second quartile, 32.5 miles per hour, and 50% of the speeds are greater than 32.5 miles per hour. 75% of the speeds are less than or equal to the third quartile, 38 miles per hour, and 25% of the speeds are greater than 38 miles per hour.

107. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-107 Objective 4 Determine and Interpret the Interquartile Range

108. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-108 The interquartile range, IQR, is the range of the middle 50% of the observations in a data set. That is, the IQR is the difference between the third and first quartiles and is found using the formula IQR = Q 3 – Q 1

109. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-109 EXAMPLE Determining and Interpreting the Interquartile Range Determine and interpret the interquartile range of the speed data. Q 1 = 28 Q 3 = 38 The range of the middle 50% of the speed of cars traveling through the construction zone is 10 miles per hour.

110. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-110 Suppose a 15 th car travels through the construction zone at 100 miles per hour. How does this value impact the mean, median, standard deviation, and interquartile range? Without 15 th carWith 15 th car Mean32.1 mph36.7 mph Median32.5 mph33 mph Standard deviation6.2 mph18.5 mph IQR10 mph11 mph

111. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-111 Objective 5 Check a Set of Data for Outliers

112. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-112 Checking for Outliers by Using Quartiles Step 1Determine the first and third quartiles of the data. Step 2Compute the interquartile range. Step 3Determine the fences. Fences serve as cutoff points for determining outliers. Lower Fence = Q 1 – 1.5(IQR) Upper Fence = Q 3 + 1.5(IQR) Step 4If a data value is less than the lower fence or greater than the upper fence, it is considered an outlier.

113. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-113 EXAMPLE Determining and Interpreting the Interquartile Range Check the speed data for outliers. Step 1: The first and third quartiles are Q 1 = 28 mph and Q 3 = 38 mph. Step 2: The interquartile range is 10 mph. Step 3: The fences are Lower Fence = Q 1 – 1.5(IQR) = 28 – 1.5(10) = 13 mph Upper Fence = Q 3 + 1.5(IQR) = 38 + 1.5(10) = 53 mph Step 4: There are no values less than 13 mph or greater than 53 mph. Therefore, there are no outliers.

114. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Section The Five-Number Summary and Boxplots 3.5

115. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-115 Objectives 1.Compute the five-number summary 2.Draw and interpret boxplots

116. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-116 Objective 1 Compute the Five-Number Summary

117. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-117 The five-number summary of a set of data consists of the smallest data value, Q1, the median, Q3, and the largest data value. We organize the five-number summary as follows:

118. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-118 EXAMPLEObtaining the Five-Number Summary Every six months, the United States Federal Reserve Board conducts a survey of credit card plans in the U.S. The following data are the interest rates charged by 10 credit card issuers randomly selected for the July 2005 survey. Determine the five-number summary of the data.

119. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-119 EXAMPLEObtaining the Five-Number Summary InstitutionRate Pulaski Bank and Trust Company6.5% Rainier Pacific Savings Bank12.0% Wells Fargo Bank NA14.4% Firstbank of Colorado14.4% Lafayette Ambassador Bank14.3% Infibank13.0% United Bank, Inc.13.3% First National Bank of The Mid-Cities13.9% Bank of Louisiana9.9% Bar Harbor Bank and Trust Company14.5% Source: http://www.federalreserve.gov/pubs/SHOP/survey.htm

120. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-120 EXAMPLEObtaining the Five-Number Summary First, we write the data in ascending order: 6.5%, 9.9%, 12.0%, 13.0%, 13.3%, 13.9%, 14.3%, 14.4%, 14.4%, 14.5% The smallest number is 6.5%. The largest number is 14.5%. The first quartile is 12.0%. The second quartile is 13.6%. The third quartile is 14.4%. Five-number Summary: 6.5% 12.0% 13.6% 14.4% 14.5%

121. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-121 Objective 2 Draw and Interpret Boxplots

122. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-122 Drawing a Boxplot Step 1Determine the lower and upper fences. Lower Fence = Q 1 – 1.5(IQR) Upper Fence = Q 3 + 1.5(IQR) where IQR = Q 3 – Q 1 Step 2Draw a number line long enough to include the maximum and minimum values. Insert vertical lines at Q 1, M, and Q 3. Enclose these vertical lines in a box. Step 3Label the lower and upper fences.

123. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-123 Drawing a Boxplot Step 4Draw a line from Q 1 to the smallest data value that is larger than the lower fence. Draw a line from Q 3 to the largest data value that is smaller than the upper fence. These lines are called whiskers. Step 5Any data values less than the lower fence or greater than the upper fence are outliers and are marked with an asterisk (*).

124. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-124 EXAMPLEObtaining the Five-Number Summary Every six months, the United States Federal Reserve Board conducts a survey of credit card plans in the U.S. The following data are the interest rates charged by 10 credit card issuers randomly selected for the July 2005 survey. Construct a boxplot of the data.

125. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-125 EXAMPLEObtaining the Five-Number Summary InstitutionRate Pulaski Bank and Trust Company6.5% Rainier Pacific Savings Bank12.0% Wells Fargo Bank NA14.4% Firstbank of Colorado14.4% Lafayette Ambassador Bank14.3% Infibank13.0% United Bank, Inc.13.3% First National Bank of The Mid-Cities13.9% Bank of Louisiana9.9% Bar Harbor Bank and Trust Company14.5% Source: http://www.federalreserve.gov/pubs/SHOP/survey.htm

126. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-126 Step 1: The interquartile range (IQR) is 14.4% - 12% = 2.4%. The lower and upper fences are: Lower Fence = Q 1 – 1.5(IQR) = 12 – 1.5(2.4) = 8.4% Upper Fence = Q 3 + 1.5(IQR) = 14.4 + 1.5(2.4) = 18.0% Step 2: [ ] *

127. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 3-127 The interest rate boxplot indicates that the distribution is skewed left. Use a boxplot and quartiles to describe the shape of a distribution.