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1 Chapter Numerically Summarizing Data © 2010 Pearson Prentice Hall. All rights reserved 3 3

2 Section 3.1 Measures of Central Tendency 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 statistics to be resistant 4.Determine the mode of a variable from raw data 3-2© 2010 Pearson Prentice Hall. All rights reserved

3 Objective 1 Determine the arithmetic mean of a variable from raw data 3-3© 2010 Pearson Prentice Hall. All rights reserved

4 The arithmetic mean of a variable is computed by determining the sum of all the values of the variable in the data set divided by the number of observations. 3-4© 2010 Pearson Prentice Hall. All rights reserved

5 The population arithmetic mean is computed using all the individuals in a population. The population mean is a parameter. The population arithmetic mean is denoted by. 3-5© 2010 Pearson Prentice Hall. All rights reserved

6 The sample arithmetic mean is computed using sample data. The sample mean is a statistic. The sample arithmetic mean is denoted by. 3-6© 2010 Pearson Prentice Hall. All rights reserved

7 If x 1, x 2, …, x N are the N observations of a variable from a population, then the population mean, µ, is 3-7© 2010 Pearson Prentice Hall. All rights reserved

8 If x 1, x 2, …, x n are the n observations of a variable from a sample, then the sample mean,, is 3-8© 2010 Pearson Prentice Hall. All rights reserved

9 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. 3-9© 2010 Pearson Prentice Hall. All rights reserved

10 EXAMPLEComputing a Population Mean and a Sample Mean (a) 3-10© 2010 Pearson Prentice Hall. All rights reserved

11 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 , 36, 23, 18, 5, 26, © 2010 Pearson Prentice Hall. All rights reserved

12 3-12© 2010 Pearson Prentice Hall. All rights reserved

13 The lengths (in minutes) of a sample of cell phone calls are shown: Find the mean. A. 54 B. 9 C. 6 D. 7 Copyright © 2010 Pearson Education, Inc. Slide 3- 13

14 The lengths (in minutes) of a sample of cell phone calls are shown: Find the mean. A. 54 B. 9 C. 6 D. 7 Copyright © 2010 Pearson Education, Inc. Slide 3- 14

15 Objective 2 Determine the median of a variable from raw data 3-15© 2010 Pearson Prentice Hall. All rights reserved

16 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. 3-16© 2010 Pearson Prentice Hall. All rights reserved

17 3-17© 2010 Pearson Prentice Hall. All rights reserved

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, © 2010 Pearson Prentice Hall. All rights reserved

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, © 2010 Pearson Prentice Hall. All rights reserved

20 The lengths (in minutes) of a sample of cell phone calls are shown: Find the median. A. 3.5 B. 9 C. 6 D. 7 Copyright © 2010 Pearson Education, Inc. Slide 3- 20

21 The lengths (in minutes) of a sample of cell phone calls are shown: Find the median. A. 3.5 B. 9 C. 6 D. 7 Copyright © 2010 Pearson Education, Inc. Slide 3- 21

22 Objective 3 Explain what it means for a statistic to be resistant 3-22© 2010 Pearson Prentice Hall. All rights reserved

23 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 3-23© 2010 Pearson Prentice Hall. All rights reserved

24 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. 3-24© 2010 Pearson Prentice Hall. All rights reserved

25 3-25© 2010 Pearson Prentice Hall. All rights reserved

26 EXAMPLE Describing the Shape of the Distribution The following data represent the asking price of homes for sale in Lincoln, NE. Source: 79,995128,950149,900189,900 99,899130,950151,350203, ,200131,800154,900217, ,000132,300159,900260, ,000134,950163,300284, ,700135,500165,000299, ,950138,500174,850309, ,900147,500180,000349, © 2010 Pearson Prentice Hall. All rights reserved

27 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. 3-27© 2010 Pearson Prentice Hall. All rights reserved

28 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. 3-28© 2010 Pearson Prentice Hall. All rights reserved

29 3-29© 2010 Pearson Prentice Hall. All rights reserved

30 Objective 4 Determine the mode of a variable from raw data 3-30© 2010 Pearson Prentice Hall. All rights reserved

31 The mode of a variable is the most frequent observation of the variable that occurs in the data set. If there is no observation that occurs with the most frequency, we say the data has no mode. 3-31© 2010 Pearson Prentice Hall. All rights reserved

32 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. 3-32© 2010 Pearson Prentice Hall. All rights reserved

33 3-33© 2010 Pearson Prentice Hall. All rights reserved

34 3-34© 2010 Pearson Prentice Hall. All rights reserved

35 The mode is New York. 3-35© 2010 Pearson Prentice Hall. All rights reserved

36 Tally data to determine most frequent observation 3-36© 2010 Pearson Prentice Hall. All rights reserved

37 The lengths (in minutes) of a sample of cell phone calls are shown: Find the mode. A. 3 B. 9 C. 6 D. 7 Copyright © 2010 Pearson Education, Inc. Slide 3- 37

38 The lengths (in minutes) of a sample of cell phone calls are shown: Find the mode. A. 3 B. 9 C. 6 D. 7 Copyright © 2010 Pearson Education, Inc. Slide 3- 38

39 Section 3.2 Measures of Dispersion Objectives 1.Compute the range of a variable from raw data 2.Compute the variance of a variable from raw data 3.Compute the standard deviation of a variable from raw data 4.Use the Empirical Rule to describe data that are bell shaped 5.Use Chebyshevs Inequality to describe any data set 3-39© 2010 Pearson Prentice Hall. All rights reserved

40 To order food at a McDonalds Restaurant, one must choose from multiple lines, while at Wendys 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 restaurants wait time. (c ) Which restaurants wait time appears more dispersed? Which line would you prefer to wait in? Why? 3-40© 2010 Pearson Prentice Hall. All rights reserved

41 Wait Time at Wendys Wait Time at McDonalds 3-41© 2010 Pearson Prentice Hall. All rights reserved

42 (a) The mean wait time in each line is 1.39 minutes. 3-42© 2010 Pearson Prentice Hall. All rights reserved

43 (b) 3-43© 2010 Pearson Prentice Hall. All rights reserved

44 Objective 1 Compute the range of a variable from raw data 3-44© 2010 Pearson Prentice Hall. All rights reserved

45 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 3-45© 2010 Pearson Prentice Hall. All rights reserved

46 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 3-46© 2010 Pearson Prentice Hall. All rights reserved

47 The lengths (in minutes) of a sample of cell phone calls are shown: Find the range. A. 2 B. 16 C. 13 D. 9 Copyright © 2010 Pearson Education, Inc. Slide 3- 47

48 The lengths (in minutes) of a sample of cell phone calls are shown: Find the range. A. 2 B. 16 C. 13 D. 9 Copyright © 2010 Pearson Education, Inc. Slide 3- 48

49 Objective 2 Compute the variance of a variable from raw data 3-49© 2010 Pearson Prentice Hall. All rights reserved

50 The population variance of a variable is the sum of squared deviations about the population mean divided by the number of observations in the population, N. That is it is the mean of the sum of the squared deviations about the population mean. 3-50© 2010 Pearson Prentice Hall. All rights reserved

51 The population variance is symbolically represented by σ 2 (lower case Greek sigma squared). Note: When using the above formula, do not round until the last computation. Use as many decimals as allowed by your calculator in order to avoid round off errors. 3-51© 2010 Pearson Prentice Hall. All rights reserved

52 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 variance of this data. Recall that 3-52© 2010 Pearson Prentice Hall. All rights reserved

53 xixi μ x i – μ(x i – μ) minutes © 2010 Pearson Prentice Hall. All rights reserved

54 The Computational Formula 3-54© 2010 Pearson Prentice Hall. All rights reserved

55 EXAMPLE Computing a Population Variance Using the Computational Formula 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 variance of this data using the computational formula. 3-55© 2010 Pearson Prentice Hall. All rights reserved

56 23, 36, 23, 18, 5, 26, © 2010 Pearson Prentice Hall. All rights reserved

57 The sample variance is computed by determining the sum of squared deviations about the sample mean and then dividing this result by n – © 2010 Pearson Prentice Hall. All rights reserved

58 Note: Whenever a statistic consistently overestimates or underestimates a parameter, it is called biased. To obtain an unbiased estimate of the population variance, we divide the sum of the squared deviations about the mean by n © 2010 Pearson Prentice Hall. All rights reserved

59 EXAMPLE Computing a Sample Variance In Section 3.1, we obtained the following simple random sample for the travel time data: 5, 36, 26. Compute the sample variance travel time. Travel Time, x i Sample Mean,Deviation about the Mean, Squared Deviations about the Mean, – = ( ) 2 = square minutes 3-59© 2010 Pearson Prentice Hall. All rights reserved

60 Objective 3 Compute the standard deviation of a variable from raw data 3-60© 2010 Pearson Prentice Hall. All rights reserved

61 The population standard deviation is denoted by It is obtained by taking the square root of the population variance, so that The sample standard deviation is denoted by s It is obtained by taking the square root of the sample variance, so that 3-61© 2010 Pearson Prentice Hall. All rights reserved

62 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. Recall, from the last objective that σ 2 = minutes 2. Therefore, 3-62© 2010 Pearson Prentice Hall. All rights reserved

63 EXAMPLEComputing a Sample Standard Deviation Recall the sample data 5, 26, 36 results in a sample variance of square minutes Use this result to determine the sample standard deviation. 3-63© 2010 Pearson Prentice Hall. All rights reserved

64 The lengths (in minutes) of a sample of cell phone calls are shown: Find the standard deviation. A. 5.7 B. 5.2 C D. 16 Copyright © 2010 Pearson Education, Inc. Slide 3- 64

65 The lengths (in minutes) of a sample of cell phone calls are shown: Find the standard deviation. A. 5.7 B. 5.2 C D. 16 Copyright © 2010 Pearson Education, Inc. Slide 3- 65

66 EXAMPLE Comparing Standard Deviations Determine the standard deviation waiting time for Wendys and McDonalds. Which is larger? Why? 3-66© 2010 Pearson Prentice Hall. All rights reserved

67 Wait Time at Wendys Wait Time at McDonalds 3-67© 2010 Pearson Prentice Hall. All rights reserved

68 EXAMPLE Comparing Standard Deviations Determine the standard deviation waiting time for Wendys and McDonalds. Which is larger? Why? Sample standard deviation for Wendys: minutes Sample standard deviation for McDonalds: minutes 3-68© 2010 Pearson Prentice Hall. All rights reserved

69 Objective 4 Use the Empirical Rule to Describe Data That Are Bell Shaped 3-69© 2010 Pearson Prentice Hall. All rights reserved

70 3-70© 2010 Pearson Prentice Hall. All rights reserved

71 3-71© 2010 Pearson Prentice Hall. All rights reserved

72 EXAMPLE Using the Empirical Rule The following data represent the serum HDL cholesterol of the 54 female patients of a family doctor © 2010 Pearson Prentice Hall. All rights reserved

73 (a) Compute the population mean and standard deviation. (b) Draw a histogram to verify the data is bell-shaped. (c) Determine the percentage of patients that have serum HDL within 3 standard deviations of the mean according to the Empirical Rule. (d) Determine the percentage of 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 © 2010 Pearson Prentice Hall. All rights reserved

74 (a) Using a TI-83 plus graphing calculator, we find (b) 3-74© 2010 Pearson Prentice Hall. All rights reserved

75 (e) 45 out of the 54 or 83.3% of the patients have a serum HDL between 34.0 and (c) According to the Empirical Rule, 99.7% of the patients that have serum HDL within 3 standard deviations of the mean. (d) 13.5% + 34% + 34% = 81.5% of patients will have a serum HDL between 34.0 and 69.1 according to the Empirical Rule. 3-75© 2010 Pearson Prentice Hall. All rights reserved

76 The mean commute time in the U.S. is 24.4 minutes with a standard deviation of 6.5 minutes. What is the minimum percentage of commuters that have commute times between 11.4 minutes and 37.4 minutes? A. 68% B. 75% C. 89% D. 95% Copyright © 2010 Pearson Education, Inc. Slide 3- 76

77 The mean commute time in the U.S. is 24.4 minutes with a standard deviation of 6.5 minutes. What is the minimum percentage of commuters that have commute times between 11.4 minutes and 37.4 minutes? A. 68% B. 75% C. 89% D. 95% Copyright © 2010 Pearson Education, Inc. Slide 3- 77

78 Objective 5 Use Chebyshevs Inequality to Describe Any Set of Data 3-78© 2010 Pearson Prentice Hall. All rights reserved

79 3-79© 2010 Pearson Prentice Hall. All rights reserved

80 EXAMPLE Using Chebyshevs Theorem Using the data from the previous example, use Chebyshevs 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 © 2010 Pearson Prentice Hall. All rights reserved

81 Section 3.3 Measures of Central Tendency and Dispersion from Grouped Data Objectives 1.Approximate the mean of a variable from grouped data 2.Compute the weighted mean 3.Approximate the variance and standard deviation of a variable from grouped data 3-81© 2010 Pearson Prentice Hall. All rights reserved

82 Objective 1 Approximate the Mean of a Variable from Grouped Data 3-82© 2010 Pearson Prentice Hall. All rights reserved

83 3-83© 2010 Pearson Prentice Hall. All rights reserved

84 Hours Frequency 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 3-84© 2010 Pearson Prentice Hall. All rights reserved

85 TimeFrequencyxixi x i f i – – © 2010 Pearson Prentice Hall. All rights reserved

86 Approximate the mean of the frequency distribution. A. 13 B C. 9.5 D ClassFrequency, f 1 – – – – Copyright © 2010 Pearson Education, Inc. Slide 3- 86

87 Approximate the mean of the frequency distribution. A. 13 B C. 9.5 D ClassFrequency, f 1 – – – – Copyright © 2010 Pearson Education, Inc. Slide 3- 87

88 Objective 2 Compute the Weighted Mean 3-88© 2010 Pearson Prentice Hall. All rights reserved

89 3-89© 2010 Pearson Prentice Hall. All rights reserved

90 EXAMPLE Computed a Weighted Mean Bob goes 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. 3-90© 2010 Pearson Prentice Hall. All rights reserved

91 Objective 3 Approximate the Variance and Standard Deviation of a Variable from Grouped Data 3-91© 2010 Pearson Prentice Hall. All rights reserved

92 3-92© 2010 Pearson Prentice Hall. All rights reserved

93 Hours Frequency 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 variance and 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 3-93© 2010 Pearson Prentice Hall. All rights reserved

94 TimeFrequency x © 2010 Pearson Prentice Hall. All rights reserved

95 Section 3.4 Measures of Position and Outliers 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 3-95© 2010 Pearson Prentice Hall. All rights reserved

96 3-96© 2010 Pearson Prentice Hall. All rights reserved

97 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 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 3-97© 2010 Pearson Prentice Hall. All rights reserved

98 Kevin Garnetts height is 4.96 standard deviations above the mean. Candace Parkers height is 4.56 standard deviations above the mean. Kevin Garnett is relatively taller. 3-98© 2010 Pearson Prentice Hall. All rights reserved

99 The mean commute time in the U.S. is 24.4 minutes with a standard deviation of 6.5 minutes. Find the z-score that corresponds to a commute time of 15 minutes. A B. –1.45 C D. –9.4 Copyright © 2010 Pearson Education, Inc. Slide 3- 99

100 The mean commute time in the U.S. is 24.4 minutes with a standard deviation of 6.5 minutes. Find the z-score that corresponds to a commute time of 15 minutes. A B. –1.45 C D. –9.4 Copyright © 2010 Pearson Education, Inc. Slide

101 The mean commute time in the U.S. is 24.4 minutes with a standard deviation of 6.5 minutes. Find the z-score that corresponds to a commute time of 15 minutes. A B. –1.45 C D. –9.4 Copyright © 2010 Pearson Education, Inc. Slide

102 The mean commute time in the U.S. is 24.4 minutes with a standard deviation of 6.5 minutes. Find the z-score that corresponds to a commute time of 15 minutes. A B. –1.45 C D. –9.4 Copyright © 2010 Pearson Education, Inc. Slide

103 Objective 2 Interpret Percentiles 3-103© 2010 Pearson Prentice Hall. All rights reserved

104 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 © 2010 Pearson Prentice Hall. All rights reserved

105 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: Interpret this admissions requirement. 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 individuals score must be in the top 30% © 2010 Pearson Prentice Hall. All rights reserved

106 Objective 3 Determine and Interpret Quartiles 3-106© 2010 Pearson Prentice Hall. All rights reserved

107 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 © 2010 Pearson Prentice Hall. All rights reserved

108 3-108© 2010 Pearson Prentice Hall. All rights reserved

109 A group of Brigham Young UniversityIdaho 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 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 = 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 = © 2010 Pearson Prentice Hall. All rights reserved

110 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 © 2010 Pearson Prentice Hall. All rights reserved

111 Objective 4 Determine and Interpret the Interquartile Range 3-111© 2010 Pearson Prentice Hall. All rights reserved

112 3-112© 2010 Pearson Prentice Hall. All rights reserved

113 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 © 2010 Pearson Prentice Hall. All rights reserved

114 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 3-114© 2010 Pearson Prentice Hall. All rights reserved

115 The closing prices for 9 telecommunications stocks are shown below. Compute the interquartile range, IQR A B C D Copyright © 2010 Pearson Education, Inc. Slide

116 The closing prices for 9 telecommunications stocks are shown below. Compute the interquartile range, IQR A B C D Copyright © 2010 Pearson Education, Inc. Slide

117 Objective 5 Check a Set of Data for Outliers 3-117© 2010 Pearson Prentice Hall. All rights reserved

118 3-118© 2010 Pearson Prentice Hall. All rights reserved

119 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) Upper Fence = Q (IQR) = 28 – 1.5(10) = (10) = 13 mph = 53 mph Step 4: There are no values less than 13 mph or greater than 53 mph. Therefore, there are no outliers © 2010 Pearson Prentice Hall. All rights reserved

120 Section 3.5 The Five-Number Summary and Boxplots Objectives 1.Compute the five-number summary 2.Draw and interpret boxplots 3-120© 2010 Pearson Prentice Hall. All rights reserved

121 3-121© 2010 Pearson Prentice Hall. All rights reserved

122 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. 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: First, we write the data is 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% 3-122© 2010 Pearson Prentice Hall. All rights reserved

123 Objective 2 Draw and interpret boxplots 3-123© 2010 Pearson Prentice Hall. All rights reserved

124 3-124© 2010 Pearson Prentice Hall. All rights reserved

125 EXAMPLEConstructing a Boxplot 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. Draw a boxplot of the data. 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: © 2010 Pearson Prentice Hall. All rights reserved

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) Upper Fence = Q (IQR) = 12 – 1.5(2.4) = (2.4) = 8.4% = 18.0% Step 2: [ ] * 3-126© 2010 Pearson Prentice Hall. All rights reserved

127 Objective 3 Use a boxplot and quartiles to describe the shape of a distribution 3-127© 2010 Pearson Prentice Hall. All rights reserved

128 The interest rate boxplot indicates that the distribution is skewed left © 2010 Pearson Prentice Hall. All rights reserved

129 Use the boxplot to identify the first quartile. A. 10 B. 18 C. 24 D. 26 | | | | | | | | | | | Copyright © 2010 Pearson Education, Inc. Slide

130 Use the boxplot to identify the first quartile. A. 10 B. 18 C. 24 D. 26 | | | | | | | | | | | Copyright © 2010 Pearson Education, Inc. Slide


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