1. Numerically Summarizing Data
Chapter 3
Numerically Summarizing Data

2. Measures of Central Tendency
Section
3.1
Measures of Central Tendency

3. Objectives Determine the arithmetic mean of a variable from raw data
Determine the median of a variable from raw data
Explain what it means for a statistic to be resistant
Determine the mode of a variable from raw data 3

4. Objective 1 Determine the Arithmetic Mean of a Variable from Raw Data4

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. 5

6. The population mean is a parameter.
The population arithmetic mean, μ (pronounced “mew”), is computed using all the individuals in a population.
The population mean is a parameter. 6

7. The sample mean is a statistic.
The sample arithmetic mean, (pronounced “x-bar”), is computed using sample data.
The sample mean is a statistic. 7

8. If x1, x2, …, xN are the N observations of a variable from a population, then the population mean, µ, is 8

9. If x1, x2, …, xn are the n observations of a variable from a sample, then the sample mean, , is9

10. Compute the population mean of this data.
EXAMPLE Computing 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
Compute the population mean of this data.
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. 10

11. EXAMPLE Computing a Population Mean and a Sample Mean11

12. EXAMPLE Computing 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.
23, 36, 23, 18, 5, 26, 43 12

13. 13

14. Objective 2
Determine the Median of a Variable from Raw Data 14

15. We use M to represent the median.
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. 15

16. Steps in Finding the Median of a Data Set
Step 1 Arrange the data in ascending order.
Step 2 Determine the number of observations, n.
Step 3 Determine the observation in the middle of the data set. 16

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. 17

18. Determine the median of this data.
EXAMPLE Computing 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.
M = 23
Step 3:
5, 18, 23, 23, 26, 36, 43 18

19. Step 2: There are n = 8 observations.
EXAMPLE Computing 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 19

20. Objective 3
Explain What it Means for a Statistic to Be Resistant 20

21. Mean before new hire: 24.9 minutes Median before new hire: 23 minutes
EXAMPLE Computing 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 21

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. 22

23. 23

24. EXAMPLE Describing the Shape of the Distribution
The following data represent the asking price of homes for sale in Lincoln, NE.
79,995
128,950
149,900
189,900
99,899
130,950
151,350
203,950
105,200
131,800
154,900
217,500
111,000
132,300
159,900
260,000
120,000
134,950
163,300
284,900
121,700
135,500
165,000
299,900
125,950
138,500
174,850
309,900
126,900
147,500
180,000
349,900
Source: 24

25. Find the mean and median
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. 25

26. Find the mean and median
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. 26

27. 27

28. Objective 4
Determine the Mode of a Variable from Raw Data 28

29. A set of data can have no mode, one mode, or more than one mode.
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. 29

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. 30

31. Joe Biden
Pennsylvania 31

32. 32

33. The mode is New York. 33

34. Tally data to determine most frequent observation34

35. Measures of Dispersion
Section
3.2
Measures of Dispersion

36. Objectives Determine the range of a variable from raw data
Determine the standard deviation of a variable from raw data
Determine the variance of a variable from raw data
Use the Empirical Rule to describe data that are bell shaped
Use Chebyshev’s Inequality to describe any data set 36

37. (a) What was the mean wait time?
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? 37

38. Wait Time at McDonald’s
Wait Time at Wendy’s
Wait Time at McDonald’s 38

39. (a) The mean wait time in each line is 1.39 minutes.

40. (b) 40

41. Objective 1
Determine the Range of a Variable from Raw Data 41

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 42

43. EXAMPLE Finding 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 43

44. Objective 2
Determine the Standard Deviation of a Variable from Raw Data 44

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). 45

46. where x1, x2, . . . , xN are the N observations in the population and μ is the population mean.46

47. A formula that is equivalent to the one on the previous slide, called the computational formula, for determining the population standard deviation is 47

48. Compute the population standard deviation of this data.
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. 48

49. xi μ
xi – μ
(xi – μ)2 23 36 18 5 26 43 49

50. xi (xi )2 Using the computational formula, yields the same result. 23
529 36
1296 18
324 5 25 26
676 43
1849
Σ xi = 174
Σ (xi)2 = 5228 50

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 x1, x2, , xn are the n observations in the sample and is the sample mean. 51

52. A formula that is equivalent to the one on the previous slide, called the computational formula, for determining the sample standard deviation is 52

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 nth value has no freedom. It must be whatever value forces the sum of the deviations about the mean to equal zero. 53

54. Find the sample standard deviation.
EXAMPLE Computing 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. 54

55. xi 5 26
3.667 36
13.667 55

56. xi (xi )2 Using the computational formula, yields the same result. 525 26
676 36
1296
Σ xi = 67
Σ (xi)2 = 1997 56

57. 57

58. EXAMPLE Comparing Standard Deviations
Determine the standard deviation waiting time for Wendy’s and McDonald’s. Which is larger? Why? 58

59. Wait Time at McDonald’s
Wait Time at Wendy’s
Wait Time at McDonald’s 59

60. Sample standard deviation for Wendy’s: 0.738 minutes
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. 60

61. Objective 3
Determine the Variance of a Variable from Raw Data 61

62. The variance of a variable is the square of the standard deviation
The variance of a variable is the square of the standard deviation. The population variance is σ2 and the sample variance is s2. 62

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. 63

64. EXAMPLE Computing a Population Variance
Recall that the population standard deviation (from slide #49) is σ = so the population variance is σ2 = minutes
and that the sample standard deviation (from slide #55) is s = 15.82, so the sample variance is s2 = minutes 64

65. Objective 4
Use the Empirical Rule to Describe Data That Are Bell Shaped 65

66. If a distribution is roughly bell shaped, then
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. 66

67. If a distribution is roughly bell shaped, then
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. 67

68. 68

69. EXAMPLE Using the Empirical Rule
The following data represent the serum HDL cholesterol of the 54 female patients of a family doctor. 69

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. 70

71. (a) Using a TI-83 plus graphing calculator, we find
(b) 71

72.
(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.
(e) 45 out of the 54 or 83.3% of the patients have a serum HDL between 34.0 and 69.1. 72

73. Objective 5
Use Chebyshev’s Inequality to Describe Any Set of Data 73

74. Chebyshev’s Inequality For any data set or distribution, at least
of the observations lie within k
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.
Note: We can also use Chebyshev’s Inequality based on sample data. 74

75. EXAMPLE Using Chebyshev’s Theorem
Using the data from the previous example, use Chebyshev’s Theorem to
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% 75

76. Measures of Central Tendency and Dispersion from Grouped Data
Section
3.3
Measures of Central Tendency and Dispersion from Grouped Data

77. Objectives Approximate the mean of a variable from grouped data
Compute the weighted mean
Approximate the standard deviation of a variable from grouped data 77

78. Objective 1
Approximate the Mean of a Variable from Grouped Data 78

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. 79

80. Approximate the Mean of a Variable from a Frequency Distribution
Population Mean
Sample Mean
where xi is the midpoint or value of the ith class
fi is the frequency of the ith class
n is the number of classes 80

81. EXAMPLE Approximating the Mean from a Relative Frequency Distribution
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.
Hours
1-5
6-10
11-15
16-20
21-25
26-30
31-35
Frequency
130
250
230
180
100 60 50
Source: 81

82. Time Frequency xi xi fi 1 - 5 130 3 390 6 - 10 250 8 2000 11 - 15 230
1 - 5
130 3
390
6 - 10
250 8
2000
230 13
2990
180 18
3240
100 23
2300
26 – 30 60 28
1680
31 – 35 50 33
1650 82

83. Objective 2
Compute the Weighted Mean 83

84. where w is the weight of the ith observation
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 ith observation
xi is the value of the ith observation 84

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. 85

86. Objective 3
Approximate the Standard Deviation of a Variable from Grouped Data 86

87. Population Standard Deviation Sample Standard Deviation
Approximate the Standard Deviation of a Variable from a Frequency Distribution
Population Standard Deviation
Sample
Standard Deviation
where xi is the midpoint or value of the ith class
fi is the frequency of the ith class 87

88. An algebraically equivalent formula for the population standard deviation is88

89. EXAMPLE Approximating the Mean from a Relative Frequency Distribution
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.
Hours
1-5
6-10
11-15
16-20
21-25
26-30
31-35
Frequency
130
250
230
180
100 60 50
Source: 89

90. Time
Frequency xi
1 - 5
130 3
–11.25
16,
6 - 10
250 8
–6.25
230 13
–1.25
180 18
3.75
100 23
8.75
26 – 30 60 28
13.75
11,343.75
31 – 35 50 33
18.75
17, 90

91. Measures of Position and Outliers
Section
3.4
Measures of Position and Outliers

92. Objectives Determine and interpret z-scores Interpret percentiles
Determine and interpret quartiles
Determine and interpret the interquartile range
Check a set of data for outliers 92

93. Objective 1
Determine and Interpret z-scores 93

94. The z-score is unitless. It has mean 0 and standard deviation 1.
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:
Population z-score
Sample z-score
The z-score is unitless. It has mean 0 and standard deviation 1. 94

95. Kevin Garnett whose height is 83 inches or
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 95

96. Kevin Garnett’s height is 4. 96 standard deviations above the mean
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. 96

97. Objective 2
Interpret Percentiles 97

98. The kth percentile, denoted, Pk , of a set of data is a value such that k percent of the observations are less than or equal to the value. 98

99. (Source: http://www.publichealth.pitt.edu/interior.php?pageID=101.)
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. 99

100. EXAMPLE Interpret a Percentile
In general, the 70th 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%.
100

101. Objective 3
Determine and Interpret Quartiles
101

102. Quartiles divide data sets into fourths, or four equal parts.
The 1st quartile, denoted Q1, divides the bottom 25% the data from the top 75%. Therefore, the 1st quartile is equivalent to the 25th percentile.
The 2nd quartile divides the bottom 50% of the data from the top 50% of the data, so that the 2nd quartile is equivalent to the 50th percentile, which is equivalent to the median.
The 3rd quartile divides the bottom 75% of the data from the top 25% of the data, so that the 3rd quartile is equivalent to the 75th percentile.
102

103. Step 1 Arrange the data in ascending order.
Finding Quartiles
Step 1 Arrange the data in ascending order.
Step 2 Determine the median, M, or second quartile, Q2 .
Step 3 Divide the data set into halves: the observations below (to the left of) M and the observations above M. The first quartile, Q1 , is the median of the bottom half, and the third quartile, Q3 , is the median of the top half.
103

104. Find and interpret the quartiles for speed in the construction zone.
EXAMPLE Finding and Interpreting Quartiles
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.
104

105. Step 1: The data is already in ascending order.
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, Q2, is the mean of the 7th and 8th observations. Therefore, M = 32.5.
Step 3: The median of the bottom half of the data is the first quartile, Q1.
20, 24, 27, 28, 29, 30, 32
The median of these seven observations is 28. Therefore, Q1 = 28. The median of the top half of the data is the third quartile, Q3. Therefore, Q3 = 38.
105

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.
106

107. Objective 4
Determine and Interpret the Interquartile Range
107

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 = Q3 – Q1
108

109. Determine and interpret the interquartile range of the speed data.
EXAMPLE Determining and Interpreting the Interquartile Range
Determine and interpret the interquartile range of the speed data.
Q1 = Q3 = 38
The range of the middle 50% of the speed of cars traveling through the construction zone is 10 miles per hour.
109

110. Suppose a 15th 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 15th car
With 15th car
Mean
32.1 mph
36.7 mph
Median
32.5 mph
33 mph
Standard deviation
6.2 mph
18.5 mph
IQR
10 mph
11 mph
110

111. Objective 5
Check a Set of Data for Outliers
111

112. Checking for Outliers by Using Quartiles
Step 1 Determine the first and third quartiles of the data.
Step 2 Compute the interquartile range.
Step 3 Determine the fences. Fences serve as cutoff points for determining outliers.
Lower Fence = Q1 – 1.5(IQR)
Upper Fence = Q (IQR)
Step 4 If a data value is less than the lower fence or greater than the upper fence, it is considered an outlier.
112

113. Check the speed data for outliers.
EXAMPLE Determining and Interpreting the Interquartile Range
Check the speed data for outliers.
Step 1: The first and third quartiles are Q1 = 28 mph and Q3 = 38 mph.
Step 2: The interquartile range is 10 mph.
Step 3: The fences are
Lower Fence = Q1 – 1.5(IQR) = 28 – 1.5(10) = 13 mph
Upper Fence = Q (IQR) = (10) = 53 mph
Step 4: There are no values less than 13 mph or greater than 53 mph. Therefore, there are no outliers.
113

114. The Five-Number Summary and Boxplots
Section
3.5
The Five-Number Summary and Boxplots

115. Objectives Compute the five-number summary Draw and interpret boxplots
115

116. Objective 1
Compute the Five-Number Summary
116

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:
117

118. EXAMPLE Obtaining 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.
118

119. Pulaski Bank and Trust Company 6.5% Rainier Pacific Savings Bank 12.0%
EXAMPLE Obtaining the Five-Number Summary
Institution
Rate
Pulaski Bank and Trust Company
6.5%
Rainier Pacific Savings Bank
12.0%
Wells Fargo Bank NA
14.4%
Firstbank of Colorado
Lafayette Ambassador Bank
14.3%
Infibank
13.0%
United Bank, Inc.
13.3%
First National Bank of The Mid-Cities
13.9%
Bank of Louisiana
9.9%
Bar Harbor Bank and Trust Company
14.5%
Source:
119

120. First, we write the data in ascending order:
EXAMPLE Obtaining 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% % % % %
120

121. Objective 2
Draw and Interpret Boxplots
121

122. Step 1 Determine the lower and upper fences.
Drawing a Boxplot
Step 1 Determine the lower and upper fences.
Lower Fence = Q1 – 1.5(IQR)
Upper Fence = Q (IQR)
where IQR = Q3 – Q1
Step 2 Draw a number line long enough to include the maximum and minimum values. Insert vertical lines at Q1, M, and Q3. Enclose these vertical lines in a box.
Step 3 Label the lower and upper fences.
122

123. Drawing a Boxplot
Step 4 Draw a line from Q1 to the smallest data value that is larger than the lower fence. Draw a line from Q3 to the largest data value that is smaller than the upper fence. These lines are called whiskers.
Step 5 Any data values less than the lower fence or greater than the upper fence are outliers and are marked with an asterisk (*).
123

124. EXAMPLE Obtaining 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.
124

125. Pulaski Bank and Trust Company 6.5% Rainier Pacific Savings Bank 12.0%
EXAMPLE Obtaining the Five-Number Summary
Institution
Rate
Pulaski Bank and Trust Company
6.5%
Rainier Pacific Savings Bank
12.0%
Wells Fargo Bank NA
14.4%
Firstbank of Colorado
Lafayette Ambassador Bank
14.3%
Infibank
13.0%
United Bank, Inc.
13.3%
First National Bank of The Mid-Cities
13.9%
Bank of Louisiana
9.9%
Bar Harbor Bank and Trust Company
14.5%
Source:
125

126. Lower Fence = Q1 – 1.5(IQR) = 12 – 1.5(2.4) = 8.4%
Step 1: The interquartile range (IQR) is 14.4% - 12% = 2.4%. The lower and upper fences are:
Lower Fence = Q1 – 1.5(IQR) = 12 – 1.5(2.4) = 8.4%
Upper Fence = Q (IQR) = (2.4) = 18.0%
Step 2: * [ ]
126

127. Use a boxplot and quartiles to describe the shape of a distribution.
The interest rate boxplot indicates that the distribution is skewed left.
127