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Chapter 3 – Numerically

Summarizing Data

OUTLINE

Putting It Together

3.1 Measures of Central Tendency

3.2 Measures of Dispersion

3.3 Measures of Central Tendency and

Dispersion from Grouped Data

3.4 Measures of Position

3.5 The Five-Number Summary and

Boxplots

When we look at a distribution of data, we

should consider three characteristics of the

distribution: shape, center, and spread. In the last

chapter, we discussed methods for organizing

raw data into tables and graphs. These graphs

(such as the histogram) allow us to identify the

shape of the distribution: symmetric (in

particular, bell shaped or uniform), skewed right,

or skewed left.

The center and spread are numerical summaries

of the data. The center of a data set is commonly

called the average. There are many ways to

describe the average value of a distribution. In

addition, there are many ways to measure the

spread of a distribution. The most appropriate

measure of center and spread depends on the

distribution’s shape.

Once these three characteristics of the

distribution are known, we can analyze the data

for interesting features, including unusual data

values, called outliers.

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Copyright © 2019 Pearson Education, Inc.

Chapter 3: Numerically Summarizing Data

Section 3.1

Measures of Central Tendency

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

Objective 1: Determine the Arithmetic Mean of a Variable from Raw Data

INTRODUCTION, PAGE 1

Answer the following after watching the video.

1) What does a measure of central tendency describe?

OBJECTIVE 1, PAGE 1

2) Explain how to compute the arithmetic mean of a variable.

3) What symbols are used to represent the population mean and the sample mean?

OBJECTIVE 1, PAGE 2

4) List the formulas used to compute the population mean and the sample mean.

Note: Throughout this course, we agree to round the mean to one more decimal place than that in the raw

data.

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Section 3.1: Measures of Central Tendency

OBJECTIVE 1, PAGE 3

Example 1

Computing a Population Mean and a Sample Mean

Table 1 shows the first exam scores of the ten students enrolled in Introductory Statistics.

Table 1

Student

1. Michelle

2. Ryanne

3. Bilal

4. Pam

5. Jennifer

6. Dave

7. Joel

8. Sam

9. Justine

10. Juan

Score

82

77

90

71

62

68

74

84

94

88

A) Compute the population mean, .

B) Find a simple random sample of size n = 4 students.

C) Compute the sample mean, x , of the sample found in part (B).

OBJECTIVE 1, PAGE 5

Answer the following after experimenting with the fulcrum animation.

5) What is the mean of the data?

6) Explain why it is helpful to think of the mean as the center of gravity.

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Chapter 3: Numerically Summarizing Data

Objective 2: Determine the Median of a Variable from Raw Data

OBJECTIVE 2, PAGE 1

7) Define the median of a variable.

OBJECTIVE 2, PAGE 2

8) List the three steps in finding the median of a data set.

OBJECTIVE 2, PAGE 3

Example 2

Determining the Median of a Data Set (Odd Number of Observations)

Table 2 shows the length (in seconds) of a random sample of songs released in the 1970s. Find the

median length of the songs.

Table 2

Song Name

Length

“Sister Golden Hair” 201

“Black Water”

257

“Free Bird”

284

“The Hustle”

208

“Southern Nights”

179

“Stayin’ Alive”

222

“We Are Family”

217

“Heart of Glass”

206

“My Sharona”

240

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Section 3.1: Measures of Central Tendency

OBJECTIVE 2, PAGE 5

Example 3

Determining the Median of a Data Set (Even Number of Observations)

Find the median score of the data in Table 1.

Table 1

Student

1. Michelle

2. Ryanne

3. Bilal

4. Pam

5. Jennifer

6. Dave

7. Joel

8. Sam

9. Justine

10. Juan

Score

82

77

90

71

62

68

74

84

94

88

Objective 3: Explain What It Means for a Statistic to be Resistant

OBJECTIVE 3, PAGE 1

Answer the following as you work through the Mean versus Median Applet.

9) When the mean and median are approximately 2, how does adding a single observation near 9 affect

the mean? How does it affect the median?

10) When the mean and median are approximately 2, how does adding a single observation near 24 affect

the mean? The median?

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Chapter 3: Numerically Summarizing Data

OBJECTIVE 3, PAGE 1 (CONTINUED)

11) When the mean and median are approximately 40, how does dragging the new observation from 35

toward 0 affect the mean? How does it affect the median?

OBJECTIVE 3, PAGE 2

Answer the following as you watch the video.

12) Which measure, the mean or the median, is least affected by extreme observations?

13) Define what it means for a numerical summary of data to be resistant.

14) Which measure, the mean or the median, is resistant?

OBJECTIVE 3, PAGE 3

15) State the reason that we compute the mean.

OBJECTIVE 3, PAGE 7

Answer the following as you work through Activity 2: Relation among the Mean, Median, and

Distribution Shape.

16) If a distribution is skewed left, what is the relation between the mean and median?

17) If a distribution is skewed right, what is the relation between the mean and median?

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Section 3.1: Measures of Central Tendency

OBJECTIVE 3, PAGE 7 (CONTINUED)

18) If a distribution is symmetric, what is the relation between the mean and median?

OBJECTIVE 3, PAGE 11

19) Sketch three graphs showing the relation between the mean and median for distributions that are

skewed left, symmetric, and skewed right.

OBJECTIVE 3, PAGE 12

Example 4

Describing the Shape of a Distribution

The data in Table 4 represent the birth weights (in pounds) of 50 randomly sampled babies.

A) Find the mean and median birth weight.

B) Describe the shape of the distribution.

C) Which measure of central tendency best describes the average birth weight?

Table 4

5.8

7.9

8.7

7.9

7.3

9.4

7.3

7.1

7.6

7.4

7.8

7.2

5.9

6.4

6.8

6.9

7.0

6.7

9.2

7.9

6.1

7.0

7.4

7.0

6.9

7.0

7.0

7.7

7.2

7.8

8.2

8.1

6.4

7.4

8.5

9.0

7.1

7.2

9.1

8.0

7.8

8.2

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7.6

7.1

7.2

7.5

7.3

7.5

8.7

7.2

Chapter 3: Numerically Summarizing Data

Objective 4: Determine the Mode of a Variable from Raw Data

OBJECTIVE 4, PAGE 1

20) Define the mode of a variable.

21) Under what conditions will a set of data have no mode?

22) Under what conditions will a set of data have two modes?

OBJECTIVE 4, PAGE 2

Example 5

Finding the Mode of Quantitative Data

The following data represent the number of O-ring failures on the shuttle Columbia for the 17 flights prior

to its fatal flight:

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 3

Find the mode number of O-ring failures.

OBJECTIVE 4, PAGE 3

Example 6

Finding the Mode of Quantitative Data

Find the mode of the exam score data listed in Table 1.

Table 1

Student

1. Michelle

2. Ryanne

3. Bilal

4. Pam

5. Jennifer

6. Dave

7. Joel

8. Sam

9. Justine

10. Juan

Score

82

77

90

71

62

68

74

84

94

88

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Section 3.1: Measures of Central Tendency

OBJECTIVE 4, PAGE 5

23) What does it mean when we say that a data set is bimodal? Multimodal?

OBJECTIVE 4, PAGE 6

Example 7

Finding the Mode of Qualitative Data

The data in Table 5 represent the location of injuries that required rehabilitation by a physical therapist.

Determine the mode location of injury.

Table 5

Back

Back

Hand

Neck

Wrist

Back

Groin

Shoulder

Elbow

Back

Back

Back

Back

Shoulder

Shoulder

Knee

Hip

Knee

Hip

Hand

Data from Krystal Catton, student at Joliet Junior College

Knee

Shoulder

Back

Knee

Back

Knee

Back

Back

Back

Wrist

OBJECTIVE 4, PAGE 8

Summary

24) List the conditions for determining when to use the following measures of central tendency.

A) Mean

B) Median

C) Mode

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Copyright © 2019 Pearson Education, Inc.

Chapter 3: Numerically Summarizing Data

Section 3.2

Measures of Dispersion

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

INTRODUCTION, PAGE 1

Measures of central tendency describe the typical value of a variable. We also want to know the amount

of dispersion (or spread) in the variable. Dispersion is the degree to which the data are spread out.

INTRODUCTION, PAGE 2

Example 1

Comparing Two Sets of Data

The data tables represent the IQ scores of a random sample of 100 students from two different

universities.

For each university, compute the mean IQ score and draw a histogram, using a lower class limit of 55 for

the first class and a class width of 15. Comment on the results.

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Section 3.2: Measures of Dispersion

Objective 1: Determine the Range of a Variable from Raw Data

OBJECTIVE 1, PAGE 1

1) What is the range of a variable?

OBJECTIVE 1, PAGE 2

Example 2

Computing the Range of a Set of Data

The data in the table represent the first exam scores of 10 students enrolled in Introductory Statistics.

Compute the range.

Student

1. Michelle

2. Ryanne

3. Bilal

4. Pam

5. Jennifer

6. Dave

7. Joel

8. Sam

9. Justine

10. Juan

Score

82

77

90

71

62

68

74

84

94

88

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

OBJECTIVE 2, PAGE 1

2) Explain how to compute the population standard deviation and list its formula.

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Chapter 3: Numerically Summarizing Data

OBJECTIVE 2, PAGE 2

Example 3

Computing a Population Standard Deviation

Compute the population standard deviation of the test scores in Table 6.

Table 6

Student

1. Michelle

2. Ryanne

3. Bilal

4. Pam

5. Jennifer

6. Dave

7. Joel

8. Sam

9. Justine

10. Juan

Score

82

77

90

71

62

68

74

84

94

88

OBJECTIVE 2, PAGE 5

3) If a data set has many values that are “far” from the mean, how is the standard deviation affected?

OBJECTIVE 2, PAGE 6

4) Explain how to compute the sample standard deviation s and list its formula.

OBJECTIVE 2, PAGE 7

5) What do we call the expression n − 1 ?

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Section 3.2: Measures of Dispersion

OBJECTIVE 2, PAGE 8

Example 4

Computing a Sample Standard Deviation

In a previous lesson we obtained a simple random sample of exam scores and computed a sample mean of

73.75. Compute the sample standard deviation of the sample of test scores for that data.

OBJECTIVE 2, PAGE 10

Answer the following after you watch the video.

6) Is standard deviation resistant? Why or why not?

OBJECTIVE 2, PAGE 11

7) When comparing two populations, what does a larger standard deviation imply about dispersion?

OBJECTIVE 2, PAGE 14

Example 5

Comparing the Standard Deviations of Two Sets of Data

The data tables represent the IQ scores of a random sample of 100 students from two different

universities.

Use the standard deviation to determine whether University A or University B has more dispersion in the

IQ scores of its students.

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Chapter 3: Numerically Summarizing Data

OBJECTIVE 2, PAGE 17

Answer the following after using the applet in Activity 1: Standard Deviation as a Measure of

Spread.

8) Compare the dispersion of the observations in Part A with the observations in Part B. Which set of data

is more spread out?

9) In Part D, how does adding a point near 10 affect the standard deviation? How is the standard deviation

affected when that point is moved near 25? What does this suggest?

OBJECTIVE 2, PAGE 18

Watch the video to reinforce the ideas from Activity 1: Standard Deviation as a Measure of Spread.

Objective 3: Determine the Variance of a Variable from Raw Data

OBJECTIVE 3, PAGE 1

10) Define variance.

OBJECTIVE 3, PAGE 2

Example 6

Determining the Variance of a Variable for a Population and a Sample

In previous examples, we considered population data of exam scores in a statistics class. For this data, we

computed a population mean of = 79 points and a population standard deviation of = 9.8 points.

Then, we obtained a simple random sample of exam scores. For this data, we computed a sample mean of

x = 73.75 points and a sample standard deviation of s = 11.3 points. Use the population standard deviation

exam score and the sample standard deviation exam score to determine the population and sample

variance of scores on the statistics exam.

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Section 3.2: Measures of Dispersion

OBJECTIVE 3, PAGE 3

Answer the following after you watch the video.

11) Using a rounded value of the standard deviation to obtain the variance results in a round-off error.

How should you deal with this issue?

OBJECTIVE 3, PAGE 5

Whenever a statistic consistently underestimates a parameter, it is said to be biased. To obtain an

unbiased estimate of the population variance, divide the sum of the squared deviations about the sample

mean by n − 1 .

Objective 4: Use the Empirical Rule to Describe Data That Are Bell-Shaped

OBJECTIVE 4, PAGE 1

12) According to the Empirical Rule, if a distribution is roughly bell shaped, then approximately what

percent of the data will lie within 1 standard deviation of the mean? What percent of the data will lie

within 2 standard deviations of the mean? What percent of the data will lie within 3 standard deviations of

the mean?

OBJECTIVE 4, PAGE 2

13) Sketch the third part of Figure 5.

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Chapter 3: Numerically Summarizing Data

OBJECTIVE 4, PAGE 3

Example 7

Using the Empirical Rule

Table 9 represents the IQs of a random sample of 100 students at a university.

A) Determine the percentage of students who have IQ scores within 3 standard deviations of the mean

according to the Empirical Rule.

B) Determine the percentage of students who have IQ scores between 67.8 and 132.2 according to the

Empirical Rule.

C) Determine the actual percentage of students who have IQ scores between 67.8 and 132.2.

D) According to the Empirical Rule, what percentage of students have IQ scores between 116.1 and

148.3?

73

108

102

103

71

102

107

102

107

118

103

93

111

110

69

109

94

109

110

85

91

91

125

84

97

105

1121

76

106

127

93

78

107

115

130

97

113

94

103

141

Table 9

136

81

80

85

91

104

115

103

93

129

108

130

90

83

62

94

106

112

110

60

92

82

122

131

85

92

97

107

125

115

104

86

101

90

94

83

106

101

101

80

90

111

82

103

110

94

85

91

91

111

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78

93

115

106

85

114

99

107

119

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Section 3.3: Measures of Central Tendency and Dispersion from Grouped Data

Section 3.3

Measures of Central Tendency and Dispersion from Grouped Data

Objectives

Approximate the Mean of a Variable from Grouped Data

Compute the Weighted Mean

Approximate the Standard Deviation from a Frequency Distribution

Objective 1: Approximate the Mean of a Variable from Grouped Data

OBJECTIVE 1, PAGE 1

1) Explain how to find the class midpoint.

2) List the formulas for approximating the population mean and sample mean from a frequency

distribution.

OBJECTIVE 1, PAGE 2

Example 1

Approximating the Mean for Continuous Quantitative Data from a Frequency

Distribution

The frequency distribution in Table 10 represents the five-year rate of return of a random sample of 40

large-blend mutual funds. Approximate the mean five-year rate of return.

Table 10

Class (5-year rate of return)

8-8.99

9-9.99

10-10.99

11-11.99

12-12.99

13-13.99

14-14.99

15-15.99

16-16.99

17-17.99

18-18.99

19-19.99

Frequency

2

2

4

1

6

13

7

3

1

0

0

1

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Chapter 3: Numerically Summarizing Data

OBJECTIVE 1, PAGE 2 (CONTINUED)

Objective 2: Compute the Weighted Mean

OBJECTIVE 2, PAGE 1

3) When data values have different importance, or weights, associated with them, we compute the

weighted mean. Explain how to compute the weighted mean and list its formula.

OBJECTIVE 2, PAGE 2

Example 2

Computing the Weighted Mean

Marissa just completed her first semester in college. She earned an A in her 4-hour statistics course, a B

in her 3-hour sociology course, an A in her 3-hour psychology course, a C in her 5-hour computer

programming course, and an A in her 1-hour drama course. Determine Marissa’s grade point average.

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Section 3.3: Measures of Central Tendency and Dispersion from Grouped Data

Objective 3: Approximate the Standard Deviation from a Frequency Distribution

OBJECTIVE 3, PAGE 1

4) List the formulas for approximating the population standard deviation and sample standard deviation of

a variable from a frequency distribution.

OBJECTIVE 3, PAGE 2

Example 3

Approximating the Standard Deviation from a Frequency Distribution

The frequency distribution in Table 11 represents the five-year rate of return of a random sample of 40

large-blend mutual funds. Approximate the standard deviation five-year rate of return.

Table 11

Class (5-year rate of return)

8-8.99

9-9.99

10-10.99

11-11.99

12-12.99

13-13.99

14-14.99

15-15.99

16-16.99

17-17.99

18-18.99

19-19.99

Frequency

2

2

4

1

6

13

7

3

1

0

0

1

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Chapter 3: Numerically Summarizing Data

Section 3.4

Measures of Position

Objective

Determine and Interpret z-Scores

Interpret Percentiles

Determine and Interpret Quartiles

Determine and Interpret the Interquartile Range

Check a Set of Data for Outliers

Objective 1: Determine and Interpret z-Scores

OBJECTIVE 1, PAGE 1

1) What does a z-score represent?

2) Explain how to find a z-score and list the formulas for computing a population z-score and a sample zscore.

3) What does a positive z-score for a data value indicate? What does a negative z-score indicate?

4) What does a z-score measure?

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Section 3.4: Measures of Position

OBJECTIVE 1, PAGE 1 (CONTINUED)

5) How are z-scores rounded?

OBJECTIVE 1, PAGE 2

Example 1

Determine and Interpret z-Scores

Determine whether the Boston Red Sox or the Colorado Rockies had a relatively better run-producing

season. The Red Sox scored 878 runs and play in the American League, where the mean number of runs

scored was = 731.3 and the standard deviation was = 54.9 runs. The Rockies scored 845 runs and

play in the National League, where the mean number of runs scored was = 718.3 and the standard

deviation was = 61.7 runs.

OBJECTIVE 1, PAGE 5

With negative z-scores, we need to be careful when deciding the better outcome. For example, when

comparing finishing times for a marathon the lower score is better because it is more standard deviations

below the mean.

Objective 2: Interpret Percentiles

OBJECTIVE 2, PAGE 1

6) What does the kth percentile represent?

OBJECTIVE 2, PAGE 2

Example 2

Interpreting a Percentile

Jennifer just received the results of her SAT exam. Her math score of 600 is at the 74th percentile.

Interpret this result.

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Chapter 3: Numerically Summarizing Data

Objective 3: Determine and Interpret Quartiles

OBJECTIVE 3, PAGE 1

7) Define the first, second, and third quartiles.

OBJECTIVE 3, PAGE 2

8) List the three steps for finding quartiles.

OBJECTIVE 3, PAGE 3

Example 3

Finding and Interpreting Quartiles

The Highway Loss Data Institute routinely collects data on collision coverage claims. Collision coverage

insures against physical damage to an insured individual’s vehicle. Table 12 represents a random sample

of 18 collision coverage claims based on data obtained from the Highway Loss Data Institute for 2007

models. Find and interpret the first, second, and third quartiles for collision coverage claims.

Table 12

$6751

$2336

$189

$1414

$10,034

$618

$9908

$21,147

$1185

$4668

$735

$180

$3461

$2332

$370

$1953

$802

$ …

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