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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.
61
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.
62
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.
63
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
64
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?
65
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?
66
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
67
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
68
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
69
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.
70
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.
71
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 ?
72
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.
73
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
8) Compare the dispersion of the observations in Part A with the observations in Part B. Which set of data
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.
74
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.
75
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
76
78
93
115
106
85
114
99
107
119
79
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
77
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.
78
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
79
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?
80
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.
81
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
\$ …