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Chapter
3
Numerically
Summarizing Data
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
Section
3.1
Measures of
Central Tendency
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
Objectives
1. Determine the arithmetic mean of a variable from
raw data
2. Determine the median of a variable from raw data
3. Explain what it means for a statistic to be resistant
4. Determine the mode of a variable from raw data
3-3
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Objective 1
• Determine the Arithmetic Mean of a Variable
from Raw Data
3-4
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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.
3-5
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The population arithmetic mean, μ
(pronounced “mew”), is computed using all the
individuals in a population.
The population mean is a parameter.
3-6
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The sample arithmetic mean, x (pronounced
“x-bar”), is computed using sample data.
The sample mean is a statistic.
3-7
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If x1, x2, …, xN are the N observations of a
variable from a population, then the population
mean, µ, is
x1  x2 L  x N  xi


N
N
3-8
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If x1, x2, …, xn are the n observations of a
variable from a sample, then the sample mean,
, is x
x1  x2 L  xn
x

n
3-9
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x
i
n
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
(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.
3-10
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EXAMPLE
(a)
Computing a Population Mean and a Sample
Mean
x


i
N
x1  x2  ...  x7

7
23  36  23  18  5  26  43

7
174

7
 24.9 minutes
3-11
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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.
Let the random sample be 5,36 and 26.
Let the second random sample be 36,23 and 26.
3-12
5  36  26
3
 22.3
x
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36  23  26
3
 28.3
x
Objective 2
• Determine the Median of a Variable from Raw
Data
3-13
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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.
3-14
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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.
3-15
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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.
3-16
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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.
n 1 7 1
M = 23

4
Step 3:
2
2
5, 18, 23, 23, 26, 36, 43
3-17
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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.
23  26
n 1 8 1
M
 24.5 minutes

 4.5
Step 3:
2
2
2
5, 18, 23, 23, 26, 36, 43, 70

M  24.5
3-18
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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.
3-19
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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
3-20
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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
3-21
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Median is resistant
whereas
Mean is not
resistant.
3-22
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EXAMPLE
Describing the Shape of the Distribution
The following data represent the asking price of homes
for sale in Lincoln, NE.
79,995
99,899
105,200
128,950
130,950
131,800
149,900
151,350
154,900
189,900
203,950
217,500
111,000
120,000
121,700
132,300
134,950
135,500
159,900
163,300
165,000
260,000
284,900
299,900
125,950
126,900
138,500
147,500
174,850
180,000
309,900
349,900
Source: http://www.homeseekers.com
3-23
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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.
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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.
3-25
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Asking Price of Homes in Lincoln, NE
12
10
Frequency
8
6
4
2
0
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100000
150000
200000
250000
Asking Price
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300000
350000
Objective 4
• Determine the Mode of a Variable from
Raw Data
3-27
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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.
3-28
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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.
3-29
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Joe Biden
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Pennsylvani
a
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The mode is
New York.
3-32
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Tally data to determine
most frequent observation
3-33
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Section
3.2
Measures of
Dispersion
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Objectives
1. Determine the range of a variable from raw data
2. Determine the standard deviation of a variable from
raw data
3. Determine the variance of a variable from raw data
4. Use the Empirical Rule to describe data that are bell
shaped
5. Use Chebyshev’s Inequality to describe any data set
3-35
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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?
3-36
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Wait Time at Wendy’s
1.50
2.53
1.88
3.99
0.90
0.79
1.20
2.94
1.90
1.23
1.01
1.46
1.40
1.00
0.92
1.66
0.89
1.33
1.54
1.09
0.94
0.95
1.20
0.99
1.72
0.67
0.90
0.84
0.35
2.00
Wait Time at McDonald’s
3.50
0.00
1.97
0.00
3.08
3-37
0.00
0.26
0.71
0.28
2.75
0.38
0.14
2.22
0.44
0.36
0.43
0.60
4.54
1.38
3.10
1.82
2.33
0.80
0.92
2.19
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3.04
2.54
0.50
1.17
0.23
(a) The mean wait time in each line is 1.39
minutes.
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(b)
3-39
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Objective 1
• Determine the Range of a Variable from Raw
Data
3-40
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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
3-41
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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
3-42
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• Although Range is easy to compute, it is
sensitive to outliers.
3-43
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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).
3-44
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x1     x2   
2


2
 L…  x N   
2
N
 x
i
 
2
N
where x1, x2, . . . , xN are the N observations in
the population and μ is the population mean.
3-45
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A formula that is equivalent to the one on the
previous slide, called the computational
formula, for determining the population
standard deviation is
x



2

3-46
x
2
i
i
N
N
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
EXAMPLE
Computing a Population Standard
Deviation
The following data represent the travel times (in
minutes) to work for all seven employees of a startup web development company.
23, 36, 23, 18, 5, 26, 43
Compute the population standard deviation of this
data.
3-47
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xi
μ
xi – μ
(xi – μ)2
23
36
23
18
24.85714
24.85714
24.85714
24.85714
-1.85714
11.14286
-1.85714
-6.85714
3.44898
124.1633
3.44898
47.02041
5
26
43
24.85714
24.85714
24.85714
-19.8571
1.142857
18.14286
394.3061
1.306122
329.1633
 x
i

3-48
 x
i
 
N
2

  
2
902.8571
902.8571
 11.36 minutes
7
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
Using the computational formula, yields the same
result.
xi
(xi )2
23
36
529
1296
23
18
5
26
529
324
25
676
43
1849

Σ xi = 174 Σ (xi)2 = 5228
3-49
x



2
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x
2
i
i
N
N
174 

5228 
2

7
7
 11.36 minutes
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.
 x
i
s
 x
2
n 1
x1  x   x2  x 
2

2
L  xn  x 
2
n 1
where x1, x2, . . . , xn are the n observations in the
sample and x is the sample mean.
3-50
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A formula that is equivalent to the one on the
previous slide, called the computational
formula, for determining the sample standard
deviation is
x



2
s
3-51
x
2
i
i
n 1
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n
The sum of deviations from the mean equal
zero.
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.
3-52
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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.
3-53
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x
xi
xi  x
xi  x 
2
5
26
22.33333
22.33333
-17.333
3.667
300.432889
13.446889
36
22.33333
13.667
186.786889
 x  x   500.66667
2
i
s
3-54
 x
i
 x
n 1
2

500.66667
 15.82 minutes
2
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Using the computational formula, yields the same
result.
xi
(xi )2
5
26
25
676
36
1296
Σ xi = 67
Σ
(xi)2
x



2

x
2
i
i
n 1
n
67 

1997 
2
= 1997

3
2
 15.82 minutes
3-55
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EXAMPLE
Comparing Standard Deviations
Determine the standard deviation waiting time
for Wendy’s and McDonald’s. Which is
larger? Why?
3-56
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Wait Time at Wendy’s
1.50
2.53
1.88
3.99
0.90
0.79
1.20
2.94
1.90
1.23
1.01
1.46
1.40
1.00
0.92
1.66
0.89
1.33
1.54
1.09
0.94
0.95
1.20
0.99
1.72
0.67
0.90
0.84
0.35
2.00
Wait Time at McDonald’s
3.50
0.00
1.97
0.00
3.08
3-57
0.00
0.26
0.71
0.28
2.75
0.38
0.14
2.22
0.44
0.36
0.43
0.60
4.54
1.38
3.10
1.82
2.33
0.80
0.92
2.19
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3.04
2.54
0.50
1.17
0.23
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.
3-58
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Objective 3
• Determine the Variance of a Variable from
Raw Data
3-59
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The variance of a variable is the square of the
standard deviation. The population variance is
σ2 and the sample variance is s2.
3-60
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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.
3-61
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EXAMPLE
Computing a Population Variance
Recall that the population standard deviation is
σ = 11.36 so the population variance is σ2 = 129.05
and that the sample standard deviation is s = 15.82,
so the sample variance is s2 = 250.27
3-62
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Note that Standard deviation has same unit of
measurement as the variable whereas Variance has
different unit.
So, if the variable is measured in dollars, the
variance is measured in dollars squared. This makes
interpreting the variance difficult. However, the
variance is important for conducting certain types of
statistical inference, which we discuss later.
Standard deviation and variance are sensitive to
outliers.
Standard deviation can be used in conjunction with
the mean to describe the variability in a single data
set.
3-63
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Objective 4
• Use the Empirical Rule to Describe Data that
are Bell Shaped
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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.
3-65
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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 x used in place of μ and s
used in place of σ.
3-66
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3-67
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EXAMPLE Using the Empirical Rule
The following data represent the serum HDL
cholesterol of the 54 female patients of a family
doctor.
41
62
67
60
54
45
3-68
48
75
69
60
54
47
43
77
69
60
55
47
38
58
70
61
56
48
35
82
65
62
56
48
37
39
72
63
56
50
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44
85
74
64
57
52
44
55
74
64
58
52
44
54
74
64
59
53
(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.
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(a) Using a TI-83 plus graphing calculator, we find
  57.4 and   11.7
(b)
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22.3
34.0
45.7
57.4
69.1
80.8
92.5
(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.
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Objective 5
• Use Chebyshev’s Inequality to Describe Any
Set of Data
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Chebyshev’s Inequality
For any data set or distribution, at least
1

 1  2  100% of the observations lie within k
k
standard deviations of the mean, where k is any
number greater than 1. That is, at least
1

 1  2  100% of the data lie between μ – kσ
k
and μ + kσ for k > 1.
Note: We can also use Chebyshev’s Inequality
based on sample data.
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EXAMPLE Using Chebyshev’s Theorem
Using the data from the previous example, use
Chebyshev’s Theorem to
(a) determine the percentage of patients that have serum
HDL within 3 standard deviations of the mean.
1

 1  2  100%  88.9%
3
(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%
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