Download Chapter 3

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  xN  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.
1
2
3
4
5
6
7
23, 36, 23, 18, 5, 26, 43
36  23  26
3
 28.3
x
3-12
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5  36  26
3
 22.3
x
3-13
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Objective 2
• Determine the Median of a Variable from Raw
Data
3-14
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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.
3-15
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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.
3-16
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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.
3-17
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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
3-18
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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
3-19
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Objective 3
• Explain What it Means for a Statistic to Be
Resistant
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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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.
3-22
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3-23
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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
3-24
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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.
3-25
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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.
3-26
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Asking Price of Homes in Lincoln, NE
12
10
Frequency
8
6
4
2
0
3-27
100000
150000
200000
250000
Asking Price
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
300000
350000
Objective 4
• Determine the Mode of a Variable from Raw
Data
3-28
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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.
3-29
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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.
3-30
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Joe Biden
3-31
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Pennsylvani
a
3-32
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The mode is
New York.
3-33
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Tally data to determine
most frequent observation
3-34
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Section
3.2
Measures of
Dispersion
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
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
3-36
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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?
3-37
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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
3-38
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.
3-39
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(b)
3-40
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Objective 1
• Determine the Range of a Variable from Raw
Data
3-41
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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
3-42
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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
3-43
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Objective 2
• Determine the Standard Deviation of a
Variable from Raw Data
3-44
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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).
3-45
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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.
3-46
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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

3-47
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.
3-48
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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

3-49
 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
3-50
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
s
 x
2
i
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.
3-51
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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
3-52
x
2
i
i
n 1
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
n
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.
3-53
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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.
3-54
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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
3-55
 x
i
 x
n 1
2

500.66667
 15.82 minutes
2
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
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
3-56
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3-57
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EXAMPLE
Comparing Standard Deviations
Determine the standard deviation waiting time
for Wendy’s and McDonald’s. Which is
larger? Why?
3-58
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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
3-59
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
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
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.
3-60
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Objective 3
• Determine the Variance of a Variable from
Raw Data
3-61
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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.
3-62
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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.
3-63
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EXAMPLE
Computing a Population Variance
Recall that the population standard deviation (from
slide #49) is σ = 11.36 so the population variance is
σ2 = 129.05 minutes
and that the sample standard deviation (from slide
#55) is s = 15.82, so the sample variance is
s2 = 250.27 minutes
3-64
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Objective 4
• Use the Empirical Rule to Describe Data That
Are Bell Shaped
3-65
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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.
3-66
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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.
3-67
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3-68
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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
3-69
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.
3-72
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Objective 5
• Use Chebyshev’s Inequality to Describe Any
Set of Data
3-73
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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.
3-74
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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%
3-75
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Section
3.3
Measures of
Central Tendency
and
Dispersion from
Grouped Data
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
Objectives
1. Approximate the mean of a variable from
grouped data
2. Compute the weighted mean
3. Approximate the standard deviation of a
variable from grouped data
3-77
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Objective 1
• Approximate the Mean of a Variable from
Grouped Data
3-78
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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.
3-79
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Approximate the Mean of a Variable from a
Frequency Distribution
Population Mean
Sample Mean
xf


f
xf

x
f
x1 f1  x2 f2  ...  xn fn

f1  f2  ...  fn
x1 f1  x2 f2  ...  xn fn

f1  f2  ...  fn
i i
i
i i
i
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
3-80
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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
0
1-5 6-10 11-15 16-20 21-25 26-30 31-35
Frequency
0 130 250
230
180
100
60
50
Source:http://nsse.iub.edu/NSSE_2007_Annual_Report/docs/withhold/NSSE_2007_Annual_Report.pdf
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Time Frequency
0
0
1-5
130
6 - 10
250
11 - 15
230
16 - 20
180
21 - 25
100
26 – 30
60
31 – 35
50
 fi  1000
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xi
xi fi
0
0
3
390
8
2000
13
2990
18
3240
23
2300
28
1680
33
1650
 xi fi  14,250
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
xf

x
f
i i
i
14, 250

1000
 14.25
Objective 2
• Compute the Weighted Mean
3-83
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The weighted mean, xw , 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
xw
wx


w
i i
i
w1 x1  w2 x2  ...  wn xn

w1  w2  ...  wn
where w is the weight of the ith observation
xi is the value of the ith observation
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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.
1($1.25)  2($3.25)  1.5($5.40)
xw 
1  2  1.5
$15.85

 $3.52
4.5
3-85
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Objective 3
• Approximate the Standard Deviation of a
Variable from Grouped Data
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Approximate the Standard Deviation of a
Variable from a Frequency Distribution
Population
Standard Deviation
 x   
f
2

i
fi
Sample
Standard Deviation
 x  x  f
 f  1
2
s
i
i
i
i
where xi is the midpoint or value of the ith class
fi is the frequency of the ith class
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An algebraically equivalent formula for the
population standard deviation is
x f


f 
2
x
i
2
i i
f
f
i
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i
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
0
1-5 6-10 11-15 16-20 21-25 26-30 31-35
Frequency
0 130 250
230
180
100
60
50
Source:http://nsse.iub.edu/NSSE_2007_Annual_Report/docs/withhold/NSSE_2007_Annual_Report.pdf
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Time
0
1-5
6 - 10
11 - 15
16 - 20
21 - 25
26 – 30
31 – 35
3-90
Frequ
ency xi
0
0
130 3
250 8
230 13
180 18
100 23
60 28
50 33
 fi  1000
xi  x
0
–11.25
–6.25
–1.25
3.75
8.75
13.75
18.75
 
 
xi  x f i s 2   x i  x f i
0
 fi  1
16,453.125
65,687.5

9765.625
1000  1
 65.8
359.375
2531.25
7656.25 s  s 2  65.8
11,343.75  8.1 hours
17,578.125
2
 xi  x fi  65,687.5

2

2

Copyright © 2013, 2010 and 2007 Pearson Education, Inc.

Section
3.4
Measures of
Position and
Outliers
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
Objectives
1.
2.
3.
4.
Determine and interpret z-scores
Interpret percentiles
Determine and interpret quartiles
Determine and interpret the interquartile
range
5. Check a set of data for outliers
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Objective 1
•
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Determine and Interpret z-scores
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
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
x
xx
z
z

s
The z-score is unitless. It has mean 0 and standard
deviation 1.
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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
3-95
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83  69.1
zkg 
2.8
 4.96
76  63.7
zcp 
2.7
 4.56
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.
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Objective 2
• Interpret Percentiles
3-97
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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.
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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:
http://www.publichealth.pitt.edu/interior.php?pageID=1
01.)
Interpret this admissions requirement.
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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%.
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Objective 3
• Determine and Interpret Quartiles
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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.
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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.
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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.
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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.
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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.
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Objective 4
• Determine and Interpret the Interquartile
Range
3-107
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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
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EXAMPLE
Determining and Interpreting the
Interquartile Range
Determine and interpret the interquartile range of the
speed data.
Q1 = 28
Q3 = 38
IQR  Q3  Q1
 38  28
 10
The range of the middle 50% of the speed of cars
traveling through the construction zone is 10 miles per
hour.
3-109
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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?
Mean
Median
Standard deviation
IQR
3-110
Without 15th car
32.1 mph
32.5 mph
6.2 mph
10 mph
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
With 15th car
36.7 mph
33 mph
18.5 mph
11 mph
Objective 5
• Check a Set of Data for Outliers
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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 = Q3 + 1.5(IQR)
Step 4 If a data value is less than the lower fence
or greater than the upper fence, it is
considered an outlier.
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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 = Q3 + 1.5(IQR) = 38 + 1.5(10) = 53 mph
Step 4: There are no values less than 13 mph or greater
than 53 mph. Therefore, there are no outliers.
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Section
3.5
The Five-Number
Summary and
Boxplots
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
Objectives
1. Compute the five-number summary
2. Draw and interpret boxplots
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Objective 1
•
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Compute the Five-Number Summary
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
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:
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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.
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EXAMPLE
Obtaining the Five-Number Summary
Institution
Pulaski Bank and Trust Company
Rainier Pacific Savings Bank
Wells Fargo Bank NA
Firstbank of Colorado
Lafayette Ambassador Bank
Infibank
United Bank, Inc.
First National Bank of The Mid-Cities
Bank of Louisiana
Bar Harbor Bank and Trust Company
Source: http://www.federalreserve.gov/pubs/SHOP/survey.htm
3-119
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Rate
6.5%
12.0%
14.4%
14.4%
14.3%
13.0%
13.3%
13.9%
9.9%
14.5%
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%
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12.0%
13.6%
14.4%
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14.5%
Objective 2
• Draw and Interpret Boxplots
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Drawing a Boxplot
Step 1 Determine the lower and upper fences.
Lower Fence = Q1 – 1.5(IQR)
Upper Fence = Q3 + 1.5(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.
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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 (*).
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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.
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EXAMPLE
Obtaining the Five-Number Summary
Institution
Pulaski Bank and Trust Company
Rainier Pacific Savings Bank
Wells Fargo Bank NA
Firstbank of Colorado
Lafayette Ambassador Bank
Infibank
United Bank, Inc.
First National Bank of The Mid-Cities
Bank of Louisiana
Bar Harbor Bank and Trust Company
Source: http://www.federalreserve.gov/pubs/SHOP/survey.htm
3-125
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Rate
6.5%
12.0%
14.4%
14.4%
14.3%
13.0%
13.3%
13.9%
9.9%
14.5%
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 = Q3 + 1.5(IQR) = 14.4 + 1.5(2.4) = 18.0%
Step 2:
*
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[
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
]
Use a boxplot and quartiles to describe the shape of a
distribution.
The interest rate boxplot indicates that the distribution
is skewed left.
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