Transcript Q - rlhawkmath

Chapter
33
Numerically
Summarizing Data
© 2010 Pearson Prentice Hall. All rights reserved
Section 3.1 Measures of Central Tendency
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 statistics to be resistant
4. Determine the mode of a variable from raw data
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3-2
Objective 1
• Determine the arithmetic mean of a
variable from raw data
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3-3
The arithmetic mean of a variable is computed by
determining the sum of all the values of the variable in
the data set divided by the number of observations.
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3-4
The population arithmetic mean is computed using
all the individuals in a population.
The population mean is a parameter.
The population arithmetic mean is denoted by .
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3-5
The sample arithmetic mean is computed using
sample data.
The sample mean is a statistic.
The sample arithmetic mean is denoted by

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x.
3-6
If x1, x2, …, xN are the N observations of a variable
from a population, then the population mean, µ, is
x1  x2 

N
 xN
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3-7
If x1, x2, …, xn are the n observations of a variable
from a sample, then the sample mean, x , is
x1  x2 
x
n
 xn
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3-8
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.
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3-9
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
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3-10
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
5  36  26
3
 22.3
x
36  23  26
3
 28.3
x
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3-11
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3-12
The lengths (in minutes) of a sample of cell
phone calls are shown:
6
19
3
6
12
Find the mean.
A. 54
B. 9
C. 6
D. 7
Slide 3- 13
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Education, Inc.
8
The lengths (in minutes) of a sample of cell
phone calls are shown:
6
19
3
6
12
Find the mean.
A. 54
B. 9
C. 6
D. 7
Slide 3- 14
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Education, Inc.
8
Objective 2
• Determine the median of a variable from
raw data
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3-15
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.
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3-16
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3-17
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
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3-18
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
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3-19
The lengths (in minutes) of a sample of cell
phone calls are shown:
6
19
3
6
12
Find the median.
A. 3.5
B. 9
C. 6
D. 7
Slide 3- 20
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Education, Inc.
8
The lengths (in minutes) of a sample of cell
phone calls are shown:
6
19
3
6
12
Find the median.
A. 3.5
B. 9
C. 6
D. 7
Slide 3- 21
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Education, Inc.
8
Objective 3
• Explain what it means for a statistic to be
resistant
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3-22
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
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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.
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3-24
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3-25
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
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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.
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3-28
Asking Price of Homes in Lincoln, NE
12
10
Frequency
8
6
4
2
0
100000
150000
200000
250000
Asking Price
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300000
350000
3-29
Objective 4
• Determine the mode of a variable from raw
data
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The mode of a variable is the most frequent
observation of the variable that occurs in the
data set.
If there is no observation that occurs with the
most frequency, we say the data has no
mode.
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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.
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The mode is
New York.
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Tally data to determine most
frequent observation
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3-36
The lengths (in minutes) of a sample of cell
phone calls are shown:
6
19
3
6
12
Find the mode.
A. 3
B. 9
C. 6
D. 7
Slide 3- 37
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Education, Inc.
8
The lengths (in minutes) of a sample of cell
phone calls are shown:
6
19
3
6
12
Find the mode.
A. 3
B. 9
C. 6
D. 7
Slide 3- 38
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Education, Inc.
8
Section 3.2 Measures of Dispersion
Objectives
1. Compute the range of a variable from raw data
2. Compute the variance of a variable from raw data
3. Compute the standard deviation 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
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3-39
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?
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3-40
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
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
3.04
2.54
0.50
1.17
0.23
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3-41
(a) The mean wait time in each line is 1.39
minutes.
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3-42
(b)
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3-43
Objective 1
• Compute the range of a variable from raw data
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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
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3-45
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
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3-46
The lengths (in minutes) of a sample of cell
phone calls are shown:
6
19
3
6
12
Find the range.
A. 2
B. 16
C. 13
D. 9
Slide 3- 47
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Education, Inc.
8
The lengths (in minutes) of a sample of cell
phone calls are shown:
6
19
3
6
12
Find the range.
A. 2
B. 16
C. 13
D. 9
Slide 3- 48
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Education, Inc.
8
Objective 2
• Compute the variance of a variable from raw
data
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The population variance of a variable is the sum of
squared deviations about the population mean divided
by the number of observations in the population, N.
That is it is the mean of the sum of the squared
deviations about the population mean.
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3-50
The population variance is symbolically represented
by σ2 (lower case Greek sigma squared).
Note: When using the above formula, do not round until the
last computation. Use as many decimals as allowed by your
calculator in order to avoid round off errors.
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3-51
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 variance of this data. Recall that
174

 24.85714
7
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3-52
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

2
x  


i
N
2
2

902.8571
902.8571

 129.0 minutes2
7
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3-53
The Computational Formula
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3-54
EXAMPLE Computing a Population Variance
Using the Computational Formula
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 variance of this data using the
computational formula.
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3-55
23, 36, 23, 18, 5, 26, 43
2
2
2
2
x

23

36

...

43
 5228
i
x
i
 23  36  ...  43  174
2 
2
x
i
x



i
N
N
2
1742
5228 
7

7
 129.0
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3-56
The sample variance is computed by determining the
sum of squared deviations about the sample mean and
then dividing this result by n – 1.
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3-57
Note: Whenever a statistic consistently overestimates or
underestimates a parameter, it is called biased. To obtain an
unbiased estimate of the population variance, we divide the
sum of the squared deviations about the mean by n - 1.
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3-58
EXAMPLE Computing a Sample Variance
In Section 3.1, we obtained the following simple random sample for the
travel time data: 5, 36, 26.
Compute the sample variance travel time.
Travel Time, xi
Sample Mean,
Deviation about the
Mean,
Squared Deviations about the
Mean,
 x  x
2
x
xi  x
5
22.333
5 – 22.333
= -17.333
(-17.333)2 = 300.432889
36
22.333
13.667
186.786889
26
22.333
3.667
13.446889
i
 x  x
i
s
2
x  x



i
n 1
2
 500.66667
2

500.66667
3 1
 250.333 square minutes
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3-59
Objective 3
• Compute the standard deviation of a
variable from raw data
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3-60
The population standard deviation is denoted by
It is obtained by taking the square root of the population
variance, so that
The sample standard deviation is denoted by
s
It is obtained by taking the square root of the sample variance, so
that
s  s2
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3-61
EXAMPLE
Computing a Population Standard
Deviation
The following data represent the travel times (in minutes) to
work for all seven employees of a start-up web development
company.
23, 36, 23, 18, 5, 26, 43
Compute the population standard deviation of this data.
Recall, from the last objective that σ2 = 129.0 minutes2.
Therefore,
  2 
902.8571
 11.4 minutes
7
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3-62
EXAMPLE Computing a Sample Standard
Deviation
Recall the sample data 5, 26, 36 results in a sample variance of
s2 

xi  x
n 1

2

500.66667
3 1
 250.333 square minutes
Use this result to determine the sample standard deviation.
s  s2 
500.666667
 15.8 minutes
3 1
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3-63
The lengths (in minutes) of a sample of cell
phone calls are shown:
6
19
3
6
12
Find the standard deviation.
A. 5.7
B. 5.2
C. 32.8
D. 16
Slide 3- 64
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Education, Inc.
8
The lengths (in minutes) of a sample of cell
phone calls are shown:
6
19
3
6
12
Find the standard deviation.
A. 5.7
B. 5.2
C. 32.8
D. 16
Slide 3- 65
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Education, Inc.
8
EXAMPLE
Comparing Standard Deviations
Determine the standard deviation waiting time
for Wendy’s and McDonald’s. Which is
larger? Why?
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3-66
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
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
3.04
2.54
0.50
1.17
0.23
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3-67
EXAMPLE
Comparing Standard Deviations
Determine the standard deviation waiting time
for Wendy’s and McDonald’s. Which is
larger? Why?
Sample standard deviation for Wendy’s:
0.738 minutes
Sample standard deviation for McDonald’s:
1.265 minutes
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3-68
Objective 4
• Use the Empirical Rule to Describe Data
That Are Bell Shaped
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3-69
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3-70
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3-71
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
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
44
85
74
64
57
52
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44
55
74
64
58
52
44
54
74
64
59
53
3-72
(a) Compute the population mean and standard
deviation.
(b) Draw a histogram to verify the data is bell-shaped.
(c) Determine the percentage of patients that have
serum HDL within 3 standard deviations of the mean
according to the Empirical Rule.
(d) Determine the percentage of 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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3-73
(a) Using a TI-83 plus graphing calculator, we find
  57.4 and  11.7
(b)
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3-74
22.3
34.0
45.7
57.4
69.1
80.8
92.5
(c) According to the Empirical Rule, 99.7% of the patients that have serum HDL
within 3 standard deviations of the mean.
(d) 13.5% + 34% + 34% = 81.5% of 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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3-75
The mean commute time in the U.S. is 24.4
minutes with a standard deviation of 6.5
minutes. What is the minimum percentage of
commuters that have commute times
between 11.4 minutes and 37.4 minutes?
A. 68%
B. 75%
C. 89%
D. 95%
Slide 3- 76
Copyright © 2010 Pearson
Education, Inc.
The mean commute time in the U.S. is 24.4
minutes with a standard deviation of 6.5
minutes. What is the minimum percentage of
commuters that have commute times
between 11.4 minutes and 37.4 minutes?
A. 68%
B. 75%
C. 89%
D. 95%
Slide 3- 77
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Education, Inc.
Objective 5
• Use Chebyshev’s Inequality to Describe
Any Set of Data
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3-78
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3-79
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
 3

100%  88.9%

(b) determine the actual percentage of patients that have serum
HDL between 34 and 80.8.
1

1  2
 2

100%  75%

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3-80
Section 3.3 Measures of Central Tendency and
Dispersion from Grouped Data
Objectives
1. Approximate the mean of a variable from grouped
data
2. Compute the weighted mean
3. Approximate the variance and standard deviation of
a variable from grouped data
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3-81
Objective 1
• Approximate the Mean of a Variable from
Grouped Data
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3-82
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3-83
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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3-84
Time
Frequency
xi
x i fi
0
0
0
0
1-5
130
3.5
455
6 - 10
250
8.5
2125
11 - 14
230
13.5
3105
16 - 20
180
18.5
3330
21 - 25
100
23.5
2350
26 – 30
60
28.5
1710
31 – 35
50
33.5
1675
f
i
x f
 1000
x
i i
 14,750
x f
f
i i
i
14, 750
1000
 14.75

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3-85
Approximate the mean of the frequency
distribution.
A. 13
B. 19.5
C. 9.5
D. 12.5
Slide 3- 86
Class
1 – 6.99
Frequency, f
21
7 – 12.99
13 – 18.99
19 – 24.99
16
28
13
Copyright © 2010 Pearson
Education, Inc.
Approximate the mean of the frequency
distribution.
A. 13
B. 19.5
C. 9.5
D. 12.5
Slide 3- 87
Class
1 – 6.99
Frequency, f
21
7 – 12.99
13 – 18.99
19 – 24.99
16
28
13
Copyright © 2010 Pearson
Education, Inc.
Objective 2
• Compute the Weighted Mean
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3-88
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3-89
EXAMPLE Computed a Weighted Mean
Bob goes 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)
1  2  1.5
$15.85

4.5
 $3.52
xw 
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3-90
Objective 3
• Approximate the Variance and Standard
Deviation of a Variable from Grouped Data
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3-91
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3-92
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 variance and 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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3-93
Time
Frequency
0
0
1-5
 x  x
i
2
fi
130
x
0
3.5
0
16453.13
6 - 10
250
8.5
9765.625
11 - 14
230
13.5
359.375
16 - 20
180
18.5
2531.25
21 - 25
100
23.5
7656.25
26 - 30
60
28.5
11343.75
31 - 35
50
33.5
17578.13
 fi  1000
x  x f



 f  1
 x  x
2
i
fi  65,687.5
2
s
2
i
i
65, 687.5
1000  1
 65.8

i
s  s2
65, 687.5
999
 8.1 hours

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3-94
Section 3.4 Measures of Position and Outliers
Objectives
1.
2.
3.
4.
5.
Determine and interpret z-scores
Interpret percentiles
Determine and interpret quartiles
Determine and interpret the interquartile range
Check a set of data for outliers
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3-95
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3-96
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 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
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3-97
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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3-98
The mean commute time in the U.S. is 24.4
minutes with a standard deviation of 6.5
minutes. Find the z-score that corresponds
to a commute time of 15 minutes.
A. 1.45
B. –1.45
C. 11.25
D. –9.4
Slide 3- 99
Copyright © 2010 Pearson
Education, Inc.
The mean commute time in the U.S. is 24.4
minutes with a standard deviation of 6.5
minutes. Find the z-score that corresponds
to a commute time of 15 minutes.
A. 1.45
B. –1.45
C. 11.25
D. –9.4
Slide 3- 100
Copyright © 2010 Pearson
Education, Inc.
The mean commute time in the U.S. is 24.4
minutes with a standard deviation of 6.5
minutes. Find the z-score that corresponds
to a commute time of 15 minutes.
A. 1.45
B. –1.45
C. 11.25
D. –9.4
Slide 3- 101
Copyright © 2010 Pearson
Education, Inc.
The mean commute time in the U.S. is 24.4
minutes with a standard deviation of 6.5
minutes. Find the z-score that corresponds
to a commute time of 15 minutes.
A. 1.45
B. –1.45
C. 11.25
D. –9.4
Slide 3- 102
Copyright © 2010 Pearson
Education, Inc.
Objective 2
• Interpret Percentiles
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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=101.)
Interpret this admissions requirement.
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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3-105
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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© 2010 Pearson Prentice Hall. All rights reserved
3-108
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.
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
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3-112
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.
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3-113
Suppose a 15th car travels through the construction zone at 100 miles per
hour. How does this value impact the mean, median, standard deviation, and
interquartile range?
Without 15th car
With 15th car
Mean
32.1 mph
36.7 mph
Median
32.5 mph
33 mph
Standard deviation
6.2 mph
18.5 mph
IQR
10 mph
11 mph
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3-114
The closing prices for 9 telecommunications
stocks are shown below. Compute the
interquartile range, IQR.
3.14 5.70 6.72
31.24 40.87 71.64
15.63
A. 29.845
B. 68.32
C. 6.21
D. 36.055
Slide 3- 115
Copyright © 2010 Pearson
Education, Inc.
17.75
28.12
The closing prices for 9 telecommunications
stocks are shown below. Compute the
interquartile range, IQR.
3.14 5.70 6.72
31.24 40.87 71.64
15.63
A. 29.845
B. 68.32
C. 6.21
D. 36.055
Slide 3- 116
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Education, Inc.
17.75
28.12
Objective 5
• Check a Set of Data for Outliers
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3-117
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3-118
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)
Upper Fence = Q3 + 1.5(IQR)
= 28 – 1.5(10)
= 38 + 1.5(10)
= 13 mph
= 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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3-119
Section 3.5 The Five-Number Summary and Boxplots
Objectives
1. Compute the five-number summary
2. Draw and interpret boxplots
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3-120
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3-121
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.
First, we write the data is
Institution
Pulaski Bank and Trust Company
Rate
6.5%
Rainier Pacific Savings Bank
12.0%
Wells Fargo Bank NA
14.4%
Firstbank of Colorado
14.4%
Lafayette Ambassador Bank
14.3%
Infibank
13.0%
United Bank, Inc.
13.3%
First National Bank of The Mid-Cities
13.9%
Bank of Louisiana
Bar Harbor Bank and Trust Company
9.9%
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:
14.5%
6.5% 12.0% 13.6% 14.4%
Source:
14.5%
http://www.federalreserve.gov/pubs/SHOP/survey.htm
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3-122
Objective 2
• Draw and interpret boxplots
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3-124
EXAMPLE
Constructing a Boxplot
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. Draw a boxplot
of the data.
Institution
Pulaski Bank and Trust Company
Rate
6.5%
Rainier Pacific Savings Bank
12.0%
Wells Fargo Bank NA
14.4%
Firstbank of Colorado
14.4%
Lafayette Ambassador Bank
14.3%
Infibank
13.0%
United Bank, Inc.
13.3%
First National Bank of The Mid-Cities
13.9%
Bank of Louisiana
Bar Harbor Bank and Trust Company
9.9%
14.5%
Source:
http://www.federalreserve.gov/pubs/SHOP/survey.htm
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3-125
Step 1: The interquartile range (IQR) is 14.4% - 12% = 2.4%. The lower and
upper fences are:
Lower Fence = Q1 – 1.5(IQR)
Upper Fence = Q3 + 1.5(IQR)
= 12 – 1.5(2.4)
= 14.4 + 1.5(2.4)
= 8.4%
= 18.0%
Step 2:
*
[
]
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3-126
Objective 3
• Use a boxplot and quartiles to describe the
shape of a distribution
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3-127
The interest rate boxplot indicates that the distribution is skewed left.
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3-128
Use the boxplot to identify the first quartile.
10
|
18
|
|
|
|
|
|
24
26
|
|
10 12 14 16 18 20 22 24 26 28 30
A. 10
B. 18
C. 24
D. 26
Slide 3- 129
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Education, Inc.
30
|
|
Use the boxplot to identify the first quartile.
10
|
18
|
|
|
|
|
|
24
26
|
|
10 12 14 16 18 20 22 24 26 28 30
A. 10
B. 18
C. 24
D. 26
Slide 3- 130
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Education, Inc.
30
|
|