Download Computing a Population Standard Deviation

Chapter
33
Numerically
Summarizing Data
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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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3-2
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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3-3
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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3-4
(a) The mean wait time in each line is 1.39
minutes.
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3-5
(b)
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3-6
Objective 1
• Compute the range of a variable from raw data
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3-7
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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3-8
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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3-9
Objective 2
• Compute the variance of a variable from raw
data
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3-10
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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3-11
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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3-12
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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3-13
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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3-14
The Computational Formula
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3-15
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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3-16
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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3-17
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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3-18
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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3-19
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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3-20
Objective 3
• Compute the standard deviation of a
variable from raw data
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3-21
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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3-22
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,
902.8571
  2 
 11.4 minutes
7
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3-23
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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3-24
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3-25
EXAMPLE
Comparing Standard Deviations
Determine the standard deviation waiting time
for Wendy’s and McDonald’s. Which is
larger? Why?
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3-26
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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3-27
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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3-28
Objective 4
• Use the Empirical Rule to Describe Data
That Are Bell Shaped
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3-31
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
3-32
(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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(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 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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3-35
Objective 5
• Use Chebyshev’s Inequality to Describe
Any Set of Data
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3-37
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

100%  88.9%

2 
 3 
(b) determine the actual percentage of patients that have serum
HDL between 34 and 80.8.
1

1  2 100%  75%
 2 
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3-38