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ELEMENTARY
STATISTICS
Section 2-5
Measures of Variation
EIGHTH
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman EDITION
MARIO F. TRIOLA
1
Objective
• Compute measures of
variability.
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
2
Waiting Times of Bank Customers
at Different Banks
in minutes
Jefferson Valley Bank
6.5
6.6
6.7
6.8
7.1
7.3
7.4
7.7
7.7
7.7
Bank of Providence
4.2
5.4
5.8
6.2
6.7
7.7
7.7
8.5
9.3
10.0
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
3
Waiting Times of Bank Customers
at Different Banks
in minutes
Jefferson Valley Bank
6.5
6.6
6.7
6.8
7.1
7.3
7.4
7.7
7.7
7.7
Bank of Providence
4.2
5.4
5.8
6.2
6.7
7.7
7.7
8.5
9.3
10.0
Jefferson Valley Bank
Bank of Providence
Mean
7.15
7.15
Median
7.20
7.20
Mode
7.7
7.7
Midrange
7.10
7.10
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
4
Section 2.5 Measures of Variation
• This last example shows that each bank
has the same measures of center, but a
closer look at the distribution of waiting
times shows that the variability of waiting
times is not the same.
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
5
Dotplots of Waiting Times
Figure 2-14
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
6
Measures of Variation
1. Range
2. Standard Deviation
3. Variance
4. Interquartile Range
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
7
Measures of Variation
Range
greatest
value
least
value
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
8
Range
Jefferson Valley Providence
6.5
4.2
6.6
5.4
6.7
5.8
6.8
6.2
7.1
6.7
7.3
7.7
7.4
7.7
7.7
8.5
7.7
9.3
7.7
10.0
• Jefferson Valley
Range = 7.7 - 6.5 = 1.2 minutes
Providence
Range = 10.0 – 4.2 = 5.8 minutes
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
9
Measures of Variation
Interquartile Range
IQR = Q3 – Q1
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
10
Interquartile Range
(IQR) = Q3-Q1
Jefferson Valley Providence
6.5
4.2
6.6
5.4
6.7
5.8
6.8
6.2
7.1
6.7
7.3
7.7
7.4
7.7
7.7
8.5
7.7
9.3
7.7
10.0
• Jefferson Valley
Median = 7.2
Q1 = 6.7 Q3 = 7.7
IQR = 7.7 - 6.7 = 1.0
Providence
Median = 7.2
Q1 = 5.8 Q3 = 8.5
IQR = 8.5 – 5.8 = 2.7 minutes
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
11
Measures of Variation
Standard Deviation
a measure of variation of the scores
about the mean
(average deviation from the mean)
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
12
Sample Standard Deviation
Formula
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
13
Sample Standard Deviation
Formula
S=
 (x - x)
n-1
2
Formula 2-4
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
14
Computing Standard Deviation
x 71.5

x

 7.15
n
10
1. Compute the mean
x
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
15
Computing
Standard
Deviation
xx
6.5  7.15  0.65
6.6  7.15  0.55
etc...
x
x-x
1. Compute the mean
2. Subtract the mean from each
data value
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
16
Computing Standard Deviation
x  x
2
 0.65   0.4225
2
 0.55   0.3025 2
x x
x etcx-x
... 
2
1. Compute the mean
2. Subtract the mean from each
data value
3. Square the differences
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
17
Computing Standard Deviation
x
x-x
x  x
 x  x 
2
2
1. Compute the mean
2. Subtract the mean from each
data value
3. Square the differences
4. Sum the squared differences
 2.045
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
18
Computing Standard Deviation
x
x-x
x  x
2
1. Compute the mean
2. Subtract the mean from each
data value
3. Square the differences
4. Sum the squared differences
5. Divide the sum by (n-1)
x  x 
2
n 1
2.045
 .22722222
9
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
19
Computing Standard Deviation
x
x-x
x  x
2
1. Compute the mean
2. Subtract the mean from each
data value
3. Square the differences
4. Sum the squared differences
5. Divide the sum by (n-1)
6. Take the square root of this
result
0.227222222  0.476678
 0.48 minutes
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
20
1.
Computing Standard Deviation
Providence Bank #5 pg81
x 71.5

Compute the mean
x

 7.15 minutes
n
10
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
21
Computing Standard Deviation
Providence Bank #5 pg81
1. Compute the mean
2. Subtract the mean from each data value
x  x
4.2  7.15  2.95
5.4  7.15  1.75
etc...
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
22
Computing Standard Deviation
Providence Bank #5 pg81
1. Compute the mean
2. Subtract the mean from
each data value
3. Square the differences
x  x
2
 2.95   8.7025
2

1
.
75

  3.0625
2
etc...
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
23
Computing Standard Deviation
Providence Bank #5 pg81
1.
2.
3.
4.
Compute the mean
Subtract the mean from each data value
Square the differences
Sum the squared differences
 x  x 
2
 29.865
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
24
Computing Standard Deviation
Providence Bank #5 pg81
1.
2.
3.
4.
5.
Compute the mean
Subtract the mean from each data value
Square the differences
Sum the squared differences
Divide the sum by (n-1)
 x  x 
n 1
2
29.865

 3.3183333
9
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
25
Computing Standard Deviation
Providence Bank #5 pg81
1.
2.
3.
4.
5.
6.
Compute the mean
Subtract the mean from each data value
Square the differences
Sum the squared differences
Divide the sum by (n-1)
Take the square root of this result
s
 x  x 
n 1
2
 1.8216  1.82 minutes
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
26
Population Standard Deviation
 =
 (x - µ)
2
N
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
27
Symbols
for Standard Deviation
Sample
Textbook
Some graphics
calculators
Some
non-graphics
calculators
s
Sx
xn-1
Population

x
x n
Book
Some graphics
calculators
Some
non-graphics
calculators
Articles in professional journals and reports often use SD
for standard deviation and VAR for variance.
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
28
Measures of Variation
Variance
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
29
Measures of Variation
Variance
standard deviation squared
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
30
Measures of Variation
Variance
standard deviation squared
}
Notation
s

2
2
use square key
on calculator
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
31
Variance
2
s =

2
=
 (x - x )
2
n-1
 (x - µ)
N
2
Sample
Variance
Population
Variance
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
32
Round-off Rule
for measures of variation
Carry one more decimal place than
is present in the original set of
values.
Round only the final answer, never in the
middle of a calculation.
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
33
• Page 81 3, 5
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
34
Standard Deviation from a
Frequency Table
s
 f  x  x 
n 1
2
where x represents the class midpoints
x is the mean
f is the frequency corresponding to class marks
n is the number of data values
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
35
Standard Deviation from a
Frequency Table
Rating
Frequency
midpoints
f x
0-2
20
1
20
3-5
14
4
56
6-8
15
7
105
9-11
2
10
20
12-14
1
13
13
  f  x   214
f  x  214


and so x 

 4.1153846
52
f
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
36
xx
Rating
Frequency
midpoints
0-2
20
1
-3.1153846
3-5
14
4
-0.11538461
6-8
15
7
2.88461538
9-11
2
10
5.88461538
12-14
1
13
8.88461538
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
37
Standard Deviation from a
Frequency Table
x  x
2
Rating
Frequency
midpoints
0-2
20
1
9.705621301
3-5
14
4
0.013313609
6-8
15
7
8.32100591
9-11
2
10
34.6286982
12-14
1
13
78.9363905
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
38
Standard Deviation from a
Frequency Table
x  x
2
f  x  x 
Rating
Frequency
midpoints
0-2
20
1
9.705621301
194.1124260
3-5
14
4
0.013313609
0.186390532
6-8
15
7
8.32100591
124.815088
9-11
2
10
34.6286982
69.2573964
12-14
1
13
78.9363905
78.9363905
2
2

  f   x  x    467.3076923
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
39
Standard Deviation from a
Frequency Table
2

f

x

x
     467.3076923

n 1
51
 9.162895928 (variance)
2


f

x

x



  9.162895928
s
n 1
 3.027027573  3.0 points
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
40
• Page 81-84 9, 11, 29
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
41
Objective: Understanding Standard
Deviation
• Apply the Empirical Rule
• Apply Chebyshev’s Rule
• Apply Range Rule of Thumb
• Identify Unusual Values
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
42
Estimation of Standard Deviation
Range Rule of Thumb
x - 2s
(minimum
usual value)
x
Range  4s
x + 2s
(maximum
usual value)
or
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
43
Estimation of Standard Deviation
Range Rule of Thumb
x - 2s
x
(minimum
usual value)
Range  4s
x + 2s
(maximum
usual value)
or
s
Range
4
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
44
Estimation of Standard Deviation
Range Rule of Thumb
x - 2s
x + 2s
x
(minimum
usual value)
(maximum
usual value)
Range  4s
or
s
Range
4
=
highest value - lowest value
4
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
45
Usual Sample Values
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
46
Usual Sample Values
minimum ‘usual’ value  (mean) - 2 (standard deviation)
minimum  x - 2(s)
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
47
Usual Sample Values
minimum ‘usual’ value  (mean) - 2 (standard deviation)
minimum  x - 2(s)
maximum ‘usual’ value  (mean) + 2 (standard deviation)
maximum  x + 2(s)
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
48
The Empirical Rule
FIGURE 2-15
(applies to bell-shaped distributions)
x
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
49
The Empirical Rule
FIGURE 2-15
(applies to bell-shaped distributions)
68% within
1 standard deviation
34%
x-s
34%
x
x+s
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
50
The Empirical Rule
FIGURE 2-15
(applies to bell-shaped distributions)
95% within
2 standard deviations
68% within
1 standard deviation
34%
34%
13.5%
x - 2s
13.5%
x-s
x
x+s
x + 2s
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
51
The Empirical Rule
FIGURE 2-15
(applies to bell-shaped distributions)
99.7% of data are within 3 standard deviations of the mean
95% within
2 standard deviations
68% within
1 standard deviation
34%
34%
2.4%
2.4%
0.1%
0.1%
13.5%
x - 3s
x - 2s
13.5%
x-s
x
x+s
x + 2s
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
x + 3s
52
Example Application of the Empirical
Rule
• A set of 1000 test scores has a symmetric,
mound-shaped distribution. The mean is 175
and the standard deviation is 10.
• Approximately what percent of the scores are
between 175 and 195?
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
53
95% of scores are between x  2 s
The shaded area is half
of the area within 2 standard deviations
of the mean .... so (.5)(.95)=.475
175
185
195
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
54
Example Application of the Empirical
Rule
• A set of 1000 test scores has a symmetric,
mound-shaped distribution. The mean is 175
and the standard deviation is 10.
• Approximately how many scores are between 155
and 165?
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
55
The area from 155 to 195 is 0.95
Example Application
The area from 165 toof
185the
is 0.68Empirical
Subtracting these valuesRule
gives the area from 155 to 165
and 185 to 195 combined. We need to divide the result by 2
because the symmetry splits this area equally.
0.95 - 0.68 = 0.27
0.27 divided by 2 = 0.135
155
165
175
185
195
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
56
Chebyshev’s Theorem
 applies to distributions of any shape.
 the proportion (or fraction) of any set of data
lying within K standard deviations of the mean is
2
always at least 1 - 1/K , where K is any positive
number greater than 1.
 at least 3/4 (75%) of all values lie within
2 standard deviations of the mean.
 at least 8/9 (89%) of all values lie within
3 standard deviations of the mean.
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
57
Measures of Variation Summary
For typical data sets, it is unusual for a
score to differ from the mean by more than
2 or 3 standard deviations.
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
58
• Page 81-84 19, 20, 21, 22, 23, 24, 25, 30 a and b
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
59