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
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
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
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
3
Dotplots of Waiting Times
Figure 2-14
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
4
Measures of Variation
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
5
Measures of Variation
Range
highest
value
lowest
value
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
6
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
7
Sample Standard Deviation
Formula
 (x - x)
n-1
2
S=
Formula 2-4
calculators can compute the
sample standard deviation of data
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
8
Sample Standard Deviation
Shortcut Formula
n (x ) - (x)
n (n - 1)
2
s=
2
Formula 2-5
calculators can compute the
sample standard deviation of data
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
9
Mean Absolute Deviation
Formula
 x-x
n
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
10
Population Standard Deviation
 =
 (x - µ)
2
N
calculators can compute the
population standard deviation
of data
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
11
Symbols
for Standard Deviation
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
12
Symbols
for Standard Deviation
Sample
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
13
Symbols
for Standard Deviation
Sample
s
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
14
Symbols
for Standard Deviation
Sample
Textbook
s
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
15
Symbols
for Standard Deviation
Sample
Textbook
s
Sx
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
16
Symbols
for Standard Deviation
Sample
Textbook
Some graphics
calculators
s
Sx
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
17
Symbols
for Standard Deviation
Sample
Textbook
Some graphics
calculators
s
Sx
xn-1
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
18
Symbols
for Standard Deviation
Sample
Textbook
Some graphics
calculators
Some
non-graphics
calculators
s
Sx
xn-1
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
19
Symbols
for Standard Deviation
Sample
Textbook
Some graphics
calculators
Some
non-graphics
calculators
Population
s
Sx
xn-1
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
20
Symbols
for Standard Deviation
Sample
Textbook
Some graphics
calculators
Some
non-graphics
calculators
s
Sx
xn-1
Population

x
xn
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
21
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
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
22
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
23
Measures of Variation
Variance
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
24
Measures of Variation
Variance
standard deviation squared
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
25
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
26
Variance
 (x - x )
2
2
s =

n-1
 (x - µ)
2
2
=
N
Sample
Variance
Population
Variance
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
27
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
28
Standard Deviation from a
Frequency Table
Formula 2-6
n [(f • x 2)] -[(f • x)]2
S=
n (n - 1)
Use the class midpoints as the x values
Calculators can compute the standard deviation for
frequency table
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
29
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
30
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
31
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
32
Usual Sample Values
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
33
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
34
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
35
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
36
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
37
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
38
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
39
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
40