Download lect2-3 - Rutgers Statistics

Measures of Variance
Section 2-5
M A R I O F. T R I O L A
Copyright © 1998, Triola, Elementary Statistics
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
Addison
Wesley
Longman
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
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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
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Measure of Variation
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Measure of Variation
Range
lowest
score
highest
score
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Measure of Variation
Standard Deviation
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Deviation from the mean: x – x
a measure of variation of the scores
about the mean
(the average deviation from the mean is zero)
x
–
x
S n =0
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Sample Standard Deviation
Formula
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Sample Standard Deviation
Formula
S=
S (x – x)
n–1
2
Formula 2 -4
calculators can calculate sample standard
deviation of data
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Same Means (x = 4)
Different Standard Deviations
FIGURE
Frequency
s=0
7
6
5
4
3
2
s = 0.8
s = 1.0
s = 3.0
1
1 2 3 4 5 6 7
1 2 3 4 5 6 7
1 2 3 4 5 6 7
1 2 3 4 5 6 7
Standard deviation gets larger as spread of data increases.
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Mean Deviation Formula
(absolute deviation)
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Mean Deviation Formula
(absolute deviation)
S x–x
n
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Population Standard Deviation
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Population Standard Deviation
s =
S (x – µ)
N
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Population Standard Deviation
s =
S (x – µ)
N
2
calculators can calculate the
population standard deviation
of data
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Examples
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Measure of Variation
Variance
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Measure of Variation
Variance
standard deviation squared
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Measure of Variation
Variance
standard deviation squared
}
Notation
s
s
2
2
use square key
on calculator
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Variance
S (x – x) Sample
s =
Variance
n–1
2
2
S (x – µ) Population
s=
Variance
N
2
2
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Round-off Rule
for measures of variation
Carry one more decimal place than
was present in the original data
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Standard Deviation
Shortcut Formula
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Standard Deviation
Shortcut Formula
n (S x ) – (S x)
n (n – 1)
2
s=
2
Formula 2 - 6
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Standard Deviation for Group Data
(Frequency Table)
S f (x – x)2
S=
n–1
where x = class mark
n [S( f • x )] – [S( f • x)]
2
S=
2
n (n – 1)
shortcut
Formula 2-7
calculators can calculate the standard deviation of grouped data
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Examples
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Range Rule of Thumb
(minimum) x – 2s
x
x + 2s (maximum)
Range  4s
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Range Rule of Thumb
(minimum) x – 2s
x + 2s (maximum)
x
Range  4s
or
s
Range
4
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FIGURE
The Empirical Rule
(applies to bell shaped distributions)
x
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FIGURE
The Empirical Rule
(applies to bell shaped distributions)
68% within
1 standard deviation
0.340
x–s
0.340
x
x+s
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FIGURE
The Empirical Rule
(applies to bell shaped distributions)
95% within
2 standard deviations
68% within
1 standard deviation
0.340
0.340
0.135
x – 2s
0.135
x–s
x
x+s
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x + 2s
30
The Empirical Rule
FIGURE
(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
0.340
0.340
0.024
0.024
0.001
0.001
0.135
x – 3s
x – 2s
0.135
x–s
x
x+s
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x + 2s
x + 3s
31
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 always at least 1 – 1/k2, where
k is any positive number greater than 1.
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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.
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