Transcript Ch 2.5 PowerPt

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
Range
highest
value
lowest
value
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
5
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
6
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
7
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
8
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
9
Measures of Variation
Variance
standard deviation squared
}
Notation
s

2
2
Sample Variance
Population Variance
Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
10
Variance Formulas
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
11
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
12
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
13
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
14
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
15
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
16
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
17
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
18
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
19
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
20
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
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