Download 1 §3.3 Measures of Variation The range is the highest value minus

§3.3
Measures of Variation
1
The range is the highest value minus the lowest value.
R = highest value - lowest value
Example
Sample A
10
60
50
30
40
20
Sample B
35
45
30
35
40
25
2
1
The population variance is the average of the squares of the
distance each data value is from the mean, given by:
The population standard deviation is the square root of the
variance:
3
The sample variance is given by the formula:
The sample standard deviation is given by:
4
2
Example: A sample of six households is made, in which
the # of automobiles of each household is given
X
1
2
2
2
3
1
5
“Shortcut calculations”
6
3
Variance and Standard Deviation for Grouped Data
Class Limits
55 - 65
66 - 76
77 - 87
88 - 98
99 - 109
110 - 120
Mid point, Xm
60
71
82
93
104
115
Frequency, f
10
8
22
10
7
3
60
f*Xm
f*Xm2
600.0
568.0
1804.0
930.0
728.0
345.0
4975.0
36000
50410
67240
86490
108160
132250
480550.0
7
The coefficient of variation is the standard deviation
divided by the mean and expressed as a percent.
For samples,
For populations,
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4
Chebyshev’s theorem
The proportion of values from a data set that fall within k standard
deviations of the mean will be at least 1 - 1/k2, where k is a number
greater than 1. (k does not have to be an integer)
Thus,
when k = 2, at least 3/4 (75%) of data lies with 2 standard
deviations of mean
when k = 3, at least 8/9 of data lies with 3 standard deviations of
mean
etc.
9
#38
The average score on a test has a mean of 53 and a standard
deviation of 6. Using Chevyshev’s theorem, find the range of
scores in which a least 75% of the scores will lie.
10
5
11
6