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Measures of Variation
Statistical measures of variation are
numerical values that indicate the variability inherent in a set
of data measurements. The most common measures of variation are
the range, variance and standard distribution.
Range
The range of a set of observations is the
absolute value of the difference between the largest and smallest
values in the set. It measures the size of the smallest
contiguous interval of real numbers that encompasses all the data
values.
EX. Given the following sorted data:
1.2, 1.5, 1.9, 2.4, 2.4, 2.5, 2.6, 3.0, 3.5, 3.8
The range of this set of data is 3.8 - 1.2 = 2.6.
Variance and Standard Deviation
The variance of a set of data is a cumulative
measure of the squares of the difference of all the data values
from the mean.
The population and sample variance are calculated as follows:
Given the set of data values x1, x2,
.... xN from a finite population of size N, the
population variance is calculated as
Given the set of data values x1, x2,
.... xn from a sample of size n, the sample
variance s2 is calculated as
Note that the population variance is simply the arithmetic
mean of the squares of the difference between each data value in
the population and the mean. On the other hand, the formula for
the sample variance is similar to the formula for the population
variance, except that the denominator in the fraction is (n-1)
instead of n. Using the above formula, the sample variance is
statistically proven to be a most effective estimator for the
variance of the population to which the sample belongs.
The standard deviation of a set of data is
the positive square root of the variance.
EX. Given the following sorted data:
1.2, 1.5, 1.9, 2.4, 2.4, 2.5, 2.6, 3.0, 3.5, 3.8
x = 2.48 as computed earlier
s2 = ( 1 / (10-1)) × ( (1.2 - 2.48)2 +
(1.5 - 2.48)2 + (1.9 - 2.48)2 + (2.4 -
2.48)2 + (2.4 - 2.48)2 + (2.5 - 2.48)2
+ (2.6 - 2.48)2 + (3.0 - 2.48)2 + (3.5 -
2.48)2 + (3.8 - 2.48)2 )
= (1 / 9) × (1.6384 + 0.9604 + 0.3364 + 0.0064 + 0.0064 +
0.0004 + 0.0144 + 0.2704 + 1.0404 + 1.7424)
= 0.6684
s = ( 0.6684 )1/2 = 0.8176
The sample variance can also be calculated as follows:
EX. Given the above data, we can calculate s2 using
the above formula:
= 1.44 + 2.25 + 3.61 + 5.76 + 5.76 + 6.25 + 6.76 + 9.00 +
12.25 + 14.44
= 67.52
s2 = ( 1 / (10 × 9) ) × ( 10 × 67.52 -
(24.8)2 )
= 0.6684
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