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.

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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 x

_{1}, x_{2}, .... x_{N}from a finite population of size N, the population variance is calculated asÂ

Given the set of data values x

_{1}, x_{2}, .... x_{n}from a sample of size n, the sample variance s_{2}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

s^{2} = ( 1 / (10-1)) Ã— ( (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) Ã— (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 s^{2} 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

s^{2 }= ( 1 / (10 Ã— 9) ) Ã— ( 10 Ã— 67.52 - (24.8)^{2} )

= 0.6684