## Variance

The variability of a set of numbers can be expressed as a variance. Take a set of numbers, such as 4, 3, 7, 2, 9.

Here is how to calculate their variance:

1 Calculate the mean:

mean = (4+3+7+2+9)/5 = 5

2 For each number, calculate the square of its deviation from the mean. For the first number, 4, it is (5-4)^{2 = 1. We do likewise for all five numbers.}

3 Add up the sum of the squared deviations from the mean; for the five numbers, it is 1 + 4 + 4 + 9 + 16 = 34.

4 Divide the sum by the n-1; n = 5 in this case.

variance = 34/4 = 8.5

The general formula for the variance of a character X is

V(x) = 1/ (n - 1) ∑ (X_{i} - X_{m})²

x_{m} is the mean, x_{i} is a standard notation for a set of numbers. Here we have five numbers. In terms of the notation, that means that i can have any value from 1 to 5 and is the value of the character for each i. Thus x_{1} = 4, x_{2} = 3, x_{5} = 9. If there had been 50 numbers, i would have varied from 1 to 50 and we should proceed as in the example for all 50 numbers.

The variance describes how variable the set of numbers is: the higher the variance, the greater the differences among the numbers. If all the numbers were the same (all x_{i} = x_{m}) then their variance is zero. See also covariance.

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