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.

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 - mean(x) )²

mean(x) 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 x₁ = 4,x₂ = 3, x₅ = 9. If there had been fifty numbers, i would have varied from 1 to 50 and we should proceed as in the example for all fifty 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 = mean x ) then their variance is zero.