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# Variance

## Description

Variance is a measure of the spread in a list of numbers. It is commonly represented as s2 or s2 (sigma-squared).

It is calculated as follows for a sample:

VAR(X) = SS/(N-1)

= (SUM(Xi - AVERAGE(X))) / (N-1)

where N is the number of items in the list.

If the list represents a complete population, then the division is by N (not N-1).

VAR(X) = SS/N

= (SUM(Xi - AVERAGE(X))) / N

## Example

A set of measures, as in Column A below, can easily have its variance calculated.

 Column A Column B Column C x x - x-bar (x - x-bar)^2 5 0.25 0.0625 6 1.25 1.5625 2 -2.75 7.5625 8 3.25 10.5625 3 -1.75 3.0625 5 0.25 0.0625 2 -2.75 7.5625 7 2.25 5.0625 number of measures, n: 8 sum of measures: 38 average, x-bar: 4.75 Sum of squares, SS: 35.5 Variance, s^2 = SS/(N-1): 5.071429

## Discussion

Variance is an improvement on Sum of the Squares, SS, as a measure of spread, as it makes the measure independent of the length of the list of numbers.

Dividing SS by (N-1) is due to the 'degrees of freedom' issue.

Variance is a part of many other calculations, from standard deviation to ANOVA.

Variance has the disadvantage that it is measured in squared units as compared with the number in the original list. This is why it is represented as s2, as sigma represents the standard deviation.

In Microsoft Excel, the formula VAR() may be used to automatically calculate the variance of a range of measures.