# Variance

The variance is a measure of the extent of variation in a set of numbers. Where all the numbers are the same, the variance is 0. The greater the variation in the numbers, the greater the variance.

The variance has three roles in survey research:

• As a way of describing the variation in data. For various technical reasons, the variation is a generally a very poor way of describing the degree of variation in the data, and alternative statistics such as the Interquartile Range and the Standard Deviation are generally superior.[note 1]
• As an input into other computations, such as the R-Square.
• As a useful theoretical concept in mathematical statistics, which greatly helps in the development of new statistical tools and in testing the properties of existing tools and techniques.

## Calculation

The most widely used formula is the formula for the sample variance: ${\displaystyle s^{2}={\frac {1}{n}}\sum _{i=1}^{n}\left(x_{i}-{\overline {x}}\right)^{2}}$ where ${\displaystyle s^{2}}$ is the estimated variance in the sample, ${\displaystyle x_{i}}$ is the observed value of the ${\displaystyle i}$ of [/itex]n[/itex] values and ${\displaystyle {\overline {x}}}$ is the average value.

Where data is weighted this needs to be reflected in the calculation of the variance.

## References

1. The scaleof the variance is the square of the scale of the values, making interpretation of the variance non-intuitive. It is also strongly effected by outliers and its interpretation is only obvious when the data is known to be normally distributed.