Table of Contents
What Is Variance?
Variance is a measure of statistical dispersion that quantifies how far a set of numbers is spread out from their average value. It is the average of the squared deviations from the mean. While standard deviation is more commonly reported (because it is in the same units as the data), variance has important mathematical properties that make it fundamental to statistical theory.
Variance is additive for independent random variables (the variance of a sum equals the sum of variances), which makes it central to portfolio theory in finance, ANOVA in experimental design, and error propagation in measurement science.
Formulas
Properties of Variance
| Property | Description |
|---|---|
| Non-negative | Variance is always ≥ 0 |
| Zero variance | All values are identical |
| Scale | Var(aX) = a² × Var(X) |
| Additivity | Var(X+Y) = Var(X) + Var(Y) if independent |
| Units | Squared units of the original data |
Frequently Asked Questions
Why divide by n-1 instead of n?
Dividing by n-1 (Bessel's correction) provides an unbiased estimate of the population variance when calculating from a sample. Using n would systematically underestimate the population variance because the sample mean is closer to the sample points than the population mean is.
What are the units of variance?
Variance is in squared units of the original data. If the data is in meters, the variance is in square meters. This is why standard deviation (the square root of variance) is often preferred for reporting, as it returns to the original units and is more interpretable.