The variance of a collection of values is the average squared deviation from the mean: Var(X) = (1/n) Σ (xᵢ − x̄)². It measures how spread out the values are — a variance of 0 means all values are identical, and larger variance means greater spread. The sample variance typically uses n − 1 in the denominator (Bessel’s correction) to produce an unbiased estimator of the population variance.
Variance is measured in squared units of the original data, which can be hard to interpret directly. The standard deviation σ = √Var(X) returns to the original units and is the more commonly reported measure of spread. For a normal distribution, about 68% of values fall within one standard deviation of the mean and about 95% within two.
Variance decomposes additively for independent random variables: Var(X + Y) = Var(X) + Var(Y) when X and Y are independent. This property underlies the central limit theorem and the analysis of variance (ANOVA). Variance is connected to the mean through the identity Var(X) = E[X²] − (E[X])², relating the spread of a distribution to its first two moments.