Variance

Variance is the mean of squared deviations from the mean. For dataset [4, 8, 6, 5, 3, 7] with mean 5.5: squared deviations are 2.25, 6.25, 0.25, 0.25, 6.25, 2.25; sum 17.5; sample variance (÷ n-1) = 3.5.

Variance is in squared units (kilograms², dollars², seconds²) which makes it awkward to interpret directly. Take the square root and you get the standard deviation, which is in the original units. The two carry the same information; variance is what you compute, standard deviation is what you report.

Why bother with variance at all? Because variances are additive across independent sources of variation. If X and Y are independent, Var(X + Y) = Var(X) + Var(Y) — a property that standard deviations don’t have. This is what makes variance the natural unit for analysis-of-variance (ANOVA), error propagation, and most theoretical statistics.

For the working-statistician’s intuition behind variance and standard deviation, see the standard deviation explained guide.