VARIANCE

Variance is a statistical quantity that measures the dispersion of a set of values. It describes how far the values are from the mean, on average, in squared units. In probability theory, Var(X) refers to the expectation of the squared deviation from the mean. There are two commonly used forms: population variance, defined for a full set of observations, and sample variance, used to estimate variance from a sample. The square root of the variance is the standard deviation, another widely used dispersion measure.

Population variance is Var(X) = E[(X − μ)^2], where μ is the mean E[X]. For a finite dataset of size N with observations x_i, the population variance is σ^2 = (1/N) Σ (x_i − μ)^2. Sample variance is s^2 = (1/(n−1)) Σ (x_i − x̄)^2, where x̄ is the sample mean. The factor n−1 makes s^2 an unbiased estimator of σ^2 under standard assumptions.

Key properties include non-negativity and zero only when all values are identical. If a constant a scales a variable, Var(aX + b) = a^2 Var(X). For two random variables, Var(X + Y) = Var(X) + Var(Y) + 2 Cov(X, Y). If X and Y are independent, Cov(X, Y) = 0, so Var(X+Y) = Var(X) + Var(Y).

Alternative formula: Var(X) = E[X^2] − (E[X])^2, and for samples, s^2 = (Σ x_i^2 − (Σ x_i)^2 / n) / (n−1). Example: for a fair six-sided die with outcomes 1–6, the mean is 3.5 and the variance is 35/12 ≈ 2.9167. Variance is widely used to quantify dispersion in data and to characterize uncertainty in probabilistic models.