variances

In statistics, variance measures how far a set of numbers or a random variable diverges from its mean. It is the average of the squared deviations from the mean. Larger variance means more dispersion.

For a random variable X with mean μ = E[X], the population variance is Var(X) = E[(X - μ)^2]. Equivalently Var(X) = E[X^2] - μ^2, provided the expectations exist. The variance is nonnegative and is expressed in the square of the variable’s units.

When data are drawn from a population, the sample variance provides an estimate: s^2 = (1/(n-1)) ∑ (xi - x̄)^2, where x̄ is the sample mean. The divisor n-1 makes s^2 an unbiased estimator of Var(X) when the data are an iid sample from a population with variance Var(X).

Key properties include: Var(aX) = a^2 Var(X); Var(X+Y) = Var(X) + Var(Y) + 2 Cov(X,Y). If X and Y are independent, Cov = 0 and Var(X+Y) = Var(X) + Var(Y). The standard deviation is the square root of the variance.

The variance of the sample mean is Var(x̄) = Var(X)/n for iid observations, illustrating how averaging reduces variability. In multivariate contexts, variance is the diagonal of the covariance matrix; the covariance matrix generalizes Var to capture pairwise relationships.