Variancës

Variancës, known in English as variance, is a statistical measure of the spread or dispersion of a random variable around its mean. It quantifies how far the values of a dataset tend to be from the average value.

There are two primary forms: the population variance and the sample variance. The population variance Var(X) of a random variable X with mean μ = E[X] is Var(X) = E[(X - μ)^2], which can also be written as Var(X) = E[X^2] - μ^2. The sample variance provides an estimate from a finite sample x1, x2, ..., xn: s^2 = (1/(n-1)) sum_{i=1}^n (xi - x̄)^2, where x̄ is the sample mean.

Key properties include that variance is non-negative and Var(X) = 0 if and only if X is almost surely constant. For constants a and b, Var(aX + b) = a^2 Var(X). If X and Y are independent, Var(X + Y) = Var(X) + Var(Y); in general, Var(X + Y) = Var(X) + Var(Y) + 2 Cov(X, Y).

Common examples help intuition: for a Bernoulli(p) variable, Var = p(1 - p). For a normal distribution N(μ, σ^2), Var = σ^2. The standard deviation, defined as sqrt(Var), provides a natural scale for dispersion. Variance is sensitive to outliers and is scale-dependent, increasing with the units of measurement.

Applications span finance for risk, quality control, measurement uncertainty, hypothesis testing, and error analysis in regression. See also standard deviation, covariance, and moments.