Varianca

Varianca is a statistical measure that describes how far a set of numbers or a random variable tends to be from its mean. It is defined as the expected value of the squared deviations from the mean: Var(X) = E[(X − μ)^2], where μ = E[X]. Equivalently, Var(X) = E[X^2] − μ^2. The concept captures dispersion rather than central tendency.

In practical terms, the population variance for a finite set of N observations {x1, x2, ..., xN} with mean μ is (1/N) Σ (xi − μ)^2. When estimating from a sample of size n drawn from a larger population, the unbiased estimator is the sample variance s^2 = (1/(n − 1)) Σ (xi − x̄)^2, where x̄ is the sample mean. The variance has the same units as the square of the variable, and the standard deviation, which is the square root of the variance, is often easier to interpret.

Key properties include that variance is nonnegative and that Var(aX + b) = a^2 Var(X). For two random variables X and Y, Var(X + Y) = Var(X) + Var(Y) + 2 Cov(X, Y); if X and Y are independent, Cov(X, Y) = 0 and Var(X + Y) = Var(X) + Var(Y). The law of total variance states Var(X) = E[Var(X|Y)] + Var(E[X|Y]).

Varianca is fundamental in fields such as statistics, finance, engineering, and research, where it informs about data spread, risk assessment, and the precision of estimates.