variancën

Variancën, commonly called the variance, is a measure of dispersion in probability and statistics that describes how far a set of values tends to be spread around the mean. For a random variable X with expected value μ, the variance is Var(X) = E[(X − μ)^2]. An equivalent expression is Var(X) = E[X^2] − μ^2 when the expectations exist.

Two common forms are used in practice: the population variance and the sample variance. The population variance uses the true mean: Var(X) = (1/n) Σ (Xi − μ)^2. The sample variance, an unbiased estimator of the population variance when the mean is unknown, is s^2 = (1/(n−1)) Σ (Xi − X̄)^2.

Key properties: Var(X) ≥ 0, with equality only if X is constant almost surely; Var(aX + b) = a^2 Var(X); if X and Y are independent, Var(X + Y) = Var(X) + Var(Y).

Relation to other measures: The standard deviation is the square root of the variance and shares the same units as X. The coefficient of variation is CV = SD/μ. The variance is sensitive to outliers and non-normality.

Interpretation and uses: Variance quantifies spread; in many fields it informs risk and reliability. In analysis of variance (ANOVA) and regression, total variance is partitioned into components attributed to factors, interactions, and error. In finance, variance is a proxy for volatility.

Computation notes: An equivalent form is s^2 = [Σ Xi^2 − n X̄^2]/(n−1).