variansen

Variansen, in statistics often called the variance, is a measure of how spread out the values of a random variable are around its mean. It captures the average squared deviation from the mean and is a central concept for describing dispersion in data.

For a random variable X with mean μ = E[X], the population variance is Var(X) = E[(X − μ)²]. In discrete form, Var(X) = Σ p(x) (x − μ)²; in continuous form, Var(X) = ∫ (x − μ)² f(x) dx. The variance can also be written as Var(X) = E[X²] − μ², relating it to the second moment and the mean.

When a sample x1, x2, ..., xn is drawn, the usual estimator for the population variance is the sample variance s² = (1/(n−1)) Σ (xi − x̄)², where x̄ is the sample mean. This is an unbiased estimator of Var(X) under mild assumptions.

Key properties include: Var(X) ≥ 0, with equality if and only if X is constant almost surely; Var(aX + b) = a² Var(X); if X and Y are independent, Var(X + Y) = Var(X) + Var(Y). The standard deviation, sd(X) = √Var(X), is the square root of the variance and shares the same units as X.

Variansen features prominently in statistical methods such as hypothesis testing, regression analysis, and analysis of variance (ANOVA). It is also sensitive to the scale of the variable because units are squared, which can affect interpretation and comparison across datasets.