Sigmasquared

Sigmasquared, denoted by σ^2, is a common symbol in statistics for the variance of a random variable. It represents the expected squared deviation from the mean and is a measure of dispersion in the data. The standard deviation, σ, is the square root of sigmasquared and shares the same units as the variable.

Formally, for a random variable X with mean μ, sigmasquared equals Var(X) = E[(X − μ)^2]. Equivalently, sigmasquared = E[X^2] − μ^2. In many probability models, such as the normal distribution N(μ, σ^2), sigmasquared serves as a fundamental parameter describing the spread of the distribution.

Properties include additivity under independence: if X and Y are independent, Var(X+Y) = Var(X) + Var(Y). The variance is nonnegative and finite when the underlying moments exist. For linear transformations, Var(aX + b) = a^2 Var(X).

Estimation and sampling: In a sample x1, ..., xn with mean x̄, the unbiased estimator of sigmasquared is s^2 = (1/(n−1)) ∑ (xi − x̄)^2. The maximum likelihood estimator for normal data is σ̂^2 = (1/n) ∑ (xi − x̄)^2, which is biased but consistent.

Applications: Sigmasquared appears in hypothesis testing, confidence intervals, analysis of variance, regression, and many stochastic models, where it quantifies the spread of a distribution relative to its mean. Its square root, the standard deviation, is often more interpretable as a measure of typical deviation.