Sigmay2

Sigmay2 is a notation sometimes used to denote the variance of a random variable Y, commonly written as Var(Y) or σ_y^2. It equals the expected squared deviation of Y from its mean: Var(Y) = E[(Y − μ_Y)^2], where μ_Y is the mean E[Y]. The standard deviation of Y is the square root of sigmay2, i.e., σ_y = sqrt(sigmay2. In many texts and software, σ_y^2 (with a superscript) is equivalent to sigmay2.

In practical use, sigmay2 measures the dispersion of Y around its mean. In regression contexts, the term is related to the error variance: if a model is Y = Xβ + ε with ε having Var(ε) = σ^2 and certain assumptions, then σ^2 is a component of the variability in Y. More generally, Var(Y) captures the total variability of Y, which may arise from both systematic structure and random error.

Estimation and interpretation: when data are observed, the population variance sigmay2 is unknown and is estimated by the sample variance s_y^2 = (1/(n−1)) ∑ (y_i − ȳ)^2. This estimator is unbiased for σ_y^2 under standard conditions. The squared value is nonnegative, with larger values indicating greater dispersion of Y. In multivariate contexts, Var(Y) extends to the covariance matrix Cov(Y), whose diagonal elements are Var(Y_i) for each component Y_i, and off-diagonal elements are covariances between components. Sigmay2, therefore, is a fundamental scalar descriptor of spread in univariate analyses and a component of more complex variance structures in multivariate settings.