VarYi

VarYi is a statistical term used in probability theory and data analysis to denote a variance associated with a quantity indexed by i. When Yi denotes a random variable corresponding to subgroup i, VarYi often means the within-group variance Var(Yi) = E[(Yi − μi)^2], with μi = E[Yi]. In other analytical settings VarYi is used to denote the conditional variance Var(Y|i) = E[(Y − E[Y|i])^2 | i], representing how dispersion depends on the conditioning index i. The exact interpretation of VarYi depends on the source and context.

In experiments and observational studies, VarYi is used in analysis of variance (ANOVA), mixed-effects models, and heteroscedastic models to describe how variability changes across groups or categories. It is also used in risk assessment where i indexes scenarios, locations, or time points.

Estimation: If data are partitioned by i, within-group variance s_i^2 provides an estimate of VarYi. The calculation for each group i with observations Yi1,...,Yi,n_i is s_i^2 = ∑(Yij − Ybar_i)^2 /(n_i − 1). For conditional variance Var(Y|i), one can estimate E[Y|i] by the group mean and use residuals to estimate conditional dispersion across i; modeling approaches may treat VarYi as a function of i to capture heteroskedasticity, using, for example, variance functions in generalized least squares or mixed-effects models.

In practice, notation varies; VarYi should be clarified in any report to avoid confusion with Var(Yi) or Var(Y|i). See also variance, conditional expectation, ANOVA, heteroscedasticity.