In statistics, explained variation measures the proportion to which a mathematical model accounts for the variation (dispersion) of a given data set. Often, variation is quantified as variance; then, the more specific term explained variance can be used.

The complementary part of the total variation is called unexplained or residual variation; likewise, when discussing variance as such, this is referred to as unexplained or residual variance.

Following Kent (1983), we use the Fraser information (Fraser 1965)

where ${\displaystyle g(r)}$ is the probability density of a random variable ${\displaystyle R\,}$, and ${\displaystyle f(r;\theta )\,}$ with ${\displaystyle \theta \in \Theta _{i}}$ (${\displaystyle i=0,1\,}$) are two families of parametric models. Model family 0 is the simpler one, with a restricted parameter space ${\displaystyle \Theta _{0}\subset \Theta _{1}}$.

Parameters are determined by maximum likelihood estimation,

The information gain of model 1 over model 0 is written as

where a factor of 2 is included for convenience. Γ is always nonnegative; it measures the extent to which the best model of family 1 is better than the best model of family 0 in explaining g(r).

Assume a two-dimensional random variable ${\displaystyle R=(X,Y)}$ where X shall be considered as an explanatory variable, and Y as a dependent variable. Models of family 1 "explain" Y in terms of X,