- is a quantity used in describing how well a model, often a regression model, represents the data being modelled. In particualr, the explained sum of squares measures how much variation there is in th modelled values and this is compared to the totoal sum of squares, which measures how much variation there is in the observed data, and to the residual sum of squares, which measures the variation in the modelling errors.
if $\hat{a}$ and $b^2$ are the estimated coefficients, then
\hat{y_i} = \hat{a}+\hat{b_1}x_{1i}+\hat{b_2}x_{2i}+...
is the $i^{th}$ predicted value of the response variable, The ESS is the sum o fthe squres of the differences of the values and the mean value of the response variable: