Explained sum of squares

In statistics, an explained sum of squares (ESS) is the sum of squared predicted values in a standard regression model (for example $$y_{i}=a+bx_{i}+\epsilon_{i}$$), where $$y_{i}$$ is the response variable, $$x_{i}$$ is the explanatory variable, $$a$$ and $$b$$ are coefficients, $$i$$ indexes the observations from $$1$$ to $$n$$, and $$\epsilon_{i}$$ is the error term.

If $$\hat{a}$$ and $$\hat{b}$$ are the estimated coefficients, then


 * $$\hat{y_{i}}=\hat{a}+\hat{b}x_{i}$$

is the predicted variable. The ESS is the sum of the squares of the differences of the predicted values and the grand mean:


 * $$\sum_{i=1}^{n}\left(\hat{y}_{i}-\bar{y}\right)^2$$

In general: total sum of squares = explained sum of squares + residual sum of squares.