Explained sum of squares

In statistics, the explained sum of squares (ESS), alternatively known as the model sum of squares or sum of squares due to regression ("SSR" – not to be confused with the residual sum of squares RSS), is a quantity used in describing how well a model, often a regression model, represents the data being modelled. In particular, the explained sum of squares measures how much variation there is in the modelled values and this is compared to the total 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.

The explained sum of squares (ESS) is the sum of the squares of the deviations of the predicted values from the mean value of a response variable, in a standard regression model — for example, y_i = a + b₁x_1i + b₂x_2i + ... + ε_i, where y_i is the i ^th observation of the response variable, x_ji is the i ^th observation of the j ^thexplanatory variable, a and b_i are coefficients, i indexes the observations from 1 to n, and ε_i is the i ^th value of the error term. In general, the greater the ESS, the better the estimated model performs.

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

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