Coefficient of Determination (R²) — How close is the fit?

Coefficient of Determination (R2) represents the proportion of the variance of a dependent variable that is explained by the predictor(s).

The sum of squared deviations, ∑(y_i − ȳ)² (SS_tot), is an unscaled, or unadjusted measure of dispersion or variability. When scaled for the number of degrees of freedom, it becomes the variance.

Partitioning of the sum of squared deviations, allows the overall variability in the data to be ascribed to that explained by the regression (SS_reg), and that not explained by the regression, i.e., the residual sum of squares, (SS_res).

$$ SS_{tot} = SS_{reg} + SS_{res} $$ $$ \text{Total SS = Explained SS + Residual SS} $$ $$ \sum(y_i - \bar y)^2=\sum(\hat y_i - \bar y)^2 + \sum (y_i - \hat y_i )^2 $$

SS_tot: Total sum of squared deviations.

SS_reg: Explained sum of squared deviation.

SS_res: Residual (error) sum of squares.

Coefficient of Determination (R²) is the proportion of the variance in the dependent variable that is predictable from the independent variables:

$$R^2=\frac{SS_{reg}}{SS_{tot}}$$

This is a commonly used statistic to evaluate model fit; it is an indicator of how well the model explains the movement in the data. For instance, an R² of 0.8 means that the regression model explains 80% of the variability in the data.