R² Measures Explained Sample Variation

For ordinary least squares with an intercept,

$R^2=1-\frac{\mathrm{SSE}}{\mathrm{SST}}.$

It is the fraction of observed response variation accounted for by the fitted model, relative to predicting every response with the sample mean.

Interpreting the value

An $R^2$ of $0.80$ means the fit accounts for 80% of the sample variation around $\bar y$. It does not mean 80% of observations are correct or that predictions are 80% accurate.

Adjusting for model size

Adding predictors cannot decrease ordinary $R^2$. Adjusted $R^2$ introduces a degrees-of-freedom penalty:

$R^2_{\rm adj}=1-\frac{\mathrm{SSE}/(n-p)}{\mathrm{SST}/(n-1)}.$

It can decrease when an added term contributes too little.

Derive the decomposition

Write $y_i-\bar y=(\hat y_i-\bar y)+e_i$. Squaring and summing produces SST = SSR + SSE because, in ordinary least squares with an intercept, residuals are orthogonal to fitted deviations. Therefore

$R^2=\frac{\mathrm{SSR}}{\mathrm{SST}}=1-\frac{\mathrm{SSE}}{\mathrm{SST}}.$

This identity can fail in familiar form when the model omits an intercept or uses a different fitting objective.

Check your understanding

Two models have $R^2=0.90$. Must they have equal RMSE?

Show the reasoning

No. $R^2$ is relative to each dataset’s SST, while RMSE is in response units. Different response scales, sample variation, or degrees of freedom can yield different RMSE values.