What R-squared does and does not tell you

Question

Define R^2 (the coefficient of determination) precisely, and name the most common ways it is misread.

Accepted Answer

Definition. For a regression fitting \hat{y}i to observed yi with mean \bar{y}, R^2 = 1 - \frac{\sumi (yi - \hat{y}i)^2}{\sumi (yi - \bar{y})^2} = 1 - \frac{	ext{residual sum of squares}}{	ext{total sum of squares}}. It is the fraction of the variance in y that the model explains relative to just predicting the mean. In an ordinary least squares fit with an intercept, it ranges from 0 (no better than the mean) to 1 (perfect fit) and equals the squared correlation between y and \hat{y}. Common misconceptions. 1. "High R^2 means the model is correct or causal." No. R^2 measures fit, not truth…

What R-squared does and does not tell you

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