R-squared › The R-squared Formula

R-squared is defined as:

Where:

• $SS_{\text{res}} = \sum_{i=1}^n (y_i - \hat{y}_i)^2$: Residual sum of squares (unexplained variance)
• $SS_{\text{tot}} = \sum_{i=1}^n (y_i - \bar{y})^2$: Total sum of squares (total variance)
• $y_i$: Actual value
• $\hat{y}_i$: Predicted value from the model
• $\bar{y}$: Mean of the actual values

The formula can be understood through variance decomposition. The total variance in the dependent variable can be split into explained variance (how much the model explains) and unexplained variance (how much remains). R-squared represents the proportion of total variance that is explained by the model. For simple linear regression, R-squared equals the square of the correlation coefficient between the predictor and response variables ($R^2 = r^2$).