What does R-squared (R^2) measure in a regression model?

R-squared (R^2) is a statistical measure that indicates the proportion of the variance in the dependent variable that can be explained by the independent variable(s) in a regression model. It is derived from two components: the Explained Sum of Squares (ESS) and the Total Sum of Squares (TSS).

The correct formulation is that R^2 equals the ratio of ESS to TSS, which is expressed as R^2 = ESS / TSS. Here, ESS measures the variation explained by the regression model, while TSS represents the total variation in the dependent variable. By dividing ESS by TSS, R^2 provides a value between 0 and 1, where a value of 0 indicates that the independent variables do not explain any of the variability of the dependent variable, and a value of 1 indicates that they explain all the variability.

This measure is key in evaluating the goodness of fit of the model, as a higher R-squared value suggests that a greater proportion of variance is accounted for by the regression model, signifying its potential effectiveness in predicting outcomes based on the given independent variables.