Adjusted R-squared

R-Square (R^2) is the proportion of variation in the dependent variable (Y) that can be explained by the predictors (X variables) in the regression model.

R²
= (regression sum of squares)/(total sum of squares)
= (regression sum of squares)/(regression sum of squares + error sum of squares)


As predictors (X variables) are added to the model, each predictor will explain some of the variance in the dependent variable (Y) simply due to chance. One could continue to add predictors to the model which would continue to improve the ability of the predictors to explain the dependent variable, although some of this increase in R-Square would be simply due to chance variation. The adjusted R-Square attempts to yield a more honest value to estimate R-Square.

Adjusted R-Square = 1-((1-R^2)*(N-1)/(N-k-1))
N: number of observations
k: number of predictors

When the number of observations (N) is small and the number of predictors (k) is large, there will be a much greater difference between R-Square and adjusted R-Square (because the ratio of (N-1)/(N-k-1) will be much less than 1).

By contrast, when the number of observations is very large compared to the number of predictors, the value of R-Square and adjusted R-Square will be much closer because the ratio of (N-1)/(N-k-1) will approach 1.