Why is adjusted R-squared less than R-squared if adjusted R-squared predicts the model better? As far as I understand, $R^2$ explains how well the model predicts the observation. Adjusted $R^2$ is the one that takes into account more observations (or degrees of freedom). So, Adjusted $R^2$ predicts the model better? Then why is this less than $R^2$? It appears it should often be more. 
 A: $R^2$ shows the linear relationship between the independent variables and the dependent variable.  It is defined as $1-\frac{SSE}{SSTO}$ which is the sum of squared errors divided by the total sum of squares.  $SSTO = SSE + SSR$ which are the total error and total sum of the regression squares.  As independent variables are added $SSR$ will continue to rise (and since $SSTO$ is fixed) $SSE$ will go down and $R^2$ will continually rise irrespective of how valuable the variables you added are.
The Adjusted $R^2$ is attempting to account for statistical shrinkage.  Models with tons of predictors tend to perform better in sample than when tested out of sample.  The adjusted $R^2$ "penalizes" you for adding the extra predictor variables that don't improve the existing model.  It can be helpful in model selection. Adjusted $R^2$ will equal $R^2$ for one predictor variable. As you add variables, it will be smaller than $R^2$.
A: R-Squared increases even when you add variables which are not related to the dependent variable, but adjusted R-Squared take care of that as it decreases whenever you add variables that are not related to the dependent variable, thus after taking care it is likely to decrease.
A: R^2 explains the proportion of the variation in your dependent variable (Y) explained by your independent variables (X) for a linear regression model.
While adjusted R^2 says the proportion of the variation in your dependent variable (Y) explained by more than 1 independent variables (X) for a linear regression model.
