What does the R^2 value represent in a regression analysis?

The R² value, or the coefficient of determination, is a key statistic in regression analysis that represents the proportion of the variance in the dependent variable that can be explained by the independent variable(s) in the model. A higher R² value indicates that a greater proportion of variance is accounted for by the regression model, which implies a better fit.

When interpreting R², if you have an R² value of 0.75, for example, it means that 75% of the variability in the outcome variable can be explained by the predictors in your regression model. This provides a useful measure to evaluate how well your chosen model captures the underlying trends present in the data.

In contrast, the correlation coefficient measures the strength and direction of the linear relationship between two variables, but it does not directly assess the proportion of variance explained like R² does. The standard error of regression quantifies the average distance that the observed values fall from the regression line, meaning it focuses more on the fit of the model rather than the proportion of explained variance. The difference between actual and predicted values refers to residuals, which do help in determining model accuracy but do not directly express how much variance is explained by the model itself.