Which statement best characterizes R-squared in regression?

Which statement best characterizes R-squared in regression?

• A. The correlation between observed and predicted values.
• B. The variance of the residuals divided by the total variance.
• C. The coefficient of determination multiplied by the residual degrees of freedom.
• D. The proportion of the variance in the dependent variable explained by the model.

Answer: (D)

R-squared shows how much of the variation in the dependent variable the regression model explains. It compares the amount of variability left after fitting the model to the total variability in the data. Specifically, it’s 1 minus the sum of squared residuals divided by the total sum of squares. When the model accounts for most of the variation, SSE is small and R-squared is close to 1; when the model explains little, SSE is large and R-squared is near 0. So it is the proportion of the variance in the dependent variable explained by the model. The other statements don’t capture this idea: the correlation between observed and predicted values is related but not the definition, the ratio described would be the unexplained portion, and multiplying by residual degrees of freedom isn’t how R-squared is formed.

R-squared shows how much of the variation in the dependent variable the regression model explains. It compares the amount of variability left after fitting the model to the total variability in the data. Specifically, it’s 1 minus the sum of squared residuals divided by the total sum of squares. When the model accounts for most of the variation, SSE is small and R-squared is close to 1; when the model explains little, SSE is large and R-squared is near 0. So it is the proportion of the variance in the dependent variable explained by the model. The other statements don’t capture this idea: the correlation between observed and predicted values is related but not the definition, the ratio described would be the unexplained portion, and multiplying by residual degrees of freedom isn’t how R-squared is formed.