In regression, what does r squared (r^2) represent?

In regression, what does r squared (r^2) represent?

• A. The correlation coefficient
• B. The slope of the regression line
• C. The proportion of variance in Y explained by X
• D. The p-value of the regression

Answer: (C)

It represents the fraction of variability in the outcome that the regression model explains. In simple linear regression, r^2 is the square of the Pearson correlation between X and Y, so a higher value means more of Y’s variation is accounted for by the predictor. For example, r^2 = 0.75 means 75% of Y’s variability is explained by X, and 25% is due to unexplained factors or random error. It is calculated as 1 minus the ratio of the sum of squared residuals to the total sum of squares (SSE/SST). It’s not the correlation coefficient itself, nor the slope of the regression line, nor the p-value. While a higher r^2 indicates a better fit in terms of explained variance, it can be inflated by adding more predictors, so adjusted r^2 is often used to assess model quality.

It represents the fraction of variability in the outcome that the regression model explains. In simple linear regression, r^2 is the square of the Pearson correlation between X and Y, so a higher value means more of Y’s variation is accounted for by the predictor. For example, r^2 = 0.75 means 75% of Y’s variability is explained by X, and 25% is due to unexplained factors or random error. It is calculated as 1 minus the ratio of the sum of squared residuals to the total sum of squares (SSE/SST). It’s not the correlation coefficient itself, nor the slope of the regression line, nor the p-value. While a higher r^2 indicates a better fit in terms of explained variance, it can be inflated by adding more predictors, so adjusted r^2 is often used to assess model quality.