What does R^2 represent in regression analysis?

What does R^2 represent in regression analysis?

• A. Proportion of variance explained by the model
• B. Intercept of the regression line
• C. Slope of the regression line
• D. Standard deviation of residuals

Answer: (A)

R^2 measures how much of the variability in the outcome the regression model captures. When you fit a line to data, you split the total variation in the observed values of the dependent variable into variation explained by the model and unexplained variation that shows up as residuals. R^2 is the fraction of the total variation that the model explains, computed as 1 minus the ratio of the sum of squared residuals to the total sum of squares around the mean. Values range from 0 to 1: 0 means the model explains none of the variability, 1 means it explains all of it (a perfect fit). This interpretation distinguishes R^2 from the intercept or slope, which are coefficients describing the line’s position and tilt, and from the standard deviation of residuals, which measures typical prediction error rather than the proportion of explained variance. Remember that a higher R^2 signals a better fit in terms of explained variability, but adding predictors can inflate R^2 even if the improvement isn’t meaningful, so adjusted R^2 or other metrics are often used to gauge true usefulness.

R^2 measures how much of the variability in the outcome the regression model captures. When you fit a line to data, you split the total variation in the observed values of the dependent variable into variation explained by the model and unexplained variation that shows up as residuals. R^2 is the fraction of the total variation that the model explains, computed as 1 minus the ratio of the sum of squared residuals to the total sum of squares around the mean. Values range from 0 to 1: 0 means the model explains none of the variability, 1 means it explains all of it (a perfect fit).

This interpretation distinguishes R^2 from the intercept or slope, which are coefficients describing the line’s position and tilt, and from the standard deviation of residuals, which measures typical prediction error rather than the proportion of explained variance. Remember that a higher R^2 signals a better fit in terms of explained variability, but adding predictors can inflate R^2 even if the improvement isn’t meaningful, so adjusted R^2 or other metrics are often used to gauge true usefulness.