In simple linear regression, R^2 measures

In simple linear regression, R^2 measures

• A. The slope of the regression line.
• B. The correlation coefficient.
• C. The standard error of the estimate.
• D. Proportion of variance in the dependent variable explained by the independent variable.

Answer: (C)

R^2 shows how much of the variation in the outcome Y is explained by the predictor X in a simple linear regression. Imagine partitioning the total variability of Y into what the regression line accounts for and what it doesn’t—the portion explained by the model is R^2, while the rest is the unexplained variability in the residuals. Mathematically, R^2 = SSR/SST = 1 − SSE/SST, where SST is the total sum of squares, SSE is the sum of squared residuals, and SSR is the sum of squares explained by the regression. In simple regression, R^2 also equals the square of the correlation between X and Y, linking it to the familiar correlation concept. So, R^2 isn’t the slope, nor the correlation coefficient itself, nor the standard error of the estimate (which measures typical residual size). It specifically represents the proportion of Y’s variance explained by the regression on X.

R^2 shows how much of the variation in the outcome Y is explained by the predictor X in a simple linear regression. Imagine partitioning the total variability of Y into what the regression line accounts for and what it doesn’t—the portion explained by the model is R^2, while the rest is the unexplained variability in the residuals. Mathematically, R^2 = SSR/SST = 1 − SSE/SST, where SST is the total sum of squares, SSE is the sum of squared residuals, and SSR is the sum of squares explained by the regression. In simple regression, R^2 also equals the square of the correlation between X and Y, linking it to the familiar correlation concept.

So, R^2 isn’t the slope, nor the correlation coefficient itself, nor the standard error of the estimate (which measures typical residual size). It specifically represents the proportion of Y’s variance explained by the regression on X.