In simple linear regression, what is the standard error of the estimate?

• A. The standard deviation of the independent variable
• B. The standard error of the mean of residuals
• C. The standard deviation of the dependent variable
• D. The standard deviation of the residuals

Answer: (D)

The standard error of the estimate is the typical size of the prediction errors in simple linear regression. It is the standard deviation of the residuals, where each residual is the difference between an observed value and its fitted value (e_i = y_i − ŷ_i). This quantity tells you, on average, how far the observed points lie from the regression line. It’s computed as s = sqrt( SSE / (n − 2) ), with SSE = sum of e_i^2 and n the sample size, and the n−2 in the denominator reflecting that two parameters (the slope and intercept) are estimated from the data. This measure is also known as the root mean square error.

Why the others don’t fit: the standard deviation of the independent variable describes how spread out the X values are, not how far Y values deviate from the fitted line. The standard deviation of the dependent variable describes the spread of Y around its mean, not the regression fit. The standard error of the mean of residuals isn’t the same thing as the spread of residuals around the line, and it doesn’t capture the prediction error you’re assessing with the regression.