Statistics for Business: Decision Making and Analysis, 3rd Edition

... the variation accounted for by the least squares regression line. The expression 1 − r² is the fraction of variation that is left in the residuals. If r² = 0, the regression line describes none of the variation in the data. In this case, the slope b₁ = r (s_y/s_x) is also zero and the fitted line is flat. If r² = 1, the line represents all of the variation and s_e = 0.

r-squared (r²)

Square of the correlation between X and Y, the percentage of “explained” variation.

For the diamonds, r² = 0.71² ≈ 0.5 and 1 − r² = 0.5. Because 0 ≤ r² ≤ 1, this summary is often described on a percentage scale. For example, we might say “The fitted line explains 50% of the variation in price.”

Summarizing the Fit of a Line

The quantity r² is a popular summary of a regression ...