Coefficient of determination (R²)

How to read it

OpenStax’s Introductory Statistics calls r² the coefficient of determination, “the square of the correlation coefficient”, usually stated as a percent: the percent of variation in the dependent variable y that can be explained by variation in x using the regression line. 1 − r², also as a percent, is the variation in y that is not explained. NIST’s Engineering Statistics Handbook describes the R² statistic in the same terms, as the fraction of the total variability in the response that is accounted for by the model. In OpenStax’s exam-score example, r = 0.663 and r² = 0.43969, so the line accounts for about 44% of the variation in final exam scores.

Also searched as: coefficient of determination symbol, r2, r², R², r squared, r squared meaning.

r² or R²

Penn State’s STAT 501 course notes use the lowercase r² for simple linear regression, with one predictor, and the capital R² for the multiple coefficient of determination, with more than one predictor. They add that Minitab does not distinguish between the two and calls both “R-sq”.

What a high R² doesn’t say

NIST’s handbook heads its page on checking a model “R² Is Not Enough!”: a high R² does not guarantee that the model fits the data well, and NIST makes graphical analysis of the residuals its main tool. Penn State’s notes list seven cautions about R² and r and call reading a large R² as a good fit the most common misuse. In one of their examples a correlation of 0.959 and an R² of 92.0% come from data on the US population by year that a curve would describe better than the straight line.

Look-alikes

R² is not a Variance (s² σ²), though both are written with a ². The variance σ² or s² is a squared standard deviation; R² is a squared correlation coefficient.