Going the other way, R-squared from a correlation

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A simple regression has a correlation of r = 0.5 between x and y. What is the model's R-squared?

Accepted Answer

The same identity works in reverse. For one predictor, R^2 = r^2 = (0.5)^2 = 0.25. So a correlation of 0.5 gives an R-squared of 0.25, meaning the model explains about 25\% of the variation in y. This direction is unambiguous: squaring a known number needs no sign clue, and a positive or negative correlation of the same size gives the same R-squared. The lesson people often miss is how squaring shrinks middling correlations. A correlation of 0.5 sounds like "halfway," but it explains only a quarter of the variation, not half. Correlations must be squared before you talk about variation explai…

Going the other way, R-squared from a correlation

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