Correlation from R-squared with a negative slope

Question

A simple regression reports R^2 = 0.64, and the fitted slope is negative. What is the correlation between x and y?

Accepted Answer

For one predictor, R-squared equals the correlation squared, so undo the square: r = \pm\sqrt{R^2} = \pm\sqrt{0.64} = \pm 0.8. Two candidates appear, +0.8 and -0.8. The correlation shares the sign of the slope, and here the slope is negative, so r = -0.8. The size (0.8) says the relationship is strong; the negative sign says it runs downhill, as x goes up, y tends to go down. R-squared would have looked identical (0.64) for a strong positive relationship, which is exactly why you must bring in the slope's sign to tell the two apart. Takeaway. r = \pm\sqrt{0.64} = \pm 0.8, and a negative slope…

Correlation from R-squared with a negative slope

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