Correlation from R-squared, a positive relationship

Question

A simple regression reports R^2 = 0.36, and the scatterplot shows an upward (positive) trend. What is the correlation between x and y?

Accepted Answer

With a single predictor, R-squared is the square of the correlation, so take the root: r = \pm\sqrt{0.36} = \pm 0.6. The two options are +0.6 and -0.6. An upward trend means the slope is positive, and the correlation carries the slope's sign, so r = +0.6. A visual trend direction is just another way of being told the sign: upward means positive, downward means negative. Either the slope's sign or the shape of the cloud of points can resolve the ambiguity that the square root leaves behind. Takeaway. r = \pm\sqrt{0.36} = \pm 0.6; an upward trend means a positive slope, so r = +0.6.

Correlation from R-squared, a positive relationship

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