English: This figure gives a sense of how the usefulness of a Pearson correlation for predicting values varies with its magnitude. Given jointly normal X, Y with correlation ρ, ${\displaystyle 1-{\sqrt {1-\rho ^{2}}}}$ (plotted here as a function of ρ) is the factor by which a given prediction interval for Y may be reduced given the corresponding value of X. For example, if ρ = .5, then the 95% prediction interval of Y|X will be about 13% smaller than the 95% prediction interval of Y.

The formula ${\displaystyle 1-{\sqrt {1-\rho ^{2}}}}$ coincides with the "forecasting efficiency" value E defined by:

Hull, C. L. (1927). The correlation coefficient and its prognostic significance. Journal of Educational Research, 15, 327–338. Retrieved from https://www.jstor.org/stable/27523485