Needed to remind myself today that there is no neat practical interpretation of the Pearson’s coefficient of correlation “r” after seeing a report that suggested otherwise. It’s true - it is useful for comparing how strong and in which direction correlations are between two variables. But that’s about it.

Or

The correlation coefficient is sometimes criticized as having no obvious intrinsic interpretation

as Schober et al put it a little more formally.

You can square it to produce an R^{2} if you like to produce a “coefficient of determination”, which does have an IRL meaning. R^{2} expresses the percent of variation in one variable that’s explained by the variation in the other one.

Just the same as R^{2} in a linear regression model. In fact a simple linear regression like “y given x” will have an R^{2} that is the square of the correlation coefficient between x and y.