Range restrictions for the correlations of 3 variables

A little follow up to my testosterone comment (written in German):
When three variables are correlated to each other, and two of the three correlations are known, the range for the third correlation is restricted according to this formula (Olkin, 1981):

Now comes the new part: here’s the graphical representation of that range restriction:

As one can see, one, or both of the two given correlations have to be fairly high to imply a positive third correlation.

Olkin, I. (1981). Range restrictions for product-moment correlation matrices. Psychometrika, 46, 469-472. doi:10.1007/BF02293804

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