Height positively correlates with IQ and foot length is a very good proxy for height.
However, “correlated with” is not a transitive relation unless the correlations are fairly substantial. Precisely, if A correlates with B with coefficient c1, and B with C by c2 (both positive or both negative), then the minimum possible correlation of A with C is cos(arccos(c1)+arccos(c2)). E.g. if c1=c2=0.5, then this minimum is −0.5. If c1=c2=0.707, the minimum is 0. In general, a positive correlation of A with C is guaranteed if and only if c1^2 + c2^2 > 1.
However, “correlated with” is not a transitive relation unless the correlations are fairly substantial. Precisely, if A correlates with B with coefficient c1, and B with C by c2 (both positive or both negative), then the minimum possible correlation of A with C is cos(arccos(c1)+arccos(c2)). E.g. if c1=c2=0.5, then this minimum is −0.5. If c1=c2=0.707, the minimum is 0. In general, a positive correlation of A with C is guaranteed if and only if c1^2 + c2^2 > 1.
googles for “correlation between height and foot length” Uh, I thought that was much stronger than it actually is.