Ah, I see your point. However, a) vocab is highly correlated with intelligence, not weakly so, b) vocab is not just highly correlated with a single intelligence metric, but is correlated with such in a variety of different metrics of intelligence. While it is possible to construct variables such that A and B are correlated, with B and C correlated, and A and C anti-correlated, it is quite difficult to do so with a large set of distinct variables that all have such correlations with each other and have a single pair be anti-correlated, especially when one has the same set of correlations even when one controls for a variety of other variables. Moreover, as a probabilistic matter if one as three variables with two pairs correlated, it is much more likely that the remaining pair will be correlated than anti-correlated, assuming that variables don’t have too pathological a distribution.
Moreover, as a probabilistic matter if one as three variables with two pairs correlated, it is much more likely that the remaining pair will be correlated than anti-correlated, assuming that variables don’t have too pathological a distribution.
Where you you get that? The intended probability space isn’t clear, but if I take three random directions in N-dimensional space for large N, I find that the chance of two pairs having an angle less than pi/2 and the third an angle greater than pi/2 is about 1.4 times the chance of all three being less than pi/2. The ratio rises to about 3 if I add the requirement that the corresponding correlations are in the range +/- 0.8 (the upper liit of correlations generally found in psychology).
Hmm, that’s a good point. I’m aware vaguely of theorems that say what I want but I don’t have any references or descriptions off hand. It may just be that one is assuming somewhat low N, but that would be in this sort of context not helpful. I do seem to remember that some version of my statement is true if the variables match bell curves, but I’m not able at the moment to construct or find a precise statement. Consider the claim withdrawn until I’ve had more time to look into the matter.
Ah, I see your point. However, a) vocab is highly correlated with intelligence, not weakly so, b) vocab is not just highly correlated with a single intelligence metric, but is correlated with such in a variety of different metrics of intelligence. While it is possible to construct variables such that A and B are correlated, with B and C correlated, and A and C anti-correlated, it is quite difficult to do so with a large set of distinct variables that all have such correlations with each other and have a single pair be anti-correlated, especially when one has the same set of correlations even when one controls for a variety of other variables. Moreover, as a probabilistic matter if one as three variables with two pairs correlated, it is much more likely that the remaining pair will be correlated than anti-correlated, assuming that variables don’t have too pathological a distribution.
Where you you get that? The intended probability space isn’t clear, but if I take three random directions in N-dimensional space for large N, I find that the chance of two pairs having an angle less than pi/2 and the third an angle greater than pi/2 is about 1.4 times the chance of all three being less than pi/2. The ratio rises to about 3 if I add the requirement that the corresponding correlations are in the range +/- 0.8 (the upper liit of correlations generally found in psychology).
Hmm, that’s a good point. I’m aware vaguely of theorems that say what I want but I don’t have any references or descriptions off hand. It may just be that one is assuming somewhat low N, but that would be in this sort of context not helpful. I do seem to remember that some version of my statement is true if the variables match bell curves, but I’m not able at the moment to construct or find a precise statement. Consider the claim withdrawn until I’ve had more time to look into the matter.