Saturday, August 08, 2009
Another look at bimodal distributionsPosted by Colin Wyers
In my last article, I looked at how players regress to the mean, and how players on the borderline between the minors and majors might not have a readily identifiable mean to regress to.
But what if we add in AA players?
It seems that adding AA players does not give us a trimodal distribution; the means for the two leagues (or at least their major league equivelencies) are pretty close to each other. It's possible that this is a selection bias from the way the MLEs are computed, of course. But this also squares pretty well with what we think we know about the difference in league quality, so even if there is such a bias I'm not sure it's large enough to give us a truly bimodal distribution between AA and AAA players.
But while that shifts our second mode over to the left a bit, it also gives us a much larger population of players in the minors than the majors. Here's a helpful illustration, at about 110 PAs:
I almost wonder if there's something I'm missing here, though, with my assumptions - if pressed I would guess that in real life the right-side part of the curve on the two distributions line up a lot better than what I'm showing here. There are estimated standard deviations, and so maybe the observed SDs for minor league talent are larger than what I'm showing. I'll have to check into that.
Colin Wyers knows exactly how much of a nerd he is. He is very interested in hearing about any other concerns you may have; you can reach him by e-mail, and he will try his best to respond in a timely fashion. He also blogs at Statistically Speaking.