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Tuesday, February 08, 2011
BPro Reader mail of the day: regression toward the mean
tbwhite asks:
If Votto has a 3 year mean TAv of .325, but posted a .350 TAv in 2010, why should he revert towards the MLB mean TAv for 1B of .280(I’m just making up a number) in 2011 instead of his own mean TAv of .325 ? Regressing back towards the MLB mean implies that even his own .325 number over the past 3 seasons doesn’t reflect his true level of ability, that the .325 is nothing more than an outlier and isn’t repeatable. That is non-sense.
It sounds like nonsense, I agree.
When Votto hits .325 year after year in a league where 1B hit .280, and, this is important, in the absence of any independent estimate of his underlying true talent level, we have no choice but to believe that Votto was the recipient of more good luck than bad luck. That his observed .325 is more likely the effect of him being a true .315-.320, and got lucky to hit .325.
We know this is true (or at least, we can estimate this to be most likely as being true), because if you look at other players like Votto, you will find that their out-of-sample performance will be around .315-.320. The out-of-sample performance, for a large enough group of players, represents that group’s true talent level (with a certain level of uncertainty).
If Votto let’s say goes .325, .325, 300, .350 instead of .325, .325, .325, .325, almost nothing changes about what we know of Votto!. Well, a bit changes, because recent performance is more indicative of his current talent. But generally speaking, he’s got a career average of .325 in either case.
It’s called regression TOWARD the mean. And in the case of veteran’s, we’re only regressing 10-15% toward the mean. We’re basically saying “Yeah, we observed he hit great, but we also know that historically, a group of players that have been observed to hit great happens to hit less great the year after.”
So, going back to his original point, that is correct, his .325 does not represent his true ability. It does indicate his true ability to a certain (great) extent, but not 100%. And, it’s much more likely that he was disproportionately lucky than he was disproportionately unlucky. In absence of any independent evaluation of his underlying talent, we have to remove the luck from the observation. And we do that by regressing his performance toward the population of players that he was drawn from.
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