Managing with no idea about random fluctuations

Ah, Jerry Manuel, "manager" of the Mets, is up to it again. Now he's suggesting that he will bench or move Jason Bay or David Wright for batting .241 and .240 after 54 and 50 at bats, respectively. Of course batting .240 after an amazingly huge sample size of 50 at bats definitely proves that you're doing terribly, right? I mean it's utterly inconceivable that a .280 or .300 batter would go through a 50 at-bat period where they batted that low.

Okay, enough sarcasm, let's run the statistics for Jason Bay. Here's a little program in python than can do it for us.

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import random
l = 0
for k in range(1000):
    i = 0.0
    for j in range(54):
        if random.random() < .300:
            i += 1.0
    if (i/54) < .241:
        l += 1
print "%2.0f%%" % (l/10.0)

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What do we learn? It turns out that if Jason Bay were a pure .300 hitter -- that is he had a 30% chance of getting a hit on every at bat, there's a 22% chance that he'd have an average as low as .241 after 54 at bats. If he were a decent .275 hitter, he'd have 35% chance of having an average that low.

The numbers are about the same (or even a bit more disappointing) for Wright, which surprised me. I would've thought 4 extra at bats would far more than cancel out one point of BA, but it doesn't because the last digit isn't significant yet! With 50 at bats, it's impossible for Wright to be slightly better than Bay by batting, say .242, because each hit still represents a BA difference of .020! If Wright had one more hit, he'd be batting a respectable .260. If one or two fewer, he'd be at .220 or .200 and probably already traded by the kinds of short-sighted people who get paid millions to run things that they don't understand. (see: Management, Mets [or most non-winning teams]).

The point isn't that Bay and Wright are great hitters going through bad times. They could have morphed into terrible terrible hitters who are lucky to even be batting the .240 that they have. The point is that we just don't know yet. And given the number of millions you're paying them based on prior performance, maybe, just maybe, it makes sense to let them play long enough so we can find out what's actually happening?

Addendum, 3 hours later: I originally decided to leave out a criticism of the utter stupidity of messing with an approach that is causing Wright to be leading the league in walks (with a .457 OBP), but nah, they deserve all the criticism they can get. So there you are.