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Kindly comments on The Fallacy of Large Numbers - Less Wrong Discussion
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Comments (31)
Fortunately, the LLN isn't just some black box. It has a proof! And we can look at that proof and get bounds on how quickly the average converges to the mean (which is basically Chebyshev's inequality, but whatever).
In cases with slightly more regularity to them, we can use Hoeffding's inequality or something similar and get even better bounds. In fact, this will give results that are almost as good as the assume-it's-normal strategy, but with the added benefit that you're actually answering the question you started with, rather than making something up.
Yeah, you can get bounds, and they are what they are.
But I grope in vain for a point to this article. The LLN doesn't converge as fast as he'd like? Yeah, and sometimes gravity is inconvenient for me, but I don't post my disgruntlement about it to the list. Somehow his expectations of the rate of convergence have been violated. I suggest he do the calculations you suggest, and educate his expectations.
Excepting those lurking in his own expectations, where's the fallacy? What is he talking about?
I think you're being unnecessarily mean.
I do tend toward leaner and meaner over kinder and gentler. I'm trying to be nicer - I actually toned down the second post, and deleted some stuff. Guess not enough for your taste.
But really, do you know what the point is?
I see it as a special case of "the fallacy of everything related to high school statistics".
(Okay, so I don't really agree with the answers the post gives. But I think it's bringing up an interesting point, and hey, this is only discussion. Possibly if we had lots of high-quality math posts, I would feel differently.)
My personal feeling is that where statisticians go wrong is that they think of their problems, not as something you solve, but something you use tools on. But I'm not sure I can articulate this feeling more precisely than that.