Thursday, April 29, 2010

Counterexample to the CLT!?

The introductory statistics text that I teach out of presents the Central Limit Theorem this way:
The Central Limit Theorem (CLT): For a relatively large sample size, the variable x' is approximately normally distributed, regardless of the distribution of the variable under consideration. The approximation becomes better with increasing sample size. [Weiss, "Introductory Statistics" 7E, p. 341]
In other words, any distribution turns into a normal curve when you're sampling (with large sample sizes). I also know off the top of my head that the formal CLT is talking about a distribution that's been standardized (converted z = (x-μ)/(σ/√(n))), and how its limit as a function is the standard-normal curve (centered at 0, standard deviation 1).

So one day I'm walking around sort of half-dozy and I'm thinking, "Wait a minute! What about a constant function? If you had a distribution that was fixed with one element (say, 100% chance that x=5), any conceivable sample mean would just be the constant x'=5, and there's no way a graph of that looks like a normal curve, right?"

Meditating...

Well, the thing that I didn't immediately have in my head, and is also entirely left out of the Weiss text -- There is one single fine-print requirement to the CLT, and it's that the standard deviation of your variable must be nonzero (and also non-infinite), i.e., 0<σ<∞. Which is sort of obvious from the fact that you need that to standardize with z = (x-μ)/(σ/√(n)), it being used to divide with in the formula and all. And of course a constant function has zero deviation, so it's indeed outside the scope of the theorem.

Guess I can't get too mad at the Weiss text for this... chances of it being useful for an introductory student to spend time parsing that is about nil. (Obviously, it hasn't come up in 5+ years of teaching the class.) Still, it might be nice to put it in a little footnote at the bottom of the page so I don't go daydreaming about possible counterexamples on my commute.

2 comments:

  1. Finite mean is also required.

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  2. Hmmm. That's even left out of my formal text from the strong version. (Casella 5.3.4). I guess that should be obvious, I guess?

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