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Asked what the central limit theorem says, a student replies, "as you take larger and larger samples from a population, the histogram of the sample values looks more and more Normal". Is the student right? Explain your answer.

My answer the student is wrong because the histogram of the sample values will look like the population distribution, whatever that distribution might look like as the sample size increases. The CLT says the sample mean follows a normal distribution with mean $u$ and variance $σ^2/n$ as the sample size goes to infinity. But CLT fails to population that has fat tails such as Cauchy Distribtion.

Is this right? If so, do you think I could add a little more?

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  • $\begingroup$ I think it is right. Explain (or define) what is $u$ and add perhaps that as the samples become larger the mean of the samples will converge to the mean of the distribution, but this again is not the CLT, but the law of large numbers. If you would take repeated samples of increasing size and make a histogram of the means of this samples, then you could use CLT to argue that the histogram would look normal. (but, your answer is ok) Actually the CLT holds for every distribution $\endgroup$
    – Jimmy R.
    Mar 8, 2014 at 18:31
  • $\begingroup$ even for Cauchy Distribution? $\endgroup$ Mar 8, 2014 at 18:40
  • $\begingroup$ Yes, see here. In the assumptions no symmetry of the distribution is needed. But, of course for a symmetrical distribution the convergence will be much faster than for a heavily skewed distribution. No, the CLT holds for samples from every distribution. $\endgroup$
    – Jimmy R.
    Mar 8, 2014 at 18:54

1 Answer 1

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The Central Limit Theorem examines sums of random variables from which we subtract the mean of the sum and then divide the whole by the standard deviation of the sum: let $Y_1,...,Y_n$ be random variables and define $S_n \equiv \sum_{i=1}^nY_i$. Then the CLT examines the random variable

$$Z_n = \frac {S_n - E[S_n]}{\sqrt {\text {Var}(S_n)}}$$

and what is its limiting distribution.

Variants of the CLT materialize as we increase the number of assumptions: the most basic variant is the assumption that the $Y_i$'s are identically and independently distributed, with existing / finite moments (but a CLT exists even for distributions that do not have finite variance, for a taste see here).

In most variants, the CLT provides sufficient but not necessary conditions for it to hold. If these conditions are satisfied, the usual CLT asserts that $Z_n \rightarrow_d N(0,1)$.

So in order to be able to apply the CLT, one must in the first place be examining a $Z_n$-variable.

Note that the work around (if the CLT holds)

$$Z_n \rightarrow_d N(0,1) \Rightarrow S_n \sim_{approx.} N\left(E[S_n],\text {Var}(S_n)\right)$$

is only an approximation, and for "large" but finite $n$.

The "student's answer" is wrong, not because it does not refer to "the sample mean", but because one cannot define a "population" in such a way so as drawing samples from it and examining their empirical distribution function becomes equivalent to examining the limiting distribution of a $Z_n$ variable as $n\rightarrow \infty$ (the latter being what the CLT does).

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  • $\begingroup$ is my answer different from yours since I cannot see the difference? $\endgroup$ Mar 9, 2014 at 2:15
  • $\begingroup$ Your answer contains two inaccuracies: a) That the CLT states something about the limiting distribution of the sample mean $\bar X$ -it doesn't, the CLT states something about the limiting distribution of the random variable $$\frac {\bar X - E(X_1)}{(\sigma/\sqrt n)}$$ -the rest are finite sample approximations, and 2) that "fat tail distributions" do not obey some (appropriate) CLT. Moreover, you asked for "a little more". $\endgroup$ Mar 9, 2014 at 2:37
  • $\begingroup$ I should take out the last sentence. However, I am still unsure how to change my answer. Could you help me revising it? $\endgroup$ Mar 9, 2014 at 2:47
  • $\begingroup$ The first sentence of your answer is correct. The third sentence should be deleted. My answer contains the correct version (i.e. the revision) of the second sentence of your answer, namely, once more, that the subject matter of the CLT is not samples of a population (neither is the "sample mean"), but standardized sums of random variables. $\endgroup$ Mar 9, 2014 at 3:01
  • $\begingroup$ For the second sentence, I revised my sentence in the following way:The CLT says the standardized sums of random variables follows a normal distribution with mean u and variance σ2/n as the sample size goes to infinity. is it right? $\endgroup$ Mar 9, 2014 at 3:07

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