# Math Help - Clarify my understanding for central limit theorem from a statement

1. ## Clarify my understanding for central limit theorem from a statement

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 distribution that has fat tails such as Cauchy Distribtion.

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

2. ## Re: Clarify my understanding for central limit theorem from a statement

You are right that it is the sample mean distribution, not the sample value distribution that tends toward a normal distribution as sample size increases. I believe that the reason CLT fails for the Cauchy distribution is because it has no mean, not because it has fat tails. I don't think that the fat tails are what causes the mean to be undefined.

3. ## Re: Clarify my understanding for central limit theorem from a statement

So...everything is right except the last sentence right...for the second sentence, "The CLT says ..." do you think I should define what $\mu$ is? If so, do you have any recommendation for defining it?