# Central Limit Theorem

Hello,

The Central Limit Theorem states that if we draw a sample from a population with a mean of mu and a variance of (variance/n), then the sampling distribution of the sample mean will be normally distributed with a men of mu and a variance of (variance/n).

My question is: Through the central limit theorem we assume that we keep drawing samples of a large n and plot their average returns, and eventually get a sampling distribution that will be normally distributed with a resemblance of the population; however in the questions I face, we are only drawing one sample of a large n. So how can we conclude that one sample drawn with a large n, is normally distributed, whereby the central limit theorem needs a lot of samples of size n?

Thank you

The central limit theorem doesn’t say that the sample will be close to normally distributed.

It says that if we take a large number of samples, the distribution of the means of those samples will be close to normally distributed.

Okay, but in most of the questions I am encountering, we are only taking one sample of usually N>30, is this sufficient?

It might be sufficient. There are cases where the central limit theorem doesn’t hold at any sample size.

Also, the CLT doesn’t say that the sampling distribution will look like the true distribution. The true distribution may be flat or exponential, while the sampling distribution may look approximately normal for samples of size 90 (arbitrary number, here). The sampling distribution doesn’t necessarily resemble the underlying distribution.

I would review the following.

Thank you guys!