# CLT and Sample statistics

As far as I understood CLT(Central Limit Theorem) specifies that the means of various samples will be Normally distribution with mean=population_mean and std. dev=population_std_dev/n (where n is size of sample).

But

1. What does CLT tells about distribution of data points/observation within sample (remember every sample has n data points/observations)?
2. What does CLT says about relation between sample and population statistics ?

Thanks in advance. Any help is appreciated.

## 1 Answer

Assuming I understand you correctly:

1. The CLT does not give any result about the individual samples. Only their sum/average.
2. Given that the sample $$X_1,X_2,\dots,X_n$$ is i.i.d. with mean $$\mu$$ and variance $$\sigma^2$$, we have $$\mathbf{E}\left[ \frac{X_1+X_2+\cdots+X_n}{n} \right]=\mu \\ Var[\overline{X}] = \frac{\sigma^2}{n}$$
• ok. Thanks. My first question is answered by your first point. Aug 1 at 12:38