Central Limit Theorem and Inference

The Central Limit Theorem

Central Limit Theorem. Given a population described by any probability distribution having mean µ and finite variance σ², the sampling distribution of the sample mean $\bar{X}$ computed from random samples of size n from this population will be approximately normal with mean µ (the population mean) and variance σ²/n (the population variance divided by n) when the sample size n is large.

Standard Error of the Sample Mean

For sample mean $\bar{X}$ calculated from a sample generated by a population with standard deviation $\sigma$, the standard error of the sample mean is given by one of two expressions:

when we know $\sigma$, the population standard deviation, or by

when we do not know the population standard deviation and need to use the sample standard deviation, s to estimate it.

The estimate of s is given by the square root of the sample variance, $s^2$, calculated as follows:

To summarise, the central limit theorem tells us that when we sample from any distribution, the distribution of the sample mean will have the following properties as long as our sample size is large:

• The distribution of the sample mean  $\bar{X}$ will be approximately normal.
• The mean of the distribution of $\bar{X}$ will be equal to the mean of the population from which the samples are drawn.
• The variance of the distribution of $\bar{X}$ will be equal to the variance of the population divided by the sample size.