Central Limit Theorem

Definition

The probability distribution of a sample statistic will be normal or nearly normal, regardless of the population distribution, given the sample size is large enough.

Generally, sample size of 30 is considered large enough.

Suppose $n$ random sample observations are taken from an infinite population, with $E(X) = \mu$ and $Var(X) = \sigma^2$. The sampling distribution of “the mean of $X$” (say $\bar x$) can be specified as:

$$\bar{x} \sim N\left(\mu, \frac{\sigma^2}{n}\right)$$

Conditions

• Sample size is large enough: $n \geq 30$.
• Sampling is random and independent.
• Population has finite variance.