# Central Limit Theorem >[!Abstract] The **Central Limit Theorem** states that if you repeatedly sample a random variable a large number of times, the distribution of the sample mean will approach a normal distribution regardless of the initial distribution of the random variable. More formally, the CLT states that: $\bar{X} = \frac{X_1 + X_2 + ... + X_n}{n} \rightarrow \sim N\bigg(\mu, \frac{\sigma^2}{n} \bigg)$ The CLT allows for studying of the properties for any statistical distribution as long as there is a large enough sample size. The CLT provides the basis for much of hypothesis testing. In order for the CLE to apply - We sample with replacement, or we simulate independent random variables from the same distribution - The statistic of interest is a sum (averages and percentages are sums in disguise) - The sample size is large enough: the more skewed the population histogram is, the large the required sample size $n$ (copied from [[Module 5 - Sampling Distributions and the Central Limit Theorem#The Law of Large Numbers|Introduction to Statistics by Stanford]]) >[!Cite] Related Concepts >[[self.stats/Statistics]]