central limit theorem

Let $X_1,X_2,\dots ,X_n$ be a random sample of size $n$ from a probability distribution with mean $\mu$ and variance ${\sigma }^2$.

If $n$ is sufficiently large, then the sample total $T_o=\sum X_i$ and sample mean $\bar{X}=\frac{T_o}{n}$ have approximately a normal distribution.

Formally,
if $X_1,X_2,\dots ,X_{10}$ are i.i.d. with mean $\mu$ and variance ${\sigma }^2$, then $\bar{X}\sim \mathcal{N}(\mu ,\frac{{\sigma }^2}{n})$ and $T_o\sim \mathcal{N}(n\mu ,n{\sigma }^2)$.

In statistical analysis, the CLT is commonly used for $n>30$.

N.b. the variance of the sampling distribution for statistic $\bar{X}$ decreases as $n$ increases. Therefore, by the law of large numbers, $\bar{X}\to \mu$ as $n\to \mathrm{\infty }$.