Central Limit Theorem

Central Limit Theorem (CLT) is one of the most powerful yet simple concept in statistics. Let’s state it formally:

\[\begin{aligned} & Let~n~independent~and~identically~distributted(iid)~random~variables~X_1, \dots, X_n~with~E(X_i)=\mu~and~Var(X_i)=\sigma^2~<\infty.\\ & Let~Y=\frac{1}{n}\sum_{i=1}^{n}X_i~i.e.,~the~mean.\\ & Then,~Y\sim~N(\mu,\frac{\sigma^2}{n}) \end{aligned}\] \[\begin{aligned} pf)& ~E(Y)=E(\frac{1}{n}\sum_{i=1}^{n}X_i)=\frac{1}{n}\sum_{i=1}^{n}E(X_i)=\frac{1}{n}n\mu=\mu~.\\ & Var(Y)=Var(\frac{1}{n}\sum_{i=1}^{n}X_i)\\ &~~~~~~~~~~~~~= \frac{1}{n^2}Var(\sum_{i=1}^{n}X_i)=\frac{1}{n^2}\sum_{i=1}^{n}Var(X_i)~~(\because all~X_i's~are~independent)\\ &~~~~~~~~~~~~~= \frac{1}{n^2}n\sigma^2=\frac{\sigma^2}{n}~\square \end{aligned}\]

Hence, the beauty of of the CLT theorem is that even though our original data does not follow a normal distribution the mean will be assuming \(n>30\).