Central Limit Theorem

The Central Limit Theorem states that no matter the shape or distribution of the original data, if you take a large enough random sample, and average the values, those sample averages will form a Normal Distribution. Furthermore, the sample mean will approach the population mean, and sample mean variance will shrink to 0.

$$\text{Let }P=X_{\mathrm{1..}n}\forall i,j\in [1,n],X_i{\sim }^{iid}{X}_{j}E[X_i]=\mu \text{Var}(X_i)={\sigma }^{2}n\to \infty$$ $$\frac{P}{Z_n=\frac{{\sum }_{k=1}^n{X}_k-n\mu }{\sigma \sqrt{n}}=\frac{\bar{X}-\mu }{\sigma /\sqrt{n}}=\frac{\bar{X}-E[\bar{X}]}{\sqrt{\text{Var}(\bar{X})}}{\to }^d\text{Nor}(0,1)}$$ $$\frac{P}{\overline{X}\approx \text{Nor}\left(\mu ,(\frac{\sigma }{\sqrt{n}}{)}^2\right)}$$