## Motivation

We know the Sample mean $X_{n}$ estimates the true population mean $μ$. We also know by the Law of large numbers that as $n$ approaches infinity, $X_{n}$ approaches $μ$.

However, our sample size is usually never infinite. Without using LLN, can we still get an accurate measurement of the error of $∣X_{n}−μ∣$ without $n→∞$?

## Claim

$n$ does not need to approach infinity. However, we can say that as the sample size increases, $X_{n}$ will approach the Normal, even if $X_{1},X_{2},…,X_{n}$ are not Normal.

In other words, for a large enough size of $n$, we have approximately

$X_{n}∼N(E(X_{n}),Var(X_{n}))$### Formally

Let $Z_{n}$ be the *standardized version* (see Standardization) of $X_{n}$. That is, given $n σ =SD(X_{n})$ and $μ=E(X_{n})$, we have

Then for every $z∈R$,

$n→∞lim P(Z_{n}≤z)=Φ(z)$We say that $Z_{n}$ *converges in distribution* (or in law) to $N(0,1)$. That is, for large enough values of $n$, $Z_{n}$ is approximately equal to $N(0,1)$.