Theorem 117.
Suppose \(X_1, \dotsc, X_n\) are independent and identically distributed (or i.i.d.) with finite expected value \(\mu\) and finite variance \(\sigma^2\text{.}\) Define:
\begin{align*}
S_n \amp = X_1 + X_2 + \dotsb + X_n \\
A_n \amp = \frac{X_1 + X_2 + \dotsb + X_n}{n} = \frac{S_n}{n}
\end{align*}
Then, for large enough \(n\text{:}\)
\begin{align*}
S_n \amp \approx \Norm(n\mu, n\sigma^2) \\
A_n \amp \approx \Norm\left(\mu, \frac{\sigma^2}{n}\right)
\end{align*}