Theorem 4.2.1.
Suppose \(X_1, X_2, \dotsc, X_n\) are independent and identically distributed (or iid) random variables with finite expected value \(\mu\) and finite variance \(\sigma^2\text{.}\) Let:
\begin{align*}
S_n \amp = X_1 + X_2 + \dotsb + X_n \\
A_n \amp = \frac{X_1 + X_2 + \dotsb + X_n}{n}
\end{align*}
Then, for sufficiently large \(n\text{:}\)
\begin{gather*}
S_n \approx \Norm(n \mu, n \sigma^2) \\
A_n \approx \Norm\left(\mu, \frac{\sigma^2}{n}\right)
\end{gather*}