Central Limit Theorem

Key idea

• $Z_n = \frac{S_n - n\mu}{\sigma\sqrt{n}}$ standardizes the sum to zero mean and unit variance.
• As n grows, the histogram of $Z_n$ converges to the standard normal $N(0,1)$, shown in green.
• This holds for any source distribution with finite mean and variance. Try switching between them.
• Symmetric distributions converge faster than skewed ones. Compare Uniform (fast) vs Exponential (slower).