Central Limit Theorem

The Central Limit Theorem

• Given any population with mean μ and standard deviation σ
• The distribution of sample means X̄ approaches N(μ, σ²/n) as n grows
• This works regardless of the original distribution's shape
• Standard error: SE = σ/√n

Why It Matters

• Foundation of confidence intervals and hypothesis testing
• Explains why the normal distribution appears everywhere in nature
• Works for n ≥ 30 as a rule of thumb (less for symmetric distributions)

Try This

• Start with Uniform at n=2 — the mean distribution is already triangular, not flat
• Increase to n=30 — it looks perfectly normal
• Try Exponential — a skewed distribution. At n=5 the means are still skewed; at n=30 they're symmetric
• Try Bimodal — two peaks merge into one bell curve
• Watch how the Std of X̄ matches σ/√n

Key Insight

Averages are predictable even when individuals are not. The CLT is why polls, quality control, and scientific experiments work.