Central Limit Theorem

The Central Limit Theorem is one of the most important results in statistics. It states that when you take sample means from any population, they form a normal (bell-shaped) distribution—regardless of the original population's shape.

Why It Matters

• Foundation of statistics: Enables reliable predictions without knowing the true distribution
• Universal applicability: Works for virtually any underlying distribution
• Sample size rule: n ≥ 30 is typically sufficient for the theorem to apply

The Mathematics

If you repeatedly sample and calculate means:

• Sample means center around the true population mean
• ~68% fall within 1 standard deviation of the mean
• ~95% fall within 2 standard deviations
• The spread narrows as sample size increases

Real-World Applications

• Election polling: Predicting outcomes from voter surveys
• Quality control: Estimating defect rates from samples
• Medical trials: Testing drug efficacy across patients

What to Observe

• Start with any distribution shape (uniform, skewed, etc.)
• Watch the histogram of sample means become bell-shaped
• Notice the distribution narrows with larger samples