Central Limit Theorem

Pick a population shape (even a heavily skewed one), set a sample size n, then draw sample means. Watch the sampling distribution of x̄ narrow and become approximately Normal as n grows. Theory: mean of x̄ = μ, and SD of x̄ = σ/√n.

■ population — the shape you sample FROM

■ sampling distribution of x̄ │ Normal μ, σ/√n

Population shape

Sample size n = 30

samples drawn0

population μ—

mean of x̄ (observed)—

SD of x̄ (observed)—

σ/√n (predicted SE)—

Try: choose Right-skewed, set n = 2, and Draw 1000 — the x̄ pile is still skewed. Now reset, slide n to 30, Draw 1000 again: the pile is bell-shaped and about √15 times narrower. The population never changed.