Key Idea: The central limit theorem says how the mean of a sample behaves. For a large sample it is close to Normal, even when the population is not, so you can find probabilities for it.
| Feature | Value |
|---|---|
| Mean | μ, the same as the population |
| Standard error | σ/√n |
| Shape for large n | X̄ ≈ N(μ, σ²/n) |
Battery example
- Data — μ = 500 h, σ = 60 h, n = 100.
- Standard error — 60/√100 = 6.
- Model — X̄ ≈ N(500, 6²).
- Probability — P(495 < X̄ < 510) = 0.750 on the GDC.
One value uses σ; a sample mean uses σ/√n.
σ = 20 and n = 25. Find the standard error. 20/5 = 4.
What happens to the standard error if n goes from 25 to 100? It halves, since √n doubles.
The population is skewed. Can you still use a Normal for X̄ when n = 100? Yes. The theorem works for any population shape once n is large.
Is the mean of X̄ bigger than μ? No. It equals μ.