What you'll learn
How sample statistics vary from sample to sample — the bridge to inference (~7–12%).
Sampling distribution
The distribution of a statistic (e.g. x̄ or p̂) over all possible samples of size n. Center, spread, and shape matter.
Sample proportion p̂
- Mean = p; SD = √(p(1−p)/n).
- Approximately normal if np ≥ 10 and n(1−p) ≥ 10.
Sample mean x̄
- Mean = μ; SD = σ/√n.
- Central Limit Theorem: for large n (commonly n ≥ 30), x̄ is approximately normal regardless of the population's shape.
Bias vs variability
- Unbiased: the statistic's mean equals the parameter (random sampling).
- Variability decreases as n increases (larger samples → smaller standard error). Population size barely matters if it's much larger than the sample.
Exam tips
- Check the normality conditions before using a normal model.
- Increasing n reduces variability, not bias.
Common mistakes
- Forgetting the CLT needs large n only when the population is non-normal.
- Confusing the SD of the population with the SD of x̄ (σ/√n).