Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- The sampling distribution of x̄ is the distribution of the sample mean over all possible random samples of size n from one population.
- It is centred at the population mean, so x̄ is unbiased for μ.
- Sample means vary less than individual values, and less again for larger samples.
- Its shape is normal if the population is normal, and approximately normal for large n (central limit theorem).
- Interpret every value for the specific population, sample size and variable.
Key relationships
| Quantity | Formula or rule | Notes |
|---|---|---|
| Mean of x̄ | μx̄ = μ | Does not depend on n |
| Standard deviation of x̄ | σx̄ = σ/√n | Needs independent values |
| Effect of n | n × 4 → σx̄ ÷ 2; n × 9 → σx̄ ÷ 3 | Square-root law |
| z-score for a sample mean | z = (x̄ − μ)/(σ/√n) | Not (x̄ − μ)/σ |
| Sample size for a target σx̄ = k | n ≥ (σ/k)² | Always round up |
Conditions
| Condition | Check | Purpose |
|---|---|---|
| Randomization | Random sample | Independence; x̄ unbiased |
| 10% | n ≤ 10% of N when sampling without replacement | σ/√n is accurate |
| Normal shape | Population normal (any n), or n ≥ 30 | Normal model for x̄ is valid |
Extremely skewed population: n may need to be well above 30. Non-normal population with n < 30: no normal model.
Mistakes to avoid
- Using σ for an average. Divide by √n.
- Dividing by n instead of √n.
- Claiming the CLT makes the sample data normal. It is about the distribution of x̄.
- Demanding n ≥ 30 when the population is already normal.
- Using the 10% condition as a shape check. It is about independence only.
- Rounding a required sample size down.
- Interpreting σx̄ as the spread of individuals. It is the typical distance of a sample mean from μ.
Quick self-check
- A population has σ = 30 units. What is σx̄ for random samples of 25? (30/√25 = 6 units)
- A random sample of 20 gives a certain σx̄. What sample size halves it? (4 × 20 = 80)
- A normal population has μ = 50 and σ = 6. For random samples of 16, find P(x̄ > 53). (σx̄ = 1.5, z = 2, P ≈ 0.0228)
Next: practice questions.