Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- A parameter describes a population and is fixed (usually unknown). A statistic comes from a sample and varies from sample to sample.
- An estimator is a statistic used to estimate a parameter: a rule such as p̂ = x / n.
- A point estimate is the single number the estimator gives for your sample.
- An estimator is unbiased if the mean of its sampling distribution equals the parameter. It neither overestimates nor underestimates on average.
- An estimator is biased if its sampling distribution is centred above or below the parameter.
Key relationships
| Parameter | Estimator | Unbiased (random samples)? |
|---|---|---|
| proportion p | sample proportion p̂ = x / n | Yes |
| mean μ | sample mean x̄ = Σxᵢ / n | Yes |
| variance σ² | sample variance s² = Σ(xᵢ − x̄)² / (n − 1) | Yes |
| variance σ² | Σ(xᵢ − x̄)² / n | No: underestimates |
| population maximum | sample maximum | No: underestimates |
| population range | sample range | No: underestimates |
How to justify: find (or simulate) the sampling distribution, find its mean, compare with the parameter. Equal (or, with simulation, very close with no consistent direction): unbiased. Clearly above or below: biased, in that direction.
Assumptions
- The samples are random. An unbiased estimator gives biased results if the sampling method is biased (convenience, voluntary response, undercoverage).
- Simulation gives an approximate sampling distribution, so expect a small difference from the parameter even for an unbiased estimator. Say “appears unbiased”.
Mistakes to avoid
- Unbiased = exact. Individual estimates still miss; they balance out on average.
- Judging bias from one sample.
- Thinking a larger sample removes bias. It reduces variability only.
- Wrong notation: writing p or μ for a sample result.
- No context: name the parameter and the population (“the proportion of all members who…”).
Quick self-check
- In a random sample of 75 commuters, 18 cycle to work. Give the point estimate and name the parameter. (p̂ = 18 ÷ 75 = 0.24 estimates p, the proportion of all commuters in the population who cycle to work.)
- An estimator’s simulated values have mean 41.9 when the parameter is 50. Biased or unbiased? (Biased: centred well below 50, so it underestimates.)
- True or false: if an estimator is unbiased, a sample of 1,000 will give exactly the parameter. (False: estimates vary; only their average equals the parameter.)
Next: practice questions.