Sampling Theory and Sampling Distributions
Sampling theory explains why a sample statistic can estimate a population parameter, and quantifies the error involved. The exam tests sampling methods, sampling versus non-sampling errors, standard error, and the sampling distribution of the mean.
Core rules
- A parameter is a population constant (μ, σ); a statistic is a sample value (x̄, s) used to estimate it.
- Methods: simple random sampling (with or without replacement), stratified (heterogeneous groups, homogeneous within), systematic (every k-th unit), cluster and multistage sampling.
- Sampling error arises from observing only part of the population; non-sampling errors include bias, non-response and measurement mistakes.
- For simple random sampling with replacement, E(x̄) = μ and V(x̄) = σ²/n, so standard error SE = σ/√n. Sampling without replacement from a finite population of size N multiplies this by the finite population correction √[(N − n)/(N − 1)].
- Standard error of a sample proportion = √(pq/n).
- The central limit theorem states that for large n the distribution of x̄ is approximately normal whatever the population shape.
Worked example
A population has σ = 10 and a sample of n = 25 is drawn. SE = 10/√25 = 2. If instead N = 100, the correction is √(75/99) = 0.870, giving SE = 2 × 0.870 = 1.74.
Traps that cost marks
Dividing by n rather than √n; ignoring the finite population correction when n is a sizeable fraction of N; describing non-response as sampling error; believing the central limit theorem requires a normal population; confusing the standard deviation of the population with the standard error of the mean.