Sampling Theory and Sampling Distributions
Domain 5: Sampling Theory and Sampling Distributions
This domain tests why samples are used, how they are drawn, and how a statistic behaves over repeated samples: standard error, the sampling distribution of the mean, and the roles of the t, χ² and F distributions.
Core rules
- A population parameter (μ, σ, P) is a constant; a sample statistic (x̄, s, p) is a random variable with its own distribution.
- Simple random sampling may be with replacement (SRSWR) or without (SRSWOR); other schemes are stratified, systematic and cluster sampling.
- Sampling error falls as n grows; non-sampling errors (non-response, faulty frame, measurement bias) do not.
- For the mean, E(x̄) = μ and SE(x̄) = σ/√n.
- In SRSWOR from a finite population, SE(x̄) = (σ/√n)·√[(N − n)/(N − 1)]; the second factor is the finite population correction and tends to 0 as n → N.
- Central limit theorem: for large n the sampling distribution of x̄ is approximately normal whatever the shape of the population.
- If σ is unknown and n is small, (x̄ − μ)/(s/√n) follows t with ν = n − 1; (n − 1)s²/σ² follows χ² with ν = n − 1; the ratio of two independent variance estimates follows F.
Worked example
A population of N = 500 has σ = 10; a sample of n = 100 is drawn without replacement. SE ignoring the fpc = 10/√100 = 1.00. fpc = √[(500 − 100)/(500 − 1)] = √(400/499) = 0.8953, so the correct SE = 0.8953. Reasoning: a fifth of the population is being observed, so the sample mean is more precise than the infinite-population formula suggests; ignoring the fpc overstates the standard error by about 10%.
Traps
- Using σ/n or σ/√(n − 1) for the standard error of the mean.
- Applying the fpc when sampling is with replacement or the population is infinite.
- Using z when σ is unknown and n < 30.
- Confusing the SD of the sample with the SD of the sample mean.
- Believing a large sample removes bias; it only reduces sampling error.