Statistics · step 6 of 13

Sampling, bias & the sampling distribution

A statistic computed from a sample varies from sample to sample, and its distribution across all possible samples is the sampling distribution. For the sample mean, E[x̄] = μ and SD(x̄) = σ/√n — the standard error. That √n is the fundamental economics of statistics: quadrupling the sample size only halves the uncertainty.

Practise sampling, bias & the sampling distribution free → All Statistics subtopics

✓ Unlimited questions · marked criterion by criterion · no card needed

Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Identify the sampling methodSimple random, stratified, cluster, systematic, or convenience. Only probability samples support inference.
Look for biasSelection bias, non-response bias and measurement bias each push estimates in a systematic direction that a bigger sample will not fix.
Compute the standard errorSE = σ/√n for a mean, or √(p(1−p)/n) for a proportion.
State the sampling distributionx̄ ~ N(μ, σ²/n) exactly for normal data, and approximately for large n by the CLT.

Worked example

A population has μ = 50 and σ = 12. Describe the distribution of x̄ for samples of size 36.

The mean of the sampling distribution is E[x̄] = μ = 50.
The standard error is σ/√n = 12/√36.
= 12/6 = 2.
With n = 36 the CLT makes the distribution approximately normal.

Answer. x̄ is approximately N(50, 2²) — centred on 50 with a standard error of 2, six times tighter than individual observations.

Where marks get dropped

These are the specific errors that cost credit on sampling, bias & the sampling distribution questions — QED's rubric penalises each of them separately.

Confusing the standard deviation of the population with the standard error of the mean. The latter is smaller by a factor of √n.
Believing a large sample fixes bias. A biased sampling method gives a precisely wrong answer — the classic 1936 Literary Digest poll had 2.4 million responses and still called the election wrongly.
Dividing by n rather than √n in the standard error.

Practise this until it is automatic

Unlimited fresh questions

QED generates new sampling, bias & the sampling distribution problems on demand at warm-up, exam and challenge level, so you can drill this one skill until it stops costing you marks.

Marked like an examiner

Every answer is scored against a point-by-point rubric with partial credit, so you see exactly which step of the method broke down — not just a tick or a cross.

Answer in real notation

A one-tap symbol palette, a visual equation editor and a truth-table builder — or photograph your handwritten working and QED converts it to LaTeX.

Saved to your library

Every question you generate is kept and re-takeable as a timed exam, and your Statistics mastery is tracked so you know when this is exam-ready.

Sampling, bias & the sampling distribution — frequently asked questions

What is the difference between σ and the standard error?

σ measures the spread of individual observations; the standard error measures the spread of the sample MEAN across repeated samples. SE = σ/√n.

Why does √n appear?

Because variances add for independent observations: Var(x̄) = Var(ΣX)/n² = nσ²/n² = σ²/n. Taking the square root gives σ/√n.

Does the population size matter?

Barely, provided the sample is a small fraction of it. A sample of 1000 works about as well for a city as for a country — the finite population correction is negligible.

The rest of Statistics

Describing data, distributions, estimation and hypothesis tests. Each subtopic below has its own method, worked example and mark-losing traps.

Ready to make sampling, bias & the sampling distribution exam-proof?

Generate your first questions free — no card, no setup, no personal data stored. Practise until the method is second nature.

Start practising free →