Sign and Wilcoxon non-parametric tests
Choose sign, signed-rank or rank-sum procedures when parametric assumptions are unsuitable, and interpret their limits.
Exam scope and assessment notes
Mapped to Cambridge International AS & A Level Further Mathematics 9231. These original explanations and self-checks are independent study support, not Cambridge questions, marks or grades. The syllabus assumes prior knowledge of the relevant 9709 Mathematics content; check each component's prerequisites with your teacher. Full-paper assessment and human subject-specialist review remain outstanding.
Before you begin
- Cambridge expects learners to know the whole Cambridge International A Level Mathematics (9709) content. Paper 1 further pure mathematics draws on 9709 Papers 1 and 3. Confirm your exam year and component route with your school.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What makes a Further Mathematics solution convincing?
What you will learn
- Explain when a non-parametric test is useful
- Use sign and Wilcoxon tests for median or population comparisons
- Apply normal approximations where appropriate and respect test conditions
Ranks can avoid a normal-population assumption
A sign test uses the direction of non-zero differences and tests a median. It makes fewer distributional assumptions but generally uses less information than a rank test.
Wilcoxon signed-rank uses the signs and ranks of absolute paired differences; Wilcoxon rank-sum compares two independent samples. Wilcoxon tests require a symmetrical distribution for their usual interpretation.
Follow the syllabus conditions before ranking
The listed questions do not involve tied ranks or observations equal to the hypothesised median for the one-sample tests; paired Wilcoxon questions do not involve zero differences or tied ranks.
For sufficiently large samples, a normal approximation may be used where appropriate. State H₀ and H₁, identify the statistic and tail, then compare with a critical value. A non-parametric result is evidence under its assumptions, not proof of a cause.
Pause and explain
Which habit best checks your work on sign and wilcoxon non-parametric tests?
Worked example
Eight non-zero paired changes are recorded; seven are positive. Describe a sign test of whether the population median change is zero.
Show the worked solution
- Under H₀, positive and negative signs are equally likely, so the number of positive signs is Binomial(8, 0.5).
- Use the observed count 7 in the appropriate one- or two-sided tail, according to H₁.
- Compare the tail probability with the stated significance level; no numerical decision is possible until that level and alternative are specified.
Answer Use X~Bin(8,0.5); the decision depends on the chosen alternative and significance level.
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Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
For 9 non-zero paired differences, 7 are positive. State the sign-test null model and test statistic.
Give me a hint
Under a zero median difference, either sign is equally likely.
Compare my reasoning
- Let X be the number of positive differences.
- Under H₀, X~Bin(9,0.5).
- The observed statistic is X=7; choose the tail from the stated alternative.
X~Bin(9,0.5), observed X=7.
Look for these in your work
- Binomial model correct
- n and p identified
- Observed statistic and tail condition stated
Two independent groups are compared using the Wilcoxon rank-sum test. Explain why the paired signed-rank procedure is inappropriate.
Give me a hint
Check whether observations are naturally matched into pairs.
Compare my reasoning
- The groups contain independent observations rather than repeated measurements or matched pairs.
- There is no meaningful within-pair difference to rank.
- Rank-sum compares the two independent samples; signed-rank is designed for paired differences.
Use rank-sum for independent groups; signed-rank requires paired differences.
Look for these in your work
- Independence recognised
- Pairwise difference requirement stated
- Procedures distinguished
A paired dataset has one zero difference and the problem asks for a Wilcoxon matched-pairs signed-rank test. What should you do before applying the syllabus procedure?
Give me a hint
The stated syllabus method excludes zero-difference pairs.
Compare my reasoning
- Check the course's required treatment and do not assign a signed rank of zero as if it were an ordinary observation.
- Confirm whether the question expects the zero pair to be omitted and how the effective sample size changes.
- If that handling is not specified, state the limitation rather than silently applying the test.
Flag the zero difference: it falls outside the stated syllabus conditions, so clarify the handling before testing.
Look for these in your work
- Zero identified
- Syllabus condition checked
- Unspecified handling not invented
Common mistakes
- Using a signed-rank test when the sample is strongly asymmetric
- Dropping zero differences without checking the test's required sample size
- Choosing a one-sided tail after seeing the sample result
What can you explain now?
Which Wilcoxon condition is important for the usual test interpretation?
Compare with the explanation
The relevant distribution is symmetrical
Wilcoxon tests are valid only for symmetrical distributions in this syllabus.
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Make it stick
How do sign, signed-rank and rank-sum tests differ in data structure and assumptions?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.