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Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
Challenge
Summary
Quick quiz
Same topic, different board
Past paper questions

Learn9231Paper 4 · Further Probability & StatisticsNon-parametric testsSign and Wilcoxon tests

Non-parametric tests · Subtopic 1 of 1

Non-parametric Tests

When you can't assume a normal population, test the median instead. The sign test counts values above and below the hypothesised median, using $B(n,0.5)$ . The Wilcoxon signed-rank test also uses the sizes of differences: rank $|d|$ , sum the ranks for each sign, and compare the smaller sum $T$ with tables.

Cambridge 9231 Paper 4≈ 45 minRevision 1 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Know when a non-parametric test is appropriate.
Carry out a sign test.
Carry out Wilcoxon signed-rank and rank-sum tests.

Pause and think

Ten taste-testers rate a new recipe. Eight prefer it. Is that convincing evidence?

Section 2 of 14

Core idea

Start hereWhen the data aren’t normal

Taste scores, pain ratings and reaction times are often skewed or full of outliers. A $t$ -test assumes normality and can mislead. Non-parametric tests ask simpler questions: which way do the differences go, and how do they rank?

The big idea

The sign test uses only the signs of differences, with $B(n,0.5)$ under $H_0$ . The Wilcoxon signed-rank test also uses the ranks of $|d|$ , making it more powerful while still assuming only symmetry.

1. The sign test

Worked exampleRecipe preference

10 tasters: 8 prefer the new recipe, 2 the old. Test at 5% (one-tailed) whether the new one is preferred.

Goal: Decision.
Strategy: Under $H_0$ , preferences are fair coin flips.

$X\sim B(10,0.5)$
Number preferring the old recipe.
$P(X\le2)=\frac{1+10+45}{1024}$
Binomial probabilities.
$=0.0547$
p-value.

Answer

0.0547>0.05

: not significant at 5%, though close.

✓ CheckWith 9 out of 10 it would be

0.0107

, significant ✓.

2. Wilcoxon signed-rank

Why

P+Q=\frac{n(n+1)}2

$\text{ranks }1,2,\dots,n$
Each $|d|$ gets one rank.
$P+Q=1+2+\dots+n$
Every rank goes to one of the two sums.
$=\frac{n(n+1)}2$
Arithmetic series.

Under

H_0

each sum is near

\frac{n(n+1)}4

; a very small

T=\min(P,Q)

is evidence against

H_0

Worked exampleSigned-rank test

Differences: $3,-1,4,6,-2,5,7,8$ . Test at 5% (one-tailed) whether the median difference is positive.

Goal: Decision.
Strategy: Rank $|d|$ , sum by sign.

$|d|:1,2,3,4,5,6,7,8\ \to\ \text{ranks }1..8$
Already distinct.
$Q=1+2=3$
Negative differences $-1,-2$ .
$P=36-3=33$
$P+Q=36$ .
$T=3\le5$
Critical value for $n=8$ , 5% one-tail.

AnswerSignificant: evidence that the median difference is positive.

✓ CheckThe sign test here gives

P(X\le2)

with

n=8

\frac{37}{256}=0.145

, not significant. Using ranks adds power.

Which test?

Drawing diagram…

More information used, more powerful test.

Common misconceptionZeros Drop zero differences and reduce

n

before looking up critical values.

Key takeawayLarge samples For large

n

T\approx N\left(\frac{n(n+1)}4,\frac{n(n+1)(2n+1)}{24}\right)

with a continuity correction.

Pause and think

Why is the sign test less powerful than Wilcoxon?

Summary

Sign test: $B(n,0.5)$ .
Signed-rank: $T=\min(P,Q)$ , reject if small.
Ties share average ranks; drop zeros.

Non-parametric: Makes no assumption about the population distribution’s form.
Rank: Position in order of size.
Median: Middle value; the parameter these tests target.

HistoryFrank Wilcoxon. Wilcoxon, a chemist testing insecticides, published his rank tests in a two-page paper in 1945. They're now among the most used tests in science.

Fun factRobust. One wild outlier can wreck a

t

-test, but in a rank test it's just "the biggest", with rank

n

Section 3 of 14

Key formulae

Sign test

X=\#\text{ positive signs}\sim B(n,0.5)\ \text{under }H_0

Signed-rank

T=\min(P,Q),\quad P+Q=\frac{n(n+1)}2

Large n

T\approx N\left(\frac{n(n+1)}4,\ \frac{n(n+1)(2n+1)}{24}\right)

TrickZeros and ties. Discard zero differences (reduce

n

). Tied

|d|

share the average of their ranks.

Section 4 of 14

Worked examples

Example 1 · Sign test0 / 2 steps

10 differences: 8 positive, 2 negative. Test at 5% (one-tailed) whether the median difference is positive.

Example 2 · Signed-rank0 / 3 steps

Differences:

3,-1,4,6,-2,5,7,8

. Find

T

Section 5 of 14

Explore the graph

Sign test distribution B(10, 0.5)

Loading the interactive graph…

Try this

The bars show $P(X=x)$ under $H_0$ .
The area left of the line is $P(X\le c)$ : the p-value for $c$ negative signs.

Under $H_0$ signs are fair coin flips.

In the real worldCustomer ratings. Ratings on a 1–5 scale aren't normal, so researchers compare products with Wilcoxon tests.

Section 6 of 14

Common mistakes

IncorrectRanking signed differences

CorrectRank

|d|

, then attach signs

Size and sign are separate.

IncorrectKeeping zero differences

CorrectDiscard them

They give no sign.

IncorrectReject if

T

is large

CorrectReject if

T\le

critical value

T

is the smaller sum.

Data: not normal

Wilcoxon: rank it and carry on.

Section 7 of 14

Exam tip

Exam tipCheck P + Q.

P+Q

must equal

\frac{n(n+1)}2

: a free check on your ranking.

Section 8 of 14

Guided practice

SA-ST-NP-G012 marksL2 StandardCalculatorSign test

In a sign test with 10 non-zero differences, 2 are negative. Find

P(X\le2)

for

X\sim B(10,0.5)

(3 s.f.).

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-ST-NP-0011 markL1 FoundationNon-calculatorWilcoxon

For a Wilcoxon signed-rank test with

n=8

, find

P+Q

Salam Academy Original Practice · Proposed mark scheme

SA-ST-NP-0022 marksL2 StandardNon-calculatorWilcoxon

Differences:

3,-1,4,6,-2,5,7,8

. Find

T

Salam Academy Original Practice · Proposed mark scheme

SA-ST-NP-0032 marksL2 StandardNon-calculatorWilcoxon

Values

|d|=2,5,5,7

are ranked. What rank does each

5

receive?

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More questions on every skill in this subtopic are in the question bank under Practise.

Section 10 of 14

Challenge

SA-ST-NP-0042 marksL3 ExamNon-calculatorWilcoxon

For large

n

T

is approximately normal with mean

\frac{n(n+1)}4

. Find this mean when

n=40

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

Sign test: $B(n,0.5)$ .
Signed-rank: rank $|d|$ , $T=$ smaller sum.
Reject when $T\le$ critical.
No normality needed.

Section 12 of 14

Quick quiz

Q1 · SA-ST-NP-Q011 markL1 FoundationNon-calculatorSign test

A non-parametric test is preferred when

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Section 13 of 14

Same topic, different board

Specification: 9231 AS & A Level Further Mathematics
Where: Paper 4 · Non-parametric tests
Expected: Sign test, Wilcoxon signed-rank test (single sample and paired), Wilcoxon rank-sum test; normal approximations for large samples; when non-parametric tests are appropriate.
Calculator: Permitted on all papers; exact answers are expected where the question asks.
Typical style: Typical: paired data with no assumption of normality; carry out a Wilcoxon signed-rank test and compare T with the tabulated critical value.
Open in: 9231 · Paper 4

Section 14 of 14

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Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Cambridge International 9231, Current syllabus (official specification).