What's the main difference between the Wilcoxon signed-rank test and the Wilcoxon rank-sum test?

The signed-rank test is for a single sample (or paired data, where you test the median of the differences). It tests if the population median is a specific value. The rank-sum test is for two independent samples, and it tests whether the two populations they come from have the same median.

Why do we use the median in these tests instead of the mean?

The median is a more 'robust' measure of central tendency than the mean. This means it is less affected by extreme values (outliers) or skewed data. Since non-parametric tests are designed for situations where the data might not be normally distributed, using the median is a more reliable approach.

What happens if there are ties in the data for a rank-sum test?

If two or more observations are identical, they are each assigned the average of the ranks they would have occupied. For example, if two values are tied for the 5th and 6th positions, they both receive a rank of (5+6)/2 = 5.5. The next value then receives rank 7. The formulae for the variance in the large sample approximations should be adjusted for ties, but this is not typically required in the 9231 syllabus unless the formula is given.

Where can I learn Non-parametric tests for Cambridge 9231?

This free lesson on MarkScheme covers syllabus point 4.4 (Non-parametric tests) for Cambridge A-Level Further Mathematics. It includes visual revision, key points, and links to real past-paper questions with mark-scheme marking at markscheme.app/courses/9231/4-4-non-parametric-tests.

Is Non-parametric tests (4.4) on 9231 free to study?

Yes. The Further Mathematics 9231 course on MarkScheme is 100% free — including premium visual lessons, exam tips, and past-paper marking. No paywall for Non-parametric tests.

How do I revise Non-parametric tests for 9231 exams?

Start with the visual steps on this page, note the key exam points, then attempt a 9231 past-paper question on Non-parametric tests using MarkScheme's /mark tool. Mark strictly against the official scheme and re-read any marks you lost.

Non-parametric tests 9231 — Free A-Level…

Non-parametric tests

In previous statistics modules, you likely used tests like the t-test or z-test. These are known as 'parametric' tests because they assume the data comes from a population that follows a specific distribution, usually the normal distribution, which is defined by parameters like the mean ( $\mu$ ) and standard deviation ( $\sigma$ ). But what if we can't make that assumption? What if our data is heavily skewed, contains significant outliers, or is only ranked? This is where non-parametric tests become essential. They provide a robust alternative by making fewer assumptions about the underlying population distribution.

BY THE END, YOU CAN…

1Understand the circumstances under which a non-parametric test is appropriate.
2Apply the Sign Test and Wilcoxon signed-rank test to investigate a population median.
3Apply the Wilcoxon rank-sum test to compare the medians of two populations.
4Use normal approximations for Wilcoxon tests with large samples, including continuity corrections.

In simple terms

A friendly intro before the formal notes — no formulas yet.

Testing Without Assumptions

Non-parametric tests are statistical methods for testing hypotheses that don't require your data to fit a specific pattern, like the bell curve of a normal distribution. They work by comparing the ranks of data points, not their exact values.

THINK OF IT LIKE…

Imagine judging a baking competition. A parametric test would be like scoring each cake out of 100, where one disastrous cake could heavily skew the average score for a team. A non-parametric test is like simply ranking the cakes from best to worst. This ranking isn't affected by how badly the worst cake failed; it's just last, giving a more robust comparison of the teams' overall baking ability.

1
State the null and alternative hypotheses regarding the population median(s).
2
Calculate the test statistic by ranking the data or counting signs, discarding any zero differences.
3
Find the critical value from the statistical tables for your chosen significance level.
4
Compare your test statistic with the critical value to decide whether to reject the null hypothesis.

Explore the concept

Use the live diagram and synced steps — play it or tap a step card to walk through.

Key formulas

Tap any symbol to reveal exactly what it means and its units.

Tap a symbol — great for exam definitions

Full topic notes

Formal explanation with the rigour you need for the exam.

The Sign Test

The Sign Test is the simplest non-parametric test. It is used to test a hypothesis about the median of a single population. The core idea is to count how many data points are above (+) and below (-) the hypothesised median. If the null hypothesis is true, we would expect a roughly equal number of pluses and minuses. Any significant deviation from this suggests the true median is different.

Hypotheses: $H_0: m = m_0$ vs $H_1: m \neq m_0$ (or $m > m_0$ , or $m < m_0$ ), where $m$ is the population median.
Procedure: For each data point $x_i$ , record a '+' if $x_i > m_0$ and a '−' if $x_i < m_0$ . Ignore any data points where $x_i = m_0$ and reduce the sample size $n$ accordingly.
Test Statistic: Let $N_+$ be the number of '+' signs and $N_-$ be the number of '−' signs. The test statistic $S$ is the smaller of $N_+$ and $N_-$ .
Distribution: Under $H_0$ , the number of pluses (or minuses) follows a binomial distribution, $X \sim B(n, 0.5)$ . We can use this to find a p-value, $P(X \le S)$ .

The Wilcoxon Signed-Rank Test

The Wilcoxon signed-rank test is also used to test the median of a single population. It is generally more powerful than the sign test because it uses more information from the data. Instead of just noting whether a value is above or below the median, it considers the magnitude of these differences by ranking them. This makes it more sensitive to changes in the data.

Procedure: Calculate the difference $d_i = x_i - m_0$ for each data point. Discard any zero differences.
Ranking: Rank the absolute values of the differences, $|d_i|$ , from smallest to largest. Average ranks for any ties.
Summing Ranks: Calculate $W_+$ , the sum of ranks corresponding to positive differences, and $W_-$ , the sum of ranks corresponding to negative differences.
Test Statistic: The test statistic is $W = \min(W_+, W_-)$ .
Critical Value: Compare $W$ to the critical value from the MF19 tables for the given $n$ and significance level. Reject $H_0$ if $W$ is less than or equal to the critical value.

The Wilcoxon Rank-Sum Test

This test is the non-parametric equivalent of the independent two-sample t-test. It is used to determine if two independent samples have been drawn from populations with the same median. The method involves combining both samples, ranking all the data points, and then summing the ranks for one of the samples.

Hypotheses: $H_0$ : The two populations have the same median. $H_1$ : The two populations do not have the same median (or one is greater than the other).
Procedure: Let the sample sizes be $n_1$ and $n_2$ . Combine all $N=n_1+n_2$ observations into a single set.
Ranking: Rank the combined data from smallest to largest. Average ranks for any ties.
Test Statistic: Calculate the sum of the ranks for the smaller sample. Let this be $R_s$ . If $n_1=n_2$ , either can be used.
Critical Region: Compare $R_s$ to the critical region from the MF19 tables for $n_1$ , $n_2$ and the significance level. Reject $H_0$ if $R_s$ falls inside the critical region (e.g., is less than or equal to the lower value, or greater than or equal to the upper value).

EXAM TIP

For the Wilcoxon rank-sum test, the tables in MF19 give the critical region for the rank sum of the smaller sample. Always calculate the rank sum for the smaller sample to use the tables directly. If you accidentally sum the ranks for the larger sample ( $R_L$ ), you can find the sum for the smaller sample ( $R_S$ ) using the formula $R_S + R_L = \frac{1}{2}N(N+1)$ , where $N$ is the total sample size.

Large Sample Approximations

The tables for Wilcoxon tests only go up to a certain sample size (e.g., $n=20$ for signed-rank). For larger samples, the distribution of the test statistic can be approximated by a normal distribution. This allows us to calculate a z-score and use standard normal tables to find a p-value.

Wilcoxon Signed-Rank Test: For large $n$ , the test statistic $W$ is approximately normal with: Mean: $E(W) = \frac{1}{4}n(n+1)$ Variance: $Var(W) = \frac{1}{24}n(n+1)(2n+1)$

Wilcoxon Rank-Sum Test: For large $n_1$ and $n_2$ , the rank sum $R$ of the sample of size $n_1$ is approximately normal with: Mean: $E(R) = \frac{1}{2}n_1(n_1+n_2+1)$ Variance: $Var(R) = \frac{1}{12}n_1 n_2 (n_1+n_2+1)$

EXAM TIP

When using a normal approximation for a discrete test statistic like $W$ or $R$ , you must apply a continuity correction. If you are calculating $P(W \le k)$ , you use $k+0.5$ . If you are calculating $P(W \ge k)$ , you use $k-0.5$ . This adjustment accounts for approximating a discrete distribution with a continuous one.

Worked examples

See the formulas applied — reveal one step at a time, like the exam.

EXAMPLE 1

A local authority claims the median response time for an ambulance is 8 minutes. A random sample of 10 emergency calls had the following response times (in minutes): 7.5, 9.1, 8.2, 10.5, 7.9, 8.0, 9.5, 11.2, 7.1, 8.9. Test the authority's claim at the 5% significance level.

1
Let $m$ be the median response time. 1. Hypotheses: $H_0: m = 8$ $H_1: m \neq 8$ (two-tailed test)

EXAMPLE 2

The manufacturer of a battery claims its median lifespan is 250 hours. A random sample of 8 batteries is tested and their lifespans are: 241, 262, 255, 238, 245, 271, 231, 258. Use a Wilcoxon signed-rank test at the 5% level to test if the median lifespan is different from 250 hours.

1
Let $m$ be the median lifespan. 1. Hypotheses: $H_0: m = 250$ $H_1: m \neq 250$ (two-tailed test)

How it all connects

The big idea sits in the middle — tap a linked idea to explore the link.

MAIN IDEANon-parametric tests

Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.

Glossary

Try to recall each definition before you reveal it.

Quick check

Answer in your head first — then tap to check. No pressure.

Revision flashcards

Flip the card. Test yourself before the exam.

QUESTION · 1 / 11

What is a non-parametric test?

TAP OR SPACE TO FLIP

ANSWER

A hypothesis test that does not assume the data is drawn from a population with a specific distribution (e.g., normal). It often uses ranks or signs instead of the original data values.

TAP OR SPACE TO FLIP BACK

1 / 11 · ← → space

Key takeaways

Review these before you close the topic — retrieval beats re-reading.

✓
Hypotheses: $H_0: m = m_0$ vs $H_1: m \neq m_0$ (or $m > m_0$ , or $m < m_0$ ), where $m$ is the population median.
✓
Procedure: For each data point $x_i$ , record a '+' if $x_i > m_0$ and a '−' if $x_i < m_0$ . Ignore any data points where $x_i = m_0$ and reduce the sample size $n$ accordingly.
✓
Test Statistic: Let $N_+$ be the number of '+' signs and $N_-$ be the number of '−' signs. The test statistic $S$ is the smaller of $N_+$ and $N_-$ .
✓
Distribution: Under $H_0$ , the number of pluses (or minuses) follows a binomial distribution, $X \sim B(n, 0.5)$ . We can use this to find a p-value, $P(X \le S)$ .

Practice — then mark it

The whole point: a real Cambridge question, marked mark-by-mark.

Real past paper[0]

Test Your Knowledge on Non-Parametric Tests

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

Extra simulations & links

PhET, GeoGebra and other curated tools — open in a new tab.

Frequently asked

✓

Checkpoint

One marked question is worth ten re-reads — close the loop before you move on.

Checkpoint

Reading it isn’t knowing it — prove it.

Before you move on: do Test Your Knowledge on Non-Parametric Tests on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.

Attempt & get marked →Takes about a minute · you’ll land back on this lesson

← PREVIOUS4.3 χ²-tests NEXT →4.5 Probability generating functions

Non-parametric tests

BY THE END, YOU CAN…

1Understand the circumstances under which a non-parametric test is appropriate.
2Apply the Sign Test and Wilcoxon signed-rank test to investigate a population median.
3Apply the Wilcoxon rank-sum test to compare the medians of two populations.
4Use normal approximations for Wilcoxon tests with large samples, including continuity corrections.

In simple terms

A friendly intro before the formal notes — no formulas yet.

Testing Without Assumptions

THINK OF IT LIKE…

1
State the null and alternative hypotheses regarding the population median(s).
2
Calculate the test statistic by ranking the data or counting signs, discarding any zero differences.
3
Find the critical value from the statistical tables for your chosen significance level.
4
Compare your test statistic with the critical value to decide whether to reject the null hypothesis.

Explore the concept

Use the live diagram and synced steps — play it or tap a step card to walk through.

Key formulas

Tap any symbol to reveal exactly what it means and its units.

Tap a symbol — great for exam definitions

Full topic notes

Formal explanation with the rigour you need for the exam.

The Sign Test

Hypotheses: $H_0: m = m_0$ vs $H_1: m \neq m_0$ (or $m > m_0$ , or $m < m_0$ ), where $m$ is the population median.
Procedure: For each data point $x_i$ , record a '+' if $x_i > m_0$ and a '−' if $x_i < m_0$ . Ignore any data points where $x_i = m_0$ and reduce the sample size $n$ accordingly.
Test Statistic: Let $N_+$ be the number of '+' signs and $N_-$ be the number of '−' signs. The test statistic $S$ is the smaller of $N_+$ and $N_-$ .
Distribution: Under $H_0$ , the number of pluses (or minuses) follows a binomial distribution, $X \sim B(n, 0.5)$ . We can use this to find a p-value, $P(X \le S)$ .

The Wilcoxon Signed-Rank Test

Procedure: Calculate the difference $d_i = x_i - m_0$ for each data point. Discard any zero differences.
Ranking: Rank the absolute values of the differences, $|d_i|$ , from smallest to largest. Average ranks for any ties.
Summing Ranks: Calculate $W_+$ , the sum of ranks corresponding to positive differences, and $W_-$ , the sum of ranks corresponding to negative differences.
Test Statistic: The test statistic is $W = \min(W_+, W_-)$ .
Critical Value: Compare $W$ to the critical value from the MF19 tables for the given $n$ and significance level. Reject $H_0$ if $W$ is less than or equal to the critical value.

The Wilcoxon Rank-Sum Test

Hypotheses: $H_0$ : The two populations have the same median. $H_1$ : The two populations do not have the same median (or one is greater than the other).
Procedure: Let the sample sizes be $n_1$ and $n_2$ . Combine all $N=n_1+n_2$ observations into a single set.
Ranking: Rank the combined data from smallest to largest. Average ranks for any ties.
Test Statistic: Calculate the sum of the ranks for the smaller sample. Let this be $R_s$ . If $n_1=n_2$ , either can be used.
Critical Region: Compare $R_s$ to the critical region from the MF19 tables for $n_1$ , $n_2$ and the significance level. Reject $H_0$ if $R_s$ falls inside the critical region (e.g., is less than or equal to the lower value, or greater than or equal to the upper value).

EXAM TIP

Large Sample Approximations

Wilcoxon Signed-Rank Test: For large $n$ , the test statistic $W$ is approximately normal with: Mean: $E(W) = \frac{1}{4}n(n+1)$ Variance: $Var(W) = \frac{1}{24}n(n+1)(2n+1)$

EXAM TIP

Worked examples

See the formulas applied — reveal one step at a time, like the exam.

EXAMPLE 1

1
Let $m$ be the median response time. 1. Hypotheses: $H_0: m = 8$ $H_1: m \neq 8$ (two-tailed test)

EXAMPLE 2

1
Let $m$ be the median lifespan. 1. Hypotheses: $H_0: m = 250$ $H_1: m \neq 250$ (two-tailed test)

How it all connects

The big idea sits in the middle — tap a linked idea to explore the link.

MAIN IDEANon-parametric tests

Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.

Glossary

Try to recall each definition before you reveal it.

Quick check

Answer in your head first — then tap to check. No pressure.

Revision flashcards

Flip the card. Test yourself before the exam.

QUESTION · 1 / 11

What is a non-parametric test?

TAP OR SPACE TO FLIP

ANSWER

A hypothesis test that does not assume the data is drawn from a population with a specific distribution (e.g., normal). It often uses ranks or signs instead of the original data values.

TAP OR SPACE TO FLIP BACK

1 / 11 · ← → space

Key takeaways

Review these before you close the topic — retrieval beats re-reading.

✓
Hypotheses: $H_0: m = m_0$ vs $H_1: m \neq m_0$ (or $m > m_0$ , or $m < m_0$ ), where $m$ is the population median.
✓
Procedure: For each data point $x_i$ , record a '+' if $x_i > m_0$ and a '−' if $x_i < m_0$ . Ignore any data points where $x_i = m_0$ and reduce the sample size $n$ accordingly.
✓
Test Statistic: Let $N_+$ be the number of '+' signs and $N_-$ be the number of '−' signs. The test statistic $S$ is the smaller of $N_+$ and $N_-$ .
✓
Distribution: Under $H_0$ , the number of pluses (or minuses) follows a binomial distribution, $X \sim B(n, 0.5)$ . We can use this to find a p-value, $P(X \le S)$ .

Practice — then mark it

The whole point: a real Cambridge question, marked mark-by-mark.

Real past paper[0]

Test Your Knowledge on Non-Parametric Tests

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

Extra simulations & links

PhET, GeoGebra and other curated tools — open in a new tab.

Frequently asked

✓

Checkpoint

One marked question is worth ten re-reads — close the loop before you move on.

Checkpoint

Reading it isn’t knowing it — prove it.

Before you move on: do Test Your Knowledge on Non-Parametric Tests on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.

Attempt & get marked →Takes about a minute · you’ll land back on this lesson

← PREVIOUS4.3 χ²-tests NEXT →4.5 Probability generating functions

Revise Non-parametric tests for Cambridge 9231 (4.4)

In simple terms

Explore the concept

Key formulas

Full topic notes

The Sign Test

The Wilcoxon Signed-Rank Test

The Wilcoxon Rank-Sum Test

Large Sample Approximations

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

Test Your Knowledge on Non-Parametric Tests

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

In simple terms

Explore the concept

Key formulas

Full topic notes

The Sign Test

The Wilcoxon Signed-Rank Test

The Wilcoxon Rank-Sum Test

Large Sample Approximations

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

Test Your Knowledge on Non-Parametric Tests

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

Discuss Non-parametric tests