Chapter map

1. 1. Question
2. 2. Pooled ranks
3. 3. Wilcoxon sum
4. 4. Wilcoxon mean
5. 5. Wilcoxon variance
6. 6. Location Z
7. 7. AB scores
8. 8. AB sum
9. 9. AB mean
10. 10. AB variance
11. 11. Scale Z
12. 12. Lepage total
13. 13. Joint reference
14. 14. Component reading
15. 15. Effect sizes
16. 16. Claim boundary

Visit the How Mathematics Works hub for the wider series and its connections to education, reasoning, science, computing, design and careers.

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1. Question

Here the geometry becomes operational. Define joint equality before analysis.

A concrete situation

Two teaching methods are compared.

What the mathematics does not license

A vague question invites reinterpretation. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Write the null. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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2. Pooled ranks

This step deserves a careful pause. Rank all observations.

A concrete situation

A occupies ranks 2,3,4.

What the mathematics does not license

Ties change both components. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Use midranks. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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3. Wilcoxon sum

The useful question is not only how to calculate, but what the calculation assumes. Add ordinary ranks for A.

A concrete situation

W=9.

What the mathematics does not license

Group coding changes sign only. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

State group. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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4. Wilcoxon mean

A small example makes the mechanism visible. Compute m(N+1)/2.

A concrete situation

Mean=10.5.

What the mathematics does not license

Unequal sizes need their own mean. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Show formula. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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5. Wilcoxon variance

This is where modelling choices begin to affect evidence. Compute mn(N+1)/12 without ties.

A concrete situation

Variance=5.25.

What the mathematics does not license

Tie correction may be required. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Apply it. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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6. Location Z

The central mathematical idea is concrete. Standardize the rank sum.

A concrete situation

ZW≈-.655.

What the mathematics does not license

Z is not an effect size. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Report median shift. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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7. AB scores

Here the geometry becomes operational. Assign 1,2,3,3,2,1.

A concrete situation

Central ranks score high.

What the mathematics does not license

Score orientation can reverse. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Document convention. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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8. AB sum

This step deserves a careful pause. Add scores for A.

A concrete situation

A sum=8.

What the mathematics does not license

Raw sum depends on m. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Standardize. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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9. AB mean

The useful question is not only how to calculate, but what the calculation assumes. Use sample fraction of total scores.

A concrete situation

Expected sum=6.

What the mathematics does not license

Ties alter score set. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Recompute. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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10. AB variance

A small example makes the mechanism visible. Use finite-population variance.

A concrete situation

Variance=1.2.

What the mathematics does not license

Independence formulas are wrong. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Verify. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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11. Scale Z

This is where modelling choices begin to affect evidence. Standardize the AB component.

A concrete situation

ZA≈1.826.

What the mathematics does not license

Shape can drive scale scores. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Plot quantiles. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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12. Lepage total

The central mathematical idea is concrete. Add squared components.

A concrete situation

L≈3.762.

What the mathematics does not license

Signs disappear in total. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Keep signed Z values. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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13. Joint reference

Here the geometry becomes operational. Use exact, permutation or chi-square calibration.

A concrete situation

Chi-square 2 is a large-sample guide.

What the mathematics does not license

Small n makes approximation coarse. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Enumerate labels. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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14. Component reading

This step deserves a careful pause. Compare squared contributions.

A concrete situation

Scale dominates this example.

What the mathematics does not license

Post-hoc component tests add multiplicity. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Use descriptive language. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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15. Effect sizes

The useful question is not only how to calculate, but what the calculation assumes. Report location and spread in units.

A concrete situation

Median and IQR accompany L.

What the mathematics does not license

Omnibus significance lacks magnitude. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Add intervals. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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16. Claim boundary

A small example makes the mechanism visible. Describe distributional evidence.

A concrete situation

Groups differ mainly in spread.

What the mathematics does not license

The test does not identify cause. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Check design. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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A seven-day practice lab

1. Day 1: define the decision, observation unit, X variables and Y or reference condition.
2. Day 2: reproduce the smallest worked example and check every dimension, unit and sign.
3. Day 3: build a transparent baseline and write what it assumes.
4. Day 4: fit the main method inside a training-only pipeline; record every parameter.
5. Day 5: challenge one assumption with a deliberately difficult dataset or operating regime.
6. Day 6: compare validation metrics with the practical workload created by errors.
7. Day 7: present one supported conclusion, one unresolved uncertainty and one safe next experiment.

Questions for a student, parent or mentor

• Which part of the model is a mathematical definition, and which part is a choice?
• What small calculation would catch a sign, scale or dimension error?
• What evidence comes from training, validation and the untouched test?
• What benign condition could look like a fault or prediction failure?
• How would the answer change for another response, asset, school project or time period?
• What should a human review before acting on the score?

A method-specific audit before deployment

The Lepage test combines two standardized rank components: a Wilcoxon component for location and an Ansari–Bradley component for scale. Squaring and adding the components creates one joint statistic that can respond when groups differ in centre, spread or both.

With group A at pooled ranks 2,3,4 and group B at ranks 1,5,6, the Wilcoxon rank sum for A is 9 rather than its null mean 10.5. Its variance is 3×3×7/12=5.25, so ZW≈-0.655. Ansari–Bradley scores 1,2,3,3,2,1 give A a sum of 8 rather than 6; the finite-population variance is 1.2, so ZA≈1.826.

The combined illustration is L=ZW²+ZA²≈3.762. Under regular large-sample conditions the reference is chi-square with two degrees of freedom, but small samples, ties and component dependence favour exact or permutation calibration. The joint result should be unpacked by showing both components.

A rejection answers a broad location–scale equality question. It does not identify which moment changed, prove a normal model or quantify an operationally important difference. Display medians, quantiles, spread measures and uncertainty so readers can see whether location, scale, shape or outliers drive the result.

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Frequently asked questions

What does Lepage test?

It tests a joint two-sample location–scale null using rank components.

What is the location component?

It is a standardized Wilcoxon rank-sum statistic.

What is the scale component?

It is a standardized Ansari–Bradley centre-out score sum.

Why square the components?

Squaring makes departures in either direction contribute to the omnibus statistic.

What is the large-sample reference?

Under regular conditions the sum is referenced to chi-square with two degrees of freedom.

Why report both components?

The total alone cannot show whether location, scale or both produced the evidence.

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Primary source and fact-check note

Yves Lepage, “A Combination of Wilcoxon’s and Ansari-Bradley’s Statistics,” Biometrika 58(1), 213–217 (1971), DOI 10.1093/biomet/58.1.213. Definitions and core claims in this article were checked against this primary publication. Calculations shown here were independently recomputed. Examples are educational illustrations; they are not reported industrial performance results.

Useful next reading

• Ansari–Bradley scale test
• Klotz scale test
• How Mathematics Works

Keep the options open. Mathematics education supports clearer questions, transferable problem-solving skills and informed choices across science, engineering, computing, AI, statistics, design and many other careers. It does not guarantee intelligence, income, admission or employment; those outcomes depend on far more than one subject or model.

Extended laboratory: topic-specific validation and responsible use

Data audit

Freeze units, labels and exclusions.

Rank audit

Reproduce pooled ordering and ties.

Score audit

Show every transformed rank.

Moment audit

Verify null mean and variance.

Calibration audit

Match exact, permutation or asymptotic rules.

Effect audit

Report direct location and spread summaries.

Sensitivity audit

Test planned alternatives without shopping.

Final claim

Describe evidence within the randomization design.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Question

Begin with this mechanism: Define joint equality before analysis. Recreate the concrete case: Two teaching methods are compared. Write units, pooled ranks, Wilcoxon sums, Ansari–Bradley scores, component means, variances, signed Z values, Lepage totals, permutations, medians and interquartile ranges with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: A vague question invites reinterpretation. Complete the transfer step: Write the null. Change exactly one observation, group label, tie rule, score convention, tail direction, location shift, scale ratio, null distribution, permutation seed or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep sampling, ranking, scoring, calibration, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete data outside tuning when testing transfer.

Workshop 2: Pooled ranks

Begin with this mechanism: Rank all observations. Recreate the concrete case: A occupies ranks 2,3,4. Write units, pooled ranks, Wilcoxon sums, Ansari–Bradley scores, component means, variances, signed Z values, Lepage totals, permutations, medians and interquartile ranges with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Ties change both components. Complete the transfer step: Use midranks. Change exactly one observation, group label, tie rule, score convention, tail direction, location shift, scale ratio, null distribution, permutation seed or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep sampling, ranking, scoring, calibration, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete data outside tuning when testing transfer.

Workshop 3: Wilcoxon sum

Begin with this mechanism: Add ordinary ranks for A. Recreate the concrete case: W=9. Write units, pooled ranks, Wilcoxon sums, Ansari–Bradley scores, component means, variances, signed Z values, Lepage totals, permutations, medians and interquartile ranges with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Group coding changes sign only. Complete the transfer step: State group. Change exactly one observation, group label, tie rule, score convention, tail direction, location shift, scale ratio, null distribution, permutation seed or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep sampling, ranking, scoring, calibration, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete data outside tuning when testing transfer.

Workshop 4: Wilcoxon mean

Begin with this mechanism: Compute m(N+1)/2. Recreate the concrete case: Mean=10.5. Write units, pooled ranks, Wilcoxon sums, Ansari–Bradley scores, component means, variances, signed Z values, Lepage totals, permutations, medians and interquartile ranges with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Unequal sizes need their own mean. Complete the transfer step: Show formula. Change exactly one observation, group label, tie rule, score convention, tail direction, location shift, scale ratio, null distribution, permutation seed or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep sampling, ranking, scoring, calibration, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete data outside tuning when testing transfer.

Workshop 5: Wilcoxon variance

Begin with this mechanism: Compute mn(N+1)/12 without ties. Recreate the concrete case: Variance=5.25. Write units, pooled ranks, Wilcoxon sums, Ansari–Bradley scores, component means, variances, signed Z values, Lepage totals, permutations, medians and interquartile ranges with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Tie correction may be required. Complete the transfer step: Apply it. Change exactly one observation, group label, tie rule, score convention, tail direction, location shift, scale ratio, null distribution, permutation seed or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep sampling, ranking, scoring, calibration, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete data outside tuning when testing transfer.

Workshop 6: Location Z

Begin with this mechanism: Standardize the rank sum. Recreate the concrete case: ZW≈-.655. Write units, pooled ranks, Wilcoxon sums, Ansari–Bradley scores, component means, variances, signed Z values, Lepage totals, permutations, medians and interquartile ranges with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Z is not an effect size. Complete the transfer step: Report median shift. Change exactly one observation, group label, tie rule, score convention, tail direction, location shift, scale ratio, null distribution, permutation seed or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep sampling, ranking, scoring, calibration, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete data outside tuning when testing transfer.

Workshop 7: AB scores

Begin with this mechanism: Assign 1,2,3,3,2,1. Recreate the concrete case: Central ranks score high. Write units, pooled ranks, Wilcoxon sums, Ansari–Bradley scores, component means, variances, signed Z values, Lepage totals, permutations, medians and interquartile ranges with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Score orientation can reverse. Complete the transfer step: Document convention. Change exactly one observation, group label, tie rule, score convention, tail direction, location shift, scale ratio, null distribution, permutation seed or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep sampling, ranking, scoring, calibration, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete data outside tuning when testing transfer.

Workshop 8: AB sum

Begin with this mechanism: Add scores for A. Recreate the concrete case: A sum=8. Write units, pooled ranks, Wilcoxon sums, Ansari–Bradley scores, component means, variances, signed Z values, Lepage totals, permutations, medians and interquartile ranges with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Raw sum depends on m. Complete the transfer step: Standardize. Change exactly one observation, group label, tie rule, score convention, tail direction, location shift, scale ratio, null distribution, permutation seed or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep sampling, ranking, scoring, calibration, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete data outside tuning when testing transfer.

Workshop 9: AB mean

Begin with this mechanism: Use sample fraction of total scores. Recreate the concrete case: Expected sum=6. Write units, pooled ranks, Wilcoxon sums, Ansari–Bradley scores, component means, variances, signed Z values, Lepage totals, permutations, medians and interquartile ranges with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Ties alter score set. Complete the transfer step: Recompute. Change exactly one observation, group label, tie rule, score convention, tail direction, location shift, scale ratio, null distribution, permutation seed or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep sampling, ranking, scoring, calibration, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete data outside tuning when testing transfer.

Workshop 10: AB variance

Begin with this mechanism: Use finite-population variance. Recreate the concrete case: Variance=1.2. Write units, pooled ranks, Wilcoxon sums, Ansari–Bradley scores, component means, variances, signed Z values, Lepage totals, permutations, medians and interquartile ranges with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Independence formulas are wrong. Complete the transfer step: Verify. Change exactly one observation, group label, tie rule, score convention, tail direction, location shift, scale ratio, null distribution, permutation seed or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep sampling, ranking, scoring, calibration, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete data outside tuning when testing transfer.

Workshop 11: Scale Z

Begin with this mechanism: Standardize the AB component. Recreate the concrete case: ZA≈1.826. Write units, pooled ranks, Wilcoxon sums, Ansari–Bradley scores, component means, variances, signed Z values, Lepage totals, permutations, medians and interquartile ranges with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Shape can drive scale scores. Complete the transfer step: Plot quantiles. Change exactly one observation, group label, tie rule, score convention, tail direction, location shift, scale ratio, null distribution, permutation seed or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep sampling, ranking, scoring, calibration, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete data outside tuning when testing transfer.

Workshop 12: Lepage total

Begin with this mechanism: Add squared components. Recreate the concrete case: L≈3.762. Write units, pooled ranks, Wilcoxon sums, Ansari–Bradley scores, component means, variances, signed Z values, Lepage totals, permutations, medians and interquartile ranges with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Signs disappear in total. Complete the transfer step: Keep signed Z values. Change exactly one observation, group label, tie rule, score convention, tail direction, location shift, scale ratio, null distribution, permutation seed or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep sampling, ranking, scoring, calibration, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete data outside tuning when testing transfer.

Workshop 13: Joint reference

Begin with this mechanism: Use exact, permutation or chi-square calibration. Recreate the concrete case: Chi-square 2 is a large-sample guide. Write units, pooled ranks, Wilcoxon sums, Ansari–Bradley scores, component means, variances, signed Z values, Lepage totals, permutations, medians and interquartile ranges with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Small n makes approximation coarse. Complete the transfer step: Enumerate labels. Change exactly one observation, group label, tie rule, score convention, tail direction, location shift, scale ratio, null distribution, permutation seed or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep sampling, ranking, scoring, calibration, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete data outside tuning when testing transfer.

Workshop 14: Component reading

Begin with this mechanism: Compare squared contributions. Recreate the concrete case: Scale dominates this example. Write units, pooled ranks, Wilcoxon sums, Ansari–Bradley scores, component means, variances, signed Z values, Lepage totals, permutations, medians and interquartile ranges with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Post-hoc component tests add multiplicity. Complete the transfer step: Use descriptive language. Change exactly one observation, group label, tie rule, score convention, tail direction, location shift, scale ratio, null distribution, permutation seed or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep sampling, ranking, scoring, calibration, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete data outside tuning when testing transfer.

Workshop 15: Effect sizes

Begin with this mechanism: Report location and spread in units. Recreate the concrete case: Median and IQR accompany L. Write units, pooled ranks, Wilcoxon sums, Ansari–Bradley scores, component means, variances, signed Z values, Lepage totals, permutations, medians and interquartile ranges with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Omnibus significance lacks magnitude. Complete the transfer step: Add intervals. Change exactly one observation, group label, tie rule, score convention, tail direction, location shift, scale ratio, null distribution, permutation seed or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep sampling, ranking, scoring, calibration, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete data outside tuning when testing transfer.

Workshop 16: Claim boundary

Begin with this mechanism: Describe distributional evidence. Recreate the concrete case: Groups differ mainly in spread. Write units, pooled ranks, Wilcoxon sums, Ansari–Bradley scores, component means, variances, signed Z values, Lepage totals, permutations, medians and interquartile ranges with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: The test does not identify cause. Complete the transfer step: Check design. Change exactly one observation, group label, tie rule, score convention, tail direction, location shift, scale ratio, null distribution, permutation seed or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep sampling, ranking, scoring, calibration, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete data outside tuning when testing transfer.