Chapter map

1. 1. Independent units
2. 2. Pooled ranks
3. 3. Squared ranks
4. 4. Contrary ranks
5. 5. Squared contrary
6. 6. Expected sum
7. 7. Standardizer
8. 8. U component
9. 9. V component
10. 10. Correlation
11. 11. Combined C
12. 12. Permutation space
13. 13. Monte Carlo
14. 14. Tie sensitivity
15. 15. Diagnostic display
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. Independent units

This step deserves a careful pause. Verify exchangeable labels.

A concrete situation

Two samples contain distinct units.

What the mathematics does not license

Clusters invalidate simple permutations. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Preserve clusters. 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

The useful question is not only how to calculate, but what the calculation assumes. Assign ranks 1 through N.

A concrete situation

A receives 2,3,4.

What the mathematics does not license

Ties need a fixed rule. Evidence becomes stronger when the limitation is tested instead of hidden.

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. Squared ranks

A small example makes the mechanism visible. Square A ranks.

A concrete situation

4+9+16=29.

What the mathematics does not license

Large ranks dominate. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Show every term. 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. Contrary ranks

This is where modelling choices begin to affect evidence. Calculate N+1-r.

A concrete situation

A contrary ranks are 5,4,3.

What the mathematics does not license

Using N-r is off by one. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Check endpoints. 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. Squared contrary

The central mathematical idea is concrete. Square and sum.

A concrete situation

25+16+9=50.

What the mathematics does not license

Lower-tail emphasis is mirrored. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Reconcile terms. 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. Expected sum

Here the geometry becomes operational. Use random-label expectations.

A concrete situation

Each group receives its sample fraction.

What the mathematics does not license

Unequal sizes alter expectation. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Calculate exactly. 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. Standardizer

This step deserves a careful pause. Use published finite-sample variance.

A concrete situation

Denominator≈98.307 in scaled formula.

What the mathematics does not license

Formula variants differ by factors. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Match one source. 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. U component

The useful question is not only how to calculate, but what the calculation assumes. Standardize squared ranks.

A concrete situation

U≈-1.007.

What the mathematics does not license

Its sign alone is not the decision. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Retain 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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9. V component

A small example makes the mechanism visible. Standardize contrary squares.

A concrete situation

V≈.275.

What the mathematics does not license

Rounding changes final C. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Keep digits. 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. Correlation

This is where modelling choices begin to affect evidence. Compute rho from N.

A concrete situation

rho≈-.917.

What the mathematics does not license

Ignoring it distorts distance. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Show rho. 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. Combined C

The central mathematical idea is concrete. Apply the quadratic form.

A concrete situation

C≈1.821.

What the mathematics does not license

C is not a probability. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Calibrate 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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12. Permutation space

Here the geometry becomes operational. Reassign labels at fixed sizes.

A concrete situation

Choose(6,3)=20 cases.

What the mathematics does not license

Data cleaning must be fixed first. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Freeze pipeline. 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. Monte Carlo

This step deserves a careful pause. Sample labels when enumeration is large.

A concrete situation

Use a recorded seed and many draws.

What the mathematics does not license

Too few draws give noisy tails. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Report error. 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. Tie sensitivity

The useful question is not only how to calculate, but what the calculation assumes. Recompute with justified tie rules.

A concrete situation

Rounded values share ranks.

What the mathematics does not license

Naive formulas may miscalibrate. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Prefer permutation. 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. Diagnostic display

A small example makes the mechanism visible. Show U, V and quantiles.

A concrete situation

Upper and lower tails move differently.

What the mathematics does not license

One C value hides direction. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Plot ECDFs. 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

This is where modelling choices begin to affect evidence. State joint evidence only.

A concrete situation

Location or scale may differ.

What the mathematics does not license

C does not establish cause. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Investigate mechanisms. 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 Cucconi test compares two samples through squared pooled ranks and squared contrary ranks. Ordinary squared ranks emphasise the upper end, while contrary ranks N+1-r emphasise the lower end. Their standardized sums are strongly correlated, so the final statistic uses that correlation rather than adding them as if independent.

For N=6 with group A at ranks 2,3,4, the squared-rank sum is 29 and the squared contrary-rank sum is 50. Using the published finite-sample standardization gives U≈-1.007, V≈0.275 and rho≈-0.917. The combined statistic C=[U²+V²-2rhoUV]/[2(1-rho²)]≈1.821.

Large C values challenge equality of location and scale. Permutation calibration is transparent: hold all values fixed, repeatedly or exhaustively reassign group labels of the observed sizes, recompute C and compare the observed value with that reference distribution. Exact enumeration is possible when the label count is small.

Cucconi is an omnibus location–scale procedure rather than a diagnostic. Ties, different shapes, dependence, unequal sampling mechanisms and multiplicity require explicit treatment. Report U and V, direct group summaries and graphics, then use subject knowledge to explain any difference.

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Frequently asked questions

What does Cucconi test?

It tests equality of two distributions against location and/or scale differences.

What are contrary ranks?

For pooled rank r among N observations, the contrary rank is N+1-r.

Why use squared ranks?

They give strong weight to one pooled tail; squared contrary ranks mirror the other tail.

Why include rho?

The two standardized sums are strongly correlated and cannot be combined as independent coordinates.

How is a p-value found?

Exact or Monte Carlo label permutations can provide the null distribution.

Does a large C identify the difference?

No. Inspect U, V, quantiles and direct effect summaries to diagnose the pattern.

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Primary source and fact-check note

Marco Marozzi, “Some Notes on the Location–Scale Cucconi Test,” Journal of Nonparametric Statistics 21(5), 629–647 (2009), DOI 10.1080/10485250902952435; developing O. Cucconi’s 1968 test. 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

• Lepage location–scale test
• Brunner–Munzel placement ranks
• 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: Independent units

Begin with this mechanism: Verify exchangeable labels. Recreate the concrete case: Two samples contain distinct units. Write units, pooled ranks, squared ranks, contrary ranks, squared contrary ranks, expected sums, standardizers, U values, V values, correlations, C statistics and permutation counts 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: Clusters invalidate simple permutations. Complete the transfer step: Preserve clusters. 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: Assign ranks 1 through N. Recreate the concrete case: A receives 2,3,4. Write units, pooled ranks, squared ranks, contrary ranks, squared contrary ranks, expected sums, standardizers, U values, V values, correlations, C statistics and permutation counts 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 need a fixed rule. 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: Squared ranks

Begin with this mechanism: Square A ranks. Recreate the concrete case: 4+9+16=29. Write units, pooled ranks, squared ranks, contrary ranks, squared contrary ranks, expected sums, standardizers, U values, V values, correlations, C statistics and permutation counts 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: Large ranks dominate. Complete the transfer step: Show every term. 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: Contrary ranks

Begin with this mechanism: Calculate N+1-r. Recreate the concrete case: A contrary ranks are 5,4,3. Write units, pooled ranks, squared ranks, contrary ranks, squared contrary ranks, expected sums, standardizers, U values, V values, correlations, C statistics and permutation counts 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: Using N-r is off by one. Complete the transfer step: Check endpoints. 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: Squared contrary

Begin with this mechanism: Square and sum. Recreate the concrete case: 25+16+9=50. Write units, pooled ranks, squared ranks, contrary ranks, squared contrary ranks, expected sums, standardizers, U values, V values, correlations, C statistics and permutation counts 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: Lower-tail emphasis is mirrored. Complete the transfer step: Reconcile terms. 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: Expected sum

Begin with this mechanism: Use random-label expectations. Recreate the concrete case: Each group receives its sample fraction. Write units, pooled ranks, squared ranks, contrary ranks, squared contrary ranks, expected sums, standardizers, U values, V values, correlations, C statistics and permutation counts 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 alter expectation. Complete the transfer step: Calculate exactly. 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: Standardizer

Begin with this mechanism: Use published finite-sample variance. Recreate the concrete case: Denominator≈98.307 in scaled formula. Write units, pooled ranks, squared ranks, contrary ranks, squared contrary ranks, expected sums, standardizers, U values, V values, correlations, C statistics and permutation counts 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: Formula variants differ by factors. Complete the transfer step: Match one source. 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: U component

Begin with this mechanism: Standardize squared ranks. Recreate the concrete case: U≈-1.007. Write units, pooled ranks, squared ranks, contrary ranks, squared contrary ranks, expected sums, standardizers, U values, V values, correlations, C statistics and permutation counts 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: Its sign alone is not the decision. Complete the transfer step: Retain 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 9: V component

Begin with this mechanism: Standardize contrary squares. Recreate the concrete case: V≈.275. Write units, pooled ranks, squared ranks, contrary ranks, squared contrary ranks, expected sums, standardizers, U values, V values, correlations, C statistics and permutation counts 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: Rounding changes final C. Complete the transfer step: Keep digits. 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: Correlation

Begin with this mechanism: Compute rho from N. Recreate the concrete case: rho≈-.917. Write units, pooled ranks, squared ranks, contrary ranks, squared contrary ranks, expected sums, standardizers, U values, V values, correlations, C statistics and permutation counts 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: Ignoring it distorts distance. Complete the transfer step: Show rho. 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: Combined C

Begin with this mechanism: Apply the quadratic form. Recreate the concrete case: C≈1.821. Write units, pooled ranks, squared ranks, contrary ranks, squared contrary ranks, expected sums, standardizers, U values, V values, correlations, C statistics and permutation counts 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: C is not a probability. Complete the transfer step: Calibrate 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 12: Permutation space

Begin with this mechanism: Reassign labels at fixed sizes. Recreate the concrete case: Choose(6,3)=20 cases. Write units, pooled ranks, squared ranks, contrary ranks, squared contrary ranks, expected sums, standardizers, U values, V values, correlations, C statistics and permutation counts 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: Data cleaning must be fixed first. Complete the transfer step: Freeze pipeline. 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: Monte Carlo

Begin with this mechanism: Sample labels when enumeration is large. Recreate the concrete case: Use a recorded seed and many draws. Write units, pooled ranks, squared ranks, contrary ranks, squared contrary ranks, expected sums, standardizers, U values, V values, correlations, C statistics and permutation counts 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: Too few draws give noisy tails. Complete the transfer step: Report error. 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: Tie sensitivity

Begin with this mechanism: Recompute with justified tie rules. Recreate the concrete case: Rounded values share ranks. Write units, pooled ranks, squared ranks, contrary ranks, squared contrary ranks, expected sums, standardizers, U values, V values, correlations, C statistics and permutation counts 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: Naive formulas may miscalibrate. Complete the transfer step: Prefer permutation. 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: Diagnostic display

Begin with this mechanism: Show U, V and quantiles. Recreate the concrete case: Upper and lower tails move differently. Write units, pooled ranks, squared ranks, contrary ranks, squared contrary ranks, expected sums, standardizers, U values, V values, correlations, C statistics and permutation counts 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: One C value hides direction. Complete the transfer step: Plot ECDFs. 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: State joint evidence only. Recreate the concrete case: Location or scale may differ. Write units, pooled ranks, squared ranks, contrary ranks, squared contrary ranks, expected sums, standardizers, U values, V values, correlations, C statistics and permutation counts 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: C does not establish cause. Complete the transfer step: Investigate mechanisms. 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.