Chapter map

1. 1. Pooled order
2. 2. Middle rank
3. 3. Rank distance
4. 4. Mood score
5. 5. Score symmetry
6. 6. Full score set
7. 7. A ranks
8. 8. A sum
9. 9. B sum
10. 10. Null mean
11. 11. Null variance
12. 12. Z value
13. 13. Exact allocations
14. 14. Location check
15. 15. Magnitude
16. 16. Claim

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. Pooled order

A small example makes the mechanism visible. Rank all six observations.

A concrete situation

Ranks are 1 through 6.

What the mathematics does not license

Sorting within groups is wrong. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Keep one pooled list. 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. Middle rank

This is where modelling choices begin to affect evidence. Compute (N+1)/2.

A concrete situation

Centre=3.5.

What the mathematics does not license

Even and odd N differ. Keeping this boundary clear makes the result easier to audit and transfer.

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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3. Rank distance

The central mathematical idea is concrete. Subtract the centre.

A concrete situation

Rank 1 is -2.5 away.

What the mathematics does not license

Signs disappear only after squaring. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Keep signed audit. 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. Mood score

Here the geometry becomes operational. Square each distance.

A concrete situation

Rank 1 score=6.25.

What the mathematics does not license

Raw values are not squared. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Name scale. 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. Score symmetry

This step deserves a careful pause. Check opposite ranks.

A concrete situation

Ranks 1 and 6 both score 6.25.

What the mathematics does not license

Tie handling can disturb pairs. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Audit. 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. Full score set

The useful question is not only how to calculate, but what the calculation assumes. List all scores.

A concrete situation

6.25,2.25,.25,.25,2.25,6.25.

What the mathematics does not license

Rounding too early changes variance. A trustworthy workflow records the choice before seeing the final test result.

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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7. A ranks

A small example makes the mechanism visible. Record group allocation.

A concrete situation

A gets 2,3,4.

What the mathematics does not license

Group labels must remain fixed. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Archive. 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. A sum

This is where modelling choices begin to affect evidence. Add its scores.

A concrete situation

2.75.

What the mathematics does not license

Sum depends on sample size. Keeping this boundary clear makes the result easier to audit and transfer.

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. B sum

The central mathematical idea is concrete. Check complement.

A concrete situation

B gets 14.75.

What the mathematics does not license

Totals must equal 17.5. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Reconcile. 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. Null mean

Here the geometry becomes operational. Allocate total by sample fraction.

A concrete situation

Each expects 8.75.

What the mathematics does not license

Unequal groups change expectation. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Calculate. 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. Null variance

This step deserves a careful pause. Use sampling-without-replacement variance.

A concrete situation

Variance=11.2.

What the mathematics does not license

IID formulas overstate information. Keeping this boundary clear makes the result easier to audit and transfer.

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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12. Z value

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

A concrete situation

ZA≈-1.793.

What the mathematics does not license

Normal tails are rough at N=6. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Use exact 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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13. Exact allocations

A small example makes the mechanism visible. Enumerate choose(6,3)=20 cases.

A concrete situation

Observed sum is compared to all.

What the mathematics does not license

Tail direction must be declared. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

State two-sided rule. 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. Location check

This is where modelling choices begin to affect evidence. Compare centres directly.

A concrete situation

Medians are similar.

What the mathematics does not license

A shift can move tail scores. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Plot data. 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. Magnitude

The central mathematical idea is concrete. Report IQR, MAD and quantile ratio.

A concrete situation

B is more spread in units.

What the mathematics does not license

Significance lacks size. A trustworthy workflow records the choice before seeing the final test result.

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

Here the geometry becomes operational. State scale evidence only.

A concrete situation

B occupies more extremes.

What the mathematics does not license

Mechanism remains unproven. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Investigate. 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

Mood’s two-sample scale statistic assigns each pooled rank a squared distance from the middle rank. Central observations receive small scores and both tails receive large scores. A group that occupies more extremes accumulates a larger score sum even when low and high departures point in opposite directions.

For N=6 the rank scores are 6.25,2.25,0.25,0.25,2.25,6.25. If A occupies ranks 2,3,4 its score sum is 2.75; B at ranks 1,5,6 has 14.75. Each group of three expects 8.75 under random labels.

The fixed-score finite-population variance for this example is 11.2, giving ZA=(2.75-8.75)/sqrt(11.2)≈-1.793. Exact permutation inference enumerates the twenty allocations of three labels among six ranks and avoids relying only on a rough normal approximation.

The scale interpretation assumes independent samples and is clearest when location and shape are comparable. A location shift, skewness, ties or unequal detection limits can redistribute tail ranks. Always report direct spread measures, graphics and the real decision threshold beside the test.

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Frequently asked questions

What are Mood scale scores?

They are squared distances of pooled ranks from the average rank.

Why do both tails score highly?

Squaring removes direction and increases with distance from the pooled centre.

Are these raw squared deviations?

No. They are squared distances on the rank scale.

What does an exact test permute?

It reallocates the fixed score set to group labels of the observed sizes.

How do ties matter?

Midranks create repeated scores and require matching calibration.

Does a large sum estimate a variance ratio?

No. It indicates relative tail occupancy; report scale magnitude separately.

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Primary source and fact-check note

Alexander M. Mood, “On the Asymptotic Efficiency of Certain Nonparametric Two-Sample Tests,” Annals of Mathematical Statistics 25(3), 514–522 (1954), DOI 10.1214/aoms/1177728719. 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

• Klotz scale test
• Siegel–Tukey spread 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

Design audit

Freeze independent units and labels.

Ranking audit

Reproduce values, ties and group IDs.

Score audit

Show every transformed rank or placement.

Moment audit

Verify finite-sample expectations and variances.

Calibration audit

Match exact, permutation or asymptotic rules.

Magnitude audit

Report direct effects in original units.

Sensitivity audit

Run only planned alternatives.

Final claim

Keep inference inside the sampling design.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Pooled order

Begin with this mechanism: Rank all six observations. Recreate the concrete case: Ranks are 1 through 6. Write units, pooled ranks, middle ranks, rank distances, squared scores, group sums, null moments, standardized statistics and exact allocations with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Sorting within groups is wrong. Complete the transfer step: Keep one pooled list. Change exactly one observation, label, tie, centre, rank, trim amount, score rule, group size, tail direction, null distribution or permutation seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result twice: first to a student using the visible ranks or placements, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from a location shift, skewness, rounding or dependence. State what extra graph, design record or held-out observation would distinguish the explanations. Finish by writing one sentence the data support, one sentence they do not support and one useful next measurement.

Workshop 2: Middle rank

Begin with this mechanism: Compute (N+1)/2. Recreate the concrete case: Centre=3.5. Write units, pooled ranks, middle ranks, rank distances, squared scores, group sums, null moments, standardized statistics and exact allocations with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Even and odd N differ. Complete the transfer step: Show formula. Change exactly one observation, label, tie, centre, rank, trim amount, score rule, group size, tail direction, null distribution or permutation seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result twice: first to a student using the visible ranks or placements, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from a location shift, skewness, rounding or dependence. State what extra graph, design record or held-out observation would distinguish the explanations. Finish by writing one sentence the data support, one sentence they do not support and one useful next measurement.

Workshop 3: Rank distance

Begin with this mechanism: Subtract the centre. Recreate the concrete case: Rank 1 is -2.5 away. Write units, pooled ranks, middle ranks, rank distances, squared scores, group sums, null moments, standardized statistics and exact allocations with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Signs disappear only after squaring. Complete the transfer step: Keep signed audit. Change exactly one observation, label, tie, centre, rank, trim amount, score rule, group size, tail direction, null distribution or permutation seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result twice: first to a student using the visible ranks or placements, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from a location shift, skewness, rounding or dependence. State what extra graph, design record or held-out observation would distinguish the explanations. Finish by writing one sentence the data support, one sentence they do not support and one useful next measurement.

Workshop 4: Mood score

Begin with this mechanism: Square each distance. Recreate the concrete case: Rank 1 score=6.25. Write units, pooled ranks, middle ranks, rank distances, squared scores, group sums, null moments, standardized statistics and exact allocations with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Raw values are not squared. Complete the transfer step: Name scale. Change exactly one observation, label, tie, centre, rank, trim amount, score rule, group size, tail direction, null distribution or permutation seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result twice: first to a student using the visible ranks or placements, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from a location shift, skewness, rounding or dependence. State what extra graph, design record or held-out observation would distinguish the explanations. Finish by writing one sentence the data support, one sentence they do not support and one useful next measurement.

Workshop 5: Score symmetry

Begin with this mechanism: Check opposite ranks. Recreate the concrete case: Ranks 1 and 6 both score 6.25. Write units, pooled ranks, middle ranks, rank distances, squared scores, group sums, null moments, standardized statistics and exact allocations with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Tie handling can disturb pairs. Complete the transfer step: Audit. Change exactly one observation, label, tie, centre, rank, trim amount, score rule, group size, tail direction, null distribution or permutation seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result twice: first to a student using the visible ranks or placements, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from a location shift, skewness, rounding or dependence. State what extra graph, design record or held-out observation would distinguish the explanations. Finish by writing one sentence the data support, one sentence they do not support and one useful next measurement.

Workshop 6: Full score set

Begin with this mechanism: List all scores. Recreate the concrete case: 6.25,2.25,.25,.25,2.25,6.25. Write units, pooled ranks, middle ranks, rank distances, squared scores, group sums, null moments, standardized statistics and exact allocations with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Rounding too early changes variance. Complete the transfer step: Keep digits. Change exactly one observation, label, tie, centre, rank, trim amount, score rule, group size, tail direction, null distribution or permutation seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result twice: first to a student using the visible ranks or placements, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from a location shift, skewness, rounding or dependence. State what extra graph, design record or held-out observation would distinguish the explanations. Finish by writing one sentence the data support, one sentence they do not support and one useful next measurement.

Workshop 7: A ranks

Begin with this mechanism: Record group allocation. Recreate the concrete case: A gets 2,3,4. Write units, pooled ranks, middle ranks, rank distances, squared scores, group sums, null moments, standardized statistics and exact allocations with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Group labels must remain fixed. Complete the transfer step: Archive. Change exactly one observation, label, tie, centre, rank, trim amount, score rule, group size, tail direction, null distribution or permutation seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result twice: first to a student using the visible ranks or placements, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from a location shift, skewness, rounding or dependence. State what extra graph, design record or held-out observation would distinguish the explanations. Finish by writing one sentence the data support, one sentence they do not support and one useful next measurement.

Workshop 8: A sum

Begin with this mechanism: Add its scores. Recreate the concrete case: 2.75. Write units, pooled ranks, middle ranks, rank distances, squared scores, group sums, null moments, standardized statistics and exact allocations with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Sum depends on sample size. Complete the transfer step: Standardize. Change exactly one observation, label, tie, centre, rank, trim amount, score rule, group size, tail direction, null distribution or permutation seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result twice: first to a student using the visible ranks or placements, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from a location shift, skewness, rounding or dependence. State what extra graph, design record or held-out observation would distinguish the explanations. Finish by writing one sentence the data support, one sentence they do not support and one useful next measurement.

Workshop 9: B sum

Begin with this mechanism: Check complement. Recreate the concrete case: B gets 14.75. Write units, pooled ranks, middle ranks, rank distances, squared scores, group sums, null moments, standardized statistics and exact allocations with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Totals must equal 17.5. Complete the transfer step: Reconcile. Change exactly one observation, label, tie, centre, rank, trim amount, score rule, group size, tail direction, null distribution or permutation seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result twice: first to a student using the visible ranks or placements, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from a location shift, skewness, rounding or dependence. State what extra graph, design record or held-out observation would distinguish the explanations. Finish by writing one sentence the data support, one sentence they do not support and one useful next measurement.

Workshop 10: Null mean

Begin with this mechanism: Allocate total by sample fraction. Recreate the concrete case: Each expects 8.75. Write units, pooled ranks, middle ranks, rank distances, squared scores, group sums, null moments, standardized statistics and exact allocations with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Unequal groups change expectation. Complete the transfer step: Calculate. Change exactly one observation, label, tie, centre, rank, trim amount, score rule, group size, tail direction, null distribution or permutation seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result twice: first to a student using the visible ranks or placements, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from a location shift, skewness, rounding or dependence. State what extra graph, design record or held-out observation would distinguish the explanations. Finish by writing one sentence the data support, one sentence they do not support and one useful next measurement.

Workshop 11: Null variance

Begin with this mechanism: Use sampling-without-replacement variance. Recreate the concrete case: Variance=11.2. Write units, pooled ranks, middle ranks, rank distances, squared scores, group sums, null moments, standardized statistics and exact allocations with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: IID formulas overstate information. Complete the transfer step: Verify. Change exactly one observation, label, tie, centre, rank, trim amount, score rule, group size, tail direction, null distribution or permutation seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result twice: first to a student using the visible ranks or placements, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from a location shift, skewness, rounding or dependence. State what extra graph, design record or held-out observation would distinguish the explanations. Finish by writing one sentence the data support, one sentence they do not support and one useful next measurement.

Workshop 12: Z value

Begin with this mechanism: Standardize the difference. Recreate the concrete case: ZA≈-1.793. Write units, pooled ranks, middle ranks, rank distances, squared scores, group sums, null moments, standardized statistics and exact allocations with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Normal tails are rough at N=6. Complete the transfer step: Use exact labels. Change exactly one observation, label, tie, centre, rank, trim amount, score rule, group size, tail direction, null distribution or permutation seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result twice: first to a student using the visible ranks or placements, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from a location shift, skewness, rounding or dependence. State what extra graph, design record or held-out observation would distinguish the explanations. Finish by writing one sentence the data support, one sentence they do not support and one useful next measurement.

Workshop 13: Exact allocations

Begin with this mechanism: Enumerate choose(6,3)=20 cases. Recreate the concrete case: Observed sum is compared to all. Write units, pooled ranks, middle ranks, rank distances, squared scores, group sums, null moments, standardized statistics and exact allocations with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Tail direction must be declared. Complete the transfer step: State two-sided rule. Change exactly one observation, label, tie, centre, rank, trim amount, score rule, group size, tail direction, null distribution or permutation seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result twice: first to a student using the visible ranks or placements, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from a location shift, skewness, rounding or dependence. State what extra graph, design record or held-out observation would distinguish the explanations. Finish by writing one sentence the data support, one sentence they do not support and one useful next measurement.

Workshop 14: Location check

Begin with this mechanism: Compare centres directly. Recreate the concrete case: Medians are similar. Write units, pooled ranks, middle ranks, rank distances, squared scores, group sums, null moments, standardized statistics and exact allocations with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: A shift can move tail scores. Complete the transfer step: Plot data. Change exactly one observation, label, tie, centre, rank, trim amount, score rule, group size, tail direction, null distribution or permutation seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result twice: first to a student using the visible ranks or placements, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from a location shift, skewness, rounding or dependence. State what extra graph, design record or held-out observation would distinguish the explanations. Finish by writing one sentence the data support, one sentence they do not support and one useful next measurement.

Workshop 15: Magnitude

Begin with this mechanism: Report IQR, MAD and quantile ratio. Recreate the concrete case: B is more spread in units. Write units, pooled ranks, middle ranks, rank distances, squared scores, group sums, null moments, standardized statistics and exact allocations with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Significance lacks size. Complete the transfer step: Add intervals. Change exactly one observation, label, tie, centre, rank, trim amount, score rule, group size, tail direction, null distribution or permutation seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result twice: first to a student using the visible ranks or placements, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from a location shift, skewness, rounding or dependence. State what extra graph, design record or held-out observation would distinguish the explanations. Finish by writing one sentence the data support, one sentence they do not support and one useful next measurement.

Workshop 16: Claim

Begin with this mechanism: State scale evidence only. Recreate the concrete case: B occupies more extremes. Write units, pooled ranks, middle ranks, rank distances, squared scores, group sums, null moments, standardized statistics and exact allocations with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Mechanism remains unproven. Complete the transfer step: Investigate. Change exactly one observation, label, tie, centre, rank, trim amount, score rule, group size, tail direction, null distribution or permutation seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result twice: first to a student using the visible ranks or placements, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from a location shift, skewness, rounding or dependence. State what extra graph, design record or held-out observation would distinguish the explanations. Finish by writing one sentence the data support, one sentence they do not support and one useful next measurement.