WHY MATHEMATICS?

Why Mathematics? | Moses Extreme-Reactions Test: Pooled Ranks, Central Spans, Trimming and Opposite Treatment Responses

How a treatment can create unusually high and low responses that cancel in the mean, and how rank spans reveal those extremes without inventing a direction.

Why mathematics is important becomes easier to see when a real decision depends on it. This guide follows the mechanism from definitions and dimensions to a worked calculation, validation, practical learning and responsible use. It keeps factual evidence separate from interpretation and treats every illustrative scenario as an example rather than a promise of results.

Choose a reading route

Pool and rank the two independent groups
Measure the prespecified control-group rank span
Apply only a declared symmetric trimming rule
Use exact label probabilities and inspect both tails

Chapter map

1. Units
2. Labels
3. Pooling
4. Ties
5. Control ranks
6. Minimum
7. Maximum
8. Span
9. Experimental tails
10. Trim rule
11. Trimmed ranks
12. Null space
13. Exact tail
14. Sensitivity
15. Effect display
16. Claim

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

1. Units

The central mathematical idea is concrete. Define independent participants.

A concrete situation

Eight units enter two groups.

What the mathematics does not license

Repeated measures need another design. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Lock IDs. 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.

2. Labels

Here the geometry becomes operational. Preserve control and experimental roles.

A concrete situation

Control has five units.

What the mathematics does not license

Swapping roles changes the statistic. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

State orientation. 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.

3. Pooling

This step deserves a careful pause. Combine values for ranking only.

A concrete situation

Ranks run 1 to 8.

What the mathematics does not license

Pooling does not erase groups. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Archive 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.

4. Ties

The useful question is not only how to calculate, but what the calculation assumes. Declare averaged ranks or exact handling.

A concrete situation

Two scores tie.

What the mathematics does not license

Discrete data reduce resolution. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Use a matching 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.

Did You Know? In this chapter, the most important number is useful only together with its construction rule. Changing scaling, sample membership or validation design can change the numerical answer without changing the arithmetic.

5. Control ranks

A small example makes the mechanism visible. List ranks in order.

A concrete situation

2,3,4,5,6.

What the mathematics does not license

One transcription error changes span. 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.

6. Minimum

This is where modelling choices begin to affect evidence. Identify the smallest selected rank.

A concrete situation

Minimum=2.

What the mathematics does not license

Post-hoc exclusion biases it. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Freeze cleaning. 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.

7. Maximum

The central mathematical idea is concrete. Identify the largest.

A concrete situation

Maximum=6.

What the mathematics does not license

Censoring hides extremes. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Audit limits. 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.

8. Span

Here the geometry becomes operational. Calculate max-min+1.

A concrete situation

Span=5.

What the mathematics does not license

It is not a raw range. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Label rank units. 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.

9. Experimental tails

This step deserves a careful pause. Inspect the other labels.

A concrete situation

Ranks 1,7,8 are experimental.

What the mathematics does not license

One-sided location can look extreme. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Plot both tails. 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.

10. Trim rule

The useful question is not only how to calculate, but what the calculation assumes. Choose equal removals from comparison ends.

A concrete situation

Trim one at each end.

What the mathematics does not license

Too much trim destroys information. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Predeclare. 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.

11. Trimmed ranks

A small example makes the mechanism visible. Recalculate remaining span.

A concrete situation

3,4,5 gives span=3.

What the mathematics does not license

Different trims answer different questions. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Report all planned. 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.

12. Null space

This is where modelling choices begin to affect evidence. Count fixed-size label allocations.

A concrete situation

Choose(8,5)=56 controls.

What the mathematics does not license

Nonrandom assignment breaks exchangeability. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Match 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.

13. Exact tail

The central mathematical idea is concrete. Count spans as short or shorter.

A concrete situation

Observed span enters one tail.

What the mathematics does not license

Direction must be chosen first. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

State 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.

14. Sensitivity

Here the geometry becomes operational. Show trimmed and untrimmed results.

A concrete situation

Both suggest experimental extremes.

What the mathematics does not license

Selective reporting misleads. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Publish both. 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.

15. Effect display

This step deserves a careful pause. Report tail quantiles in original units.

A concrete situation

Experimental group has low and high values.

What the mathematics does not license

Ranks hide magnitude. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Add units. 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.

16. Claim

The useful question is not only how to calculate, but what the calculation assumes. Describe extreme-reaction evidence only.

A concrete situation

Treatment responses may be heterogeneous.

What the mathematics does not license

Cause and mechanism remain open. Keeping this boundary clear makes the result easier to audit and transfer.

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.

A seven-day practice lab

Day 1: define the decision, observation unit, X variables and Y or reference condition.
Day 2: reproduce the smallest worked example and check every dimension, unit and sign.
Day 3: build a transparent baseline and write what it assumes.
Day 4: fit the main method inside a training-only pipeline; record every parameter.
Day 5: challenge one assumption with a deliberately difficult dataset or operating regime.
Day 6: compare validation metrics with the practical workload created by errors.
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 Moses extreme-reactions test was designed for treatments that may push some observations unusually high and others unusually low. Instead of comparing means, it asks whether one group occupies more of the pooled extremes. That question is useful when opposite responses could cancel in an average.

Pool the observations and assign ordinary ranks. Choose the comparison group exactly as the design specifies, then calculate its rank span as largest rank minus smallest rank plus one. If the other group owns many extremes, the comparison group tends to be compressed into a short central span.

With eight pooled ranks, suppose the control group occupies 2,3,4,5,6 and the experimental group occupies 1,7,8. The control span is 6-2+1=5. Trimming one control observation from each end leaves 3,4,5 and span 3. Trimming is a prespecified robustness rule, not permission to delete inconvenient points.

The exact null treats group labels as exchangeable at fixed sample sizes. A short span can indicate more experimental extremes, but dependence, unequal measurement limits, location shifts, selective missingness and post-hoc trimming can create the same appearance. Show the ranked strip plot and direct tail summaries.

Frequently asked questions

What problem does the Moses test address?

It detects whether one independent group contains more unusually high and low observations than another.

What is the span?

It is the largest selected pooled rank minus the smallest plus one.

Why can means miss the effect?

Positive and negative treatment responses can cancel.

Why trim?

A prespecified symmetric trim reduces sensitivity to a few comparison-group extremes.

Is trimming optional after seeing data?

No. The trimming amount must be declared before examining the result.

Does a short span prove treatment heterogeneity?

No. It is rank evidence requiring design and measurement checks.

Primary source and fact-check note

Lincoln E. Moses, “A Two-Sample Test,” Psychometrika 17(3), 239–247 (1952), DOI 10.1007/BF02288755. 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

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.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

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.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Units

Begin with this mechanism: Define independent participants. Recreate the concrete case: Eight units enter two groups. Write units, labels, pooled ranks, minima, maxima, spans, trimming amounts, exact allocations and tail summaries with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Repeated measures need another design. Complete the transfer step: Lock IDs. 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: Labels

Begin with this mechanism: Preserve control and experimental roles. Recreate the concrete case: Control has five units. Write units, labels, pooled ranks, minima, maxima, spans, trimming amounts, exact allocations and tail summaries with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Swapping roles changes the statistic. Complete the transfer step: State orientation. 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: Pooling

Begin with this mechanism: Combine values for ranking only. Recreate the concrete case: Ranks run 1 to 8. Write units, labels, pooled ranks, minima, maxima, spans, trimming amounts, exact allocations and tail summaries with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Pooling does not erase groups. Complete the transfer step: Archive 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 4: Ties

Begin with this mechanism: Declare averaged ranks or exact handling. Recreate the concrete case: Two scores tie. Write units, labels, pooled ranks, minima, maxima, spans, trimming amounts, exact allocations and tail summaries with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Discrete data reduce resolution. Complete the transfer step: Use a matching null. 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.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Workshop 5: Control ranks

Begin with this mechanism: List ranks in order. Recreate the concrete case: 2,3,4,5,6. Write units, labels, pooled ranks, minima, maxima, spans, trimming amounts, exact allocations and tail summaries with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: One transcription error changes span. 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 6: Minimum

Begin with this mechanism: Identify the smallest selected rank. Recreate the concrete case: Minimum=2. Write units, labels, pooled ranks, minima, maxima, spans, trimming amounts, exact allocations and tail summaries with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Post-hoc exclusion biases it. Complete the transfer step: Freeze cleaning. 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: Maximum

Begin with this mechanism: Identify the largest. Recreate the concrete case: Maximum=6. Write units, labels, pooled ranks, minima, maxima, spans, trimming amounts, exact allocations and tail summaries with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Censoring hides extremes. Complete the transfer step: Audit limits. 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: Span

Begin with this mechanism: Calculate max-min+1. Recreate the concrete case: Span=5. Write units, labels, pooled ranks, minima, maxima, spans, trimming amounts, exact allocations and tail summaries with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: It is not a raw range. Complete the transfer step: Label rank units. 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.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Workshop 9: Experimental tails

Begin with this mechanism: Inspect the other labels. Recreate the concrete case: Ranks 1,7,8 are experimental. Write units, labels, pooled ranks, minima, maxima, spans, trimming amounts, exact allocations and tail summaries with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: One-sided location can look extreme. Complete the transfer step: Plot both tails. 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: Trim rule

Begin with this mechanism: Choose equal removals from comparison ends. Recreate the concrete case: Trim one at each end. Write units, labels, pooled ranks, minima, maxima, spans, trimming amounts, exact allocations and tail summaries with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Too much trim destroys information. Complete the transfer step: Predeclare. 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: Trimmed ranks

Begin with this mechanism: Recalculate remaining span. Recreate the concrete case: 3,4,5 gives span=3. Write units, labels, pooled ranks, minima, maxima, spans, trimming amounts, exact allocations and tail summaries with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Different trims answer different questions. Complete the transfer step: Report all planned. 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: Null space

Begin with this mechanism: Count fixed-size label allocations. Recreate the concrete case: Choose(8,5)=56 controls. Write units, labels, pooled ranks, minima, maxima, spans, trimming amounts, exact allocations and tail summaries with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Nonrandom assignment breaks exchangeability. Complete the transfer step: Match design. 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.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Workshop 13: Exact tail

Begin with this mechanism: Count spans as short or shorter. Recreate the concrete case: Observed span enters one tail. Write units, labels, pooled ranks, minima, maxima, spans, trimming amounts, exact allocations and tail summaries with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Direction must be chosen first. Complete the transfer step: State it. 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: Sensitivity

Begin with this mechanism: Show trimmed and untrimmed results. Recreate the concrete case: Both suggest experimental extremes. Write units, labels, pooled ranks, minima, maxima, spans, trimming amounts, exact allocations and tail summaries with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Selective reporting misleads. Complete the transfer step: Publish both. 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: Effect display

Begin with this mechanism: Report tail quantiles in original units. Recreate the concrete case: Experimental group has low and high values. Write units, labels, pooled ranks, minima, maxima, spans, trimming amounts, exact allocations and tail summaries with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Ranks hide magnitude. Complete the transfer step: Add units. 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: Describe extreme-reaction evidence only. Recreate the concrete case: Treatment responses may be heterogeneous. Write units, labels, pooled ranks, minima, maxima, spans, trimming amounts, exact allocations and tail summaries with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Cause and mechanism remain open. 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.