WHY MATHEMATICS?

Why Mathematics? | Fligner–Policello Test: Cross-Sample Placements, Unequal Variances, Median Comparisons and Robust Rank Evidence

How counting cross-sample orderings separately for each group relaxes equal-shape assumptions in the two-sample location problem.

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

Compute cross-sample placements with half credit for ties
Summarize placement means and within-sample variability
Standardize the directional difference without pooled variance
Interpret medians only under the method’s stated conditions

Chapter map

1. Samples
2. Orientation
3. X1 placement
4. X2 placement
5. X3 placement
6. X vector
7. Y placements
8. Placement means
9. X variability
10. Y variability
11. Denominator
12. Z statistic
13. Median target
14. Probability view
15. Sensitivity
16. Claim

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

1. Samples

This is where modelling choices begin to affect evidence. Define two independent groups.

A concrete situation

X and Y each have three values.

What the mathematics does not license

Pairing changes inference. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Record 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.

2. Orientation

The central mathematical idea is concrete. Fix X minus Y direction.

A concrete situation

Negative means X tends lower.

What the mathematics does not license

Swapping groups flips sign. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Label output. 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. X1 placement

Here the geometry becomes operational. Count Y below X=1.

A concrete situation

Zero Y values are below.

What the mathematics does not license

Equal values need half credit. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

List comparisons. 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. X2 placement

This step deserves a careful pause. Evaluate X=2.

A concrete situation

Y=2 contributes .5.

What the mathematics does not license

Ignoring ties biases order. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Use .5. 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. X3 placement

The useful question is not only how to calculate, but what the calculation assumes. Evaluate X=3.

A concrete situation

Only Y=2 is lower.

What the mathematics does not license

Every cross-pair must appear once. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Audit matrix. 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. X vector

A small example makes the mechanism visible. Collect placements.

A concrete situation

0,.5,1 sum to 1.5.

What the mathematics does not license

A sum alone lacks variance. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Keep vector. 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. Y placements

This is where modelling choices begin to affect evidence. Count X below each Y.

A concrete situation

1.5,3,3 sum to 7.5.

What the mathematics does not license

Direction must be symmetric. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Cross-check totals. 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. Placement means

The central mathematical idea is concrete. Average within groups.

A concrete situation

.5 and 2.5.

What the mathematics does not license

Means are rank probabilities, not units. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Explain. 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. X variability

Here the geometry becomes operational. Sum squared deviations.

A concrete situation

VX=.5.

What the mathematics does not license

Small n is unstable. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Report n. 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. Y variability

This step deserves a careful pause. Repeat for Y.

A concrete situation

VY=1.5.

What the mathematics does not license

One tied placement affects it. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Show 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.

11. Denominator

The useful question is not only how to calculate, but what the calculation assumes. Combine published variance terms.

A concrete situation

Denominator≈3.606.

What the mathematics does not license

Formula conventions matter. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Match 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.

12. Z statistic

A small example makes the mechanism visible. Divide the placement difference.

A concrete situation

Z≈-1.664.

What the mathematics does not license

Normal calibration is approximate. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Use small-sample checks. 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. Median target

This is where modelling choices begin to affect evidence. Check paper conditions for median language.

A concrete situation

Group shapes differ in spread.

What the mathematics does not license

Arbitrary shape differences complicate medians. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

State assumptions. 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. Probability view

The central mathematical idea is concrete. Report cross-ordering probability.

A concrete situation

Y exceeds X often.

What the mathematics does not license

Probability and median are distinct. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Keep 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. Sensitivity

Here the geometry becomes operational. Compare Brunner–Munzel and plots.

A concrete situation

Direction agrees.

What the mathematics does not license

Method shopping inflates certainty. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Preplan. 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

This step deserves a careful pause. Describe robust location evidence.

A concrete situation

Y tends larger than X.

What the mathematics does not license

Cause is not identified. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Use design evidence. 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 Fligner–Policello procedure addresses a two-sample location question when distributions may have unequal shapes or spreads. It replaces one pooled rank sum with cross-sample placement scores: each observation counts how many observations in the other group are below it, while ties contribute one half.

For X=1,2,3 and Y=2,4,6, X placements in Y are 0,0.5,1. Y placements in X are 1.5,3,3. Their sums are 1.5 and 7.5. Using the published variance components gives an illustrative standardized statistic near -1.664 under the stated orientation.

Placements preserve the probability-of-ordering idea while estimating variability separately in the two samples. The procedure was developed for the Behrens–Fisher problem of comparing centres without assuming equal variances. Small samples and heavy ties still need careful calibration.

The target is commonly expressed as equality of medians under the paper’s conditions, or more cautiously as a location comparison with fewer equal-shape assumptions. It does not turn arbitrary distribution differences into a pure median claim. Report medians, probability-of-superiority summaries and full distribution plots.

Frequently asked questions

What is a placement score?

It counts observations in the other sample below a value, with half credit for ties.

Why not use one pooled variance?

The procedure estimates placement variability separately in groups.

What problem motivated the test?

It addresses the nonparametric Behrens–Fisher location problem under unequal spreads.

Does it require identical shapes?

It was designed with fewer equal-shape restrictions than ordinary rank-sum median claims.

How do ties contribute?

An equal cross-sample observation contributes one half.

Does the sign have a fixed meaning?

It depends on group order, so orientation must be stated.

Primary source and fact-check note

Michael A. Fligner and George E. Policello II, “Robust Rank Procedures for the Behrens-Fisher Problem,” Journal of the American Statistical Association 76(373), 162–168 (1981), DOI 10.1080/01621459.1981.10477623. 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: Samples

Begin with this mechanism: Define two independent groups. Recreate the concrete case: X and Y each have three values. Write units, cross-pairs, placements, tie halves, placement sums, means, variance components, standardized statistics and probability summaries with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Pairing changes inference. Complete the transfer step: Record 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.

Workshop 2: Orientation

Begin with this mechanism: Fix X minus Y direction. Recreate the concrete case: Negative means X tends lower. Write units, cross-pairs, placements, tie halves, placement sums, means, variance components, standardized statistics and probability summaries with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Swapping groups flips sign. Complete the transfer step: Label output. 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: X1 placement

Begin with this mechanism: Count Y below X=1. Recreate the concrete case: Zero Y values are below. Write units, cross-pairs, placements, tie halves, placement sums, means, variance components, standardized statistics and probability summaries with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Equal values need half credit. Complete the transfer step: List comparisons. 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: X2 placement

Begin with this mechanism: Evaluate X=2. Recreate the concrete case: Y=2 contributes .5. Write units, cross-pairs, placements, tie halves, placement sums, means, variance components, standardized statistics and probability summaries with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Ignoring ties biases order. Complete the transfer step: Use .5. 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: X3 placement

Begin with this mechanism: Evaluate X=3. Recreate the concrete case: Only Y=2 is lower. Write units, cross-pairs, placements, tie halves, placement sums, means, variance components, standardized statistics and probability summaries with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Every cross-pair must appear once. Complete the transfer step: Audit matrix. 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: X vector

Begin with this mechanism: Collect placements. Recreate the concrete case: 0,.5,1 sum to 1.5. Write units, cross-pairs, placements, tie halves, placement sums, means, variance components, standardized statistics and probability summaries with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: A sum alone lacks variance. Complete the transfer step: Keep vector. 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: Y placements

Begin with this mechanism: Count X below each Y. Recreate the concrete case: 1.5,3,3 sum to 7.5. Write units, cross-pairs, placements, tie halves, placement sums, means, variance components, standardized statistics and probability summaries with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Direction must be symmetric. Complete the transfer step: Cross-check totals. 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: Placement means

Begin with this mechanism: Average within groups. Recreate the concrete case: .5 and 2.5. Write units, cross-pairs, placements, tie halves, placement sums, means, variance components, standardized statistics and probability summaries with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Means are rank probabilities, not units. Complete the transfer step: Explain. 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: X variability

Begin with this mechanism: Sum squared deviations. Recreate the concrete case: VX=.5. Write units, cross-pairs, placements, tie halves, placement sums, means, variance components, standardized statistics and probability summaries with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Small n is unstable. Complete the transfer step: Report n. 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: Y variability

Begin with this mechanism: Repeat for Y. Recreate the concrete case: VY=1.5. Write units, cross-pairs, placements, tie halves, placement sums, means, variance components, standardized statistics and probability summaries with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: One tied placement affects it. Complete the transfer step: Show terms. 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: Denominator

Begin with this mechanism: Combine published variance terms. Recreate the concrete case: Denominator≈3.606. Write units, cross-pairs, placements, tie halves, placement sums, means, variance components, standardized statistics and probability summaries with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Formula conventions matter. Complete the transfer step: Match source. 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 statistic

Begin with this mechanism: Divide the placement difference. Recreate the concrete case: Z≈-1.664. Write units, cross-pairs, placements, tie halves, placement sums, means, variance components, standardized statistics and probability summaries with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Normal calibration is approximate. Complete the transfer step: Use small-sample checks. 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: Median target

Begin with this mechanism: Check paper conditions for median language. Recreate the concrete case: Group shapes differ in spread. Write units, cross-pairs, placements, tie halves, placement sums, means, variance components, standardized statistics and probability summaries with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Arbitrary shape differences complicate medians. Complete the transfer step: State assumptions. 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: Probability view

Begin with this mechanism: Report cross-ordering probability. Recreate the concrete case: Y exceeds X often. Write units, cross-pairs, placements, tie halves, placement sums, means, variance components, standardized statistics and probability summaries with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Probability and median are distinct. Complete the transfer step: Keep 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: Sensitivity

Begin with this mechanism: Compare Brunner–Munzel and plots. Recreate the concrete case: Direction agrees. Write units, cross-pairs, placements, tie halves, placement sums, means, variance components, standardized statistics and probability summaries with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Method shopping inflates certainty. Complete the transfer step: Preplan. 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 robust location evidence. Recreate the concrete case: Y tends larger than X. Write units, cross-pairs, placements, tie halves, placement sums, means, variance components, standardized statistics and probability summaries with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Cause is not identified. Complete the transfer step: Use design evidence. 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.