WHY MATHEMATICS?

Why Mathematics? | Rousseeuw–Croux Sn Scale: Median-of-Medians Distances, Location-Free Spread and Asymmetric Robustness

How local median distances become a global robust scale, why the construction can suit asymmetric data, and why nested median conventions must remain reproducible.

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

Build one distance row around every observation
Take each row median and then their outer median
Apply 1.1926 and the appropriate finite-sample factor
Use the local-distance map for sensitivity and subgroup questions

Chapter map

1. Sample
2. Distance row
3. First row
4. Second row
5. Third row
6. Fourth row
7. Fifth row
8. Inner vector
9. Outer median
10. Consistency constant
11. Finite correction
12. Translation
13. Breakdown
14. Efficiency
15. Local diagnostic
16. Claim

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

1. Sample

A small example makes the mechanism visible. Freeze observations and IDs.

A concrete situation

Use 1,2,3,4,8.

What the mathematics does not license

Clustered data need separate treatment. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Audit 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. Distance row

This is where modelling choices begin to affect evidence. For each i compute distances to all j.

A concrete situation

Each row includes one zero self-distance.

What the mathematics does not license

Omitting self-distance changes small-n medians. Keeping this boundary clear makes the result easier to audit and transfer.

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.

3. First row

The central mathematical idea is concrete. Sort distances from x=1.

A concrete situation

0,1,2,3,7 gives median 2.

What the mathematics does not license

Endpoint rows can be wide. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Keep row. 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. Second row

Here the geometry becomes operational. Sort distances from x=2.

A concrete situation

0,1,1,2,6 gives median 1.

What the mathematics does not license

Ties carry real information. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Record ties. 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. Third row

This step deserves a careful pause. Sort distances from x=3.

A concrete situation

0,1,1,2,5 gives median 1.

What the mathematics does not license

The sample centre need not be estimated. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Note property. 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. Fourth row

The useful question is not only how to calculate, but what the calculation assumes. Sort distances from x=4.

A concrete situation

0,1,2,3,4 gives median 2.

What the mathematics does not license

Copy errors change the outer median. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Use 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.

7. Fifth row

A small example makes the mechanism visible. Sort distances from x=8.

A concrete situation

0,4,5,6,7 gives median 5.

What the mathematics does not license

A remote point has a large local radius. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Inspect provenance. 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. Inner vector

This is where modelling choices begin to affect evidence. Collect all local medians.

A concrete situation

2,1,1,2,5.

What the mathematics does not license

Do not average these values. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Keep order. 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. Outer median

The central mathematical idea is concrete. Take the median over i.

A concrete situation

The raw Sn core is 2.

What the mathematics does not license

Even-n conventions must be explicit. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Name median 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.

10. Consistency constant

Here the geometry becomes operational. Multiply by 1.1926.

A concrete situation

Teaching value≈2.3852.

What the mathematics does not license

The constant targets the normal model. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

State target. 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. Finite correction

This step deserves a careful pause. Apply the n-specific adjustment.

A concrete situation

Separate it from 1.1926.

What the mathematics does not license

Small samples otherwise remain biased. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Record version. 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. Translation

The useful question is not only how to calculate, but what the calculation assumes. Shift every observation equally.

A concrete situation

Every distance and Sn remain unchanged.

What the mathematics does not license

Nonlinear transformations do change it. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Explain 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.

13. Breakdown

A small example makes the mechanism visible. Assess worst-case replacement contamination.

A concrete situation

The breakdown point is 50%.

What the mathematics does not license

Structured contamination can still mislead. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Use sensitivity. 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. Efficiency

This is where modelling choices begin to affect evidence. Compare normal-model sampling variance.

A concrete situation

Asymptotic Gaussian efficiency is about 58%.

What the mathematics does not license

This is not universal. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

State model. 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. Local diagnostic

The central mathematical idea is concrete. Plot each inner median by observation.

A concrete situation

The value 8 has local radius 5.

What the mathematics does not license

A large radius can be genuine subgroup structure. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Investigate context. 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

Here the geometry becomes operational. State median-of-local-distances scale evidence.

A concrete situation

Typical local separation is summarized.

What the mathematics does not license

Sn is not an outlier classifier by itself. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Keep scope. 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 Rousseeuw–Croux Sn estimator asks a local question around every observation and then takes a global median. For each x_i, calculate the median over j of |x_i−x_j|. Then take the median of those n local medians and multiply by the normal-consistency constant 1.1926 plus the relevant finite-sample correction. No sample mean or sample median is required.

For 1,2,3,4,8, the five local median distances are 2,1,1,2 and 5 when the zero self-distance is included as defined. Their outer median is 2. Multiplying by 1.1926 gives Sn≈2.3852 before the finite-n correction. The final point has a large local distance, yet one large local summary does not control the outer median.

Sn has a 50% breakdown point and about 58% asymptotic Gaussian efficiency in the original paper. It can behave attractively for skewed models because it does not measure deviations from a single centre. Its local-median construction also gives a useful diagnostic map: unusually large inner medians identify observations sitting far from most of the sample.

The nested medians create conventions that must be explicit for even sample sizes, duplicated observations and finite-sample correction. A software number is not reproducible without the precise median rule. Robustness also does not remove the need to investigate data provenance, clustering, dependence or genuine multimodality.

Frequently asked questions

What is the inner statistic?

For each observation, the median distance to all observations, including its zero self-distance.

What is the outer statistic?

The median of those local median distances.

Does Sn require a location estimate?

No.

What constant gives normal consistency?

Approximately 1.1926 before the separate finite-n correction.

What are its headline robustness properties?

A 50% breakdown point and about 58% asymptotic Gaussian efficiency.

Can a large local median label an outlier automatically?

No. It may also indicate a real subgroup, skewness or measurement issue.

Primary source and fact-check note

Peter J. Rousseeuw and Christophe Croux, “Alternatives to the Median Absolute Deviation,” Journal of the American Statistical Association 88(424), 1273–1283 (1993), DOI 10.1080/01621459.1993.10476408. 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 observations, units, target quantile, dependence and missing-value rules.

Ordering audit

Reproduce pair generation, distance rows, order-statistic ranks or beta intervals.

Statistic audit

Show every gap, local median, weight, moment, constant and correction factor.

Calibration audit

Match finite-sample correction, quantile convention and uncertainty method.

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.

Graph audit

Preserve distance maps, ordered-value weights, influence displays and bootstrap views.

Magnitude audit

Explain spread or uncertainty in original units.

Sensitivity audit

Use only prespecified measurement, contamination and subgroup checks.

Final claim

Keep the conclusion inside the sampled population and measured process.

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: Sample

Begin with this mechanism: Freeze observations and IDs. Recreate the concrete case: Use 1,2,3,4,8. Write observations, full distance rows, inner medians, outer medians, consistency constants, finite-sample factors and scale units with units, indices and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Clustered data need separate treatment. Complete the transfer step: Audit design. Change exactly one observation, tie, missing-value rule, pair count, inner-median convention, selection index, consistency constant, finite-sample correction, target probability, beta shape, CDF boundary, tail value, jackknife deletion or confidence multiplier. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, ordering, statistic, correction, uncertainty, magnitude, interpretation and validation on separate notebook lines. Explain the result once to a student with the calculation table and once to a parent with the real decision and original units. Add a counterexample in which the same estimate comes from rounding, censoring, mixture structure, dependence or contamination. State what extra graph or measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 2: Distance row

Begin with this mechanism: For each i compute distances to all j. Recreate the concrete case: Each row includes one zero self-distance. Write observations, full distance rows, inner medians, outer medians, consistency constants, finite-sample factors and scale units with units, indices and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Omitting self-distance changes small-n medians. Complete the transfer step: Match source. Change exactly one observation, tie, missing-value rule, pair count, inner-median convention, selection index, consistency constant, finite-sample correction, target probability, beta shape, CDF boundary, tail value, jackknife deletion or confidence multiplier. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, ordering, statistic, correction, uncertainty, magnitude, interpretation and validation on separate notebook lines. Explain the result once to a student with the calculation table and once to a parent with the real decision and original units. Add a counterexample in which the same estimate comes from rounding, censoring, mixture structure, dependence or contamination. State what extra graph or measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 3: First row

Begin with this mechanism: Sort distances from x=1. Recreate the concrete case: 0,1,2,3,7 gives median 2. Write observations, full distance rows, inner medians, outer medians, consistency constants, finite-sample factors and scale units with units, indices and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Endpoint rows can be wide. Complete the transfer step: Keep row. Change exactly one observation, tie, missing-value rule, pair count, inner-median convention, selection index, consistency constant, finite-sample correction, target probability, beta shape, CDF boundary, tail value, jackknife deletion or confidence multiplier. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, ordering, statistic, correction, uncertainty, magnitude, interpretation and validation on separate notebook lines. Explain the result once to a student with the calculation table and once to a parent with the real decision and original units. Add a counterexample in which the same estimate comes from rounding, censoring, mixture structure, dependence or contamination. State what extra graph or measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 4: Second row

Begin with this mechanism: Sort distances from x=2. Recreate the concrete case: 0,1,1,2,6 gives median 1. Write observations, full distance rows, inner medians, outer medians, consistency constants, finite-sample factors and scale units with units, indices and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Ties carry real information. Complete the transfer step: Record ties. Change exactly one observation, tie, missing-value rule, pair count, inner-median convention, selection index, consistency constant, finite-sample correction, target probability, beta shape, CDF boundary, tail value, jackknife deletion or confidence multiplier. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, ordering, statistic, correction, uncertainty, magnitude, interpretation and validation on separate notebook lines. Explain the result once to a student with the calculation table and once to a parent with the real decision and original units. Add a counterexample in which the same estimate comes from rounding, censoring, mixture structure, dependence or contamination. State what extra graph or measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence 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: Third row

Begin with this mechanism: Sort distances from x=3. Recreate the concrete case: 0,1,1,2,5 gives median 1. Write observations, full distance rows, inner medians, outer medians, consistency constants, finite-sample factors and scale units with units, indices and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: The sample centre need not be estimated. Complete the transfer step: Note property. Change exactly one observation, tie, missing-value rule, pair count, inner-median convention, selection index, consistency constant, finite-sample correction, target probability, beta shape, CDF boundary, tail value, jackknife deletion or confidence multiplier. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, ordering, statistic, correction, uncertainty, magnitude, interpretation and validation on separate notebook lines. Explain the result once to a student with the calculation table and once to a parent with the real decision and original units. Add a counterexample in which the same estimate comes from rounding, censoring, mixture structure, dependence or contamination. State what extra graph or measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 6: Fourth row

Begin with this mechanism: Sort distances from x=4. Recreate the concrete case: 0,1,2,3,4 gives median 2. Write observations, full distance rows, inner medians, outer medians, consistency constants, finite-sample factors and scale units with units, indices and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Copy errors change the outer median. Complete the transfer step: Use matrix. Change exactly one observation, tie, missing-value rule, pair count, inner-median convention, selection index, consistency constant, finite-sample correction, target probability, beta shape, CDF boundary, tail value, jackknife deletion or confidence multiplier. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, ordering, statistic, correction, uncertainty, magnitude, interpretation and validation on separate notebook lines. Explain the result once to a student with the calculation table and once to a parent with the real decision and original units. Add a counterexample in which the same estimate comes from rounding, censoring, mixture structure, dependence or contamination. State what extra graph or measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 7: Fifth row

Begin with this mechanism: Sort distances from x=8. Recreate the concrete case: 0,4,5,6,7 gives median 5. Write observations, full distance rows, inner medians, outer medians, consistency constants, finite-sample factors and scale units with units, indices and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: A remote point has a large local radius. Complete the transfer step: Inspect provenance. Change exactly one observation, tie, missing-value rule, pair count, inner-median convention, selection index, consistency constant, finite-sample correction, target probability, beta shape, CDF boundary, tail value, jackknife deletion or confidence multiplier. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, ordering, statistic, correction, uncertainty, magnitude, interpretation and validation on separate notebook lines. Explain the result once to a student with the calculation table and once to a parent with the real decision and original units. Add a counterexample in which the same estimate comes from rounding, censoring, mixture structure, dependence or contamination. State what extra graph or measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 8: Inner vector

Begin with this mechanism: Collect all local medians. Recreate the concrete case: 2,1,1,2,5. Write observations, full distance rows, inner medians, outer medians, consistency constants, finite-sample factors and scale units with units, indices and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Do not average these values. Complete the transfer step: Keep order. Change exactly one observation, tie, missing-value rule, pair count, inner-median convention, selection index, consistency constant, finite-sample correction, target probability, beta shape, CDF boundary, tail value, jackknife deletion or confidence multiplier. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, ordering, statistic, correction, uncertainty, magnitude, interpretation and validation on separate notebook lines. Explain the result once to a student with the calculation table and once to a parent with the real decision and original units. Add a counterexample in which the same estimate comes from rounding, censoring, mixture structure, dependence or contamination. State what extra graph or measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence 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: Outer median

Begin with this mechanism: Take the median over i. Recreate the concrete case: The raw Sn core is 2. Write observations, full distance rows, inner medians, outer medians, consistency constants, finite-sample factors and scale units with units, indices and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Even-n conventions must be explicit. Complete the transfer step: Name median rule. Change exactly one observation, tie, missing-value rule, pair count, inner-median convention, selection index, consistency constant, finite-sample correction, target probability, beta shape, CDF boundary, tail value, jackknife deletion or confidence multiplier. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, ordering, statistic, correction, uncertainty, magnitude, interpretation and validation on separate notebook lines. Explain the result once to a student with the calculation table and once to a parent with the real decision and original units. Add a counterexample in which the same estimate comes from rounding, censoring, mixture structure, dependence or contamination. State what extra graph or measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 10: Consistency constant

Begin with this mechanism: Multiply by 1.1926. Recreate the concrete case: Teaching value≈2.3852. Write observations, full distance rows, inner medians, outer medians, consistency constants, finite-sample factors and scale units with units, indices and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: The constant targets the normal model. Complete the transfer step: State target. Change exactly one observation, tie, missing-value rule, pair count, inner-median convention, selection index, consistency constant, finite-sample correction, target probability, beta shape, CDF boundary, tail value, jackknife deletion or confidence multiplier. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, ordering, statistic, correction, uncertainty, magnitude, interpretation and validation on separate notebook lines. Explain the result once to a student with the calculation table and once to a parent with the real decision and original units. Add a counterexample in which the same estimate comes from rounding, censoring, mixture structure, dependence or contamination. State what extra graph or measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 11: Finite correction

Begin with this mechanism: Apply the n-specific adjustment. Recreate the concrete case: Separate it from 1.1926. Write observations, full distance rows, inner medians, outer medians, consistency constants, finite-sample factors and scale units with units, indices and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Small samples otherwise remain biased. Complete the transfer step: Record version. Change exactly one observation, tie, missing-value rule, pair count, inner-median convention, selection index, consistency constant, finite-sample correction, target probability, beta shape, CDF boundary, tail value, jackknife deletion or confidence multiplier. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, ordering, statistic, correction, uncertainty, magnitude, interpretation and validation on separate notebook lines. Explain the result once to a student with the calculation table and once to a parent with the real decision and original units. Add a counterexample in which the same estimate comes from rounding, censoring, mixture structure, dependence or contamination. State what extra graph or measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 12: Translation

Begin with this mechanism: Shift every observation equally. Recreate the concrete case: Every distance and Sn remain unchanged. Write observations, full distance rows, inner medians, outer medians, consistency constants, finite-sample factors and scale units with units, indices and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Nonlinear transformations do change it. Complete the transfer step: Explain units. Change exactly one observation, tie, missing-value rule, pair count, inner-median convention, selection index, consistency constant, finite-sample correction, target probability, beta shape, CDF boundary, tail value, jackknife deletion or confidence multiplier. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, ordering, statistic, correction, uncertainty, magnitude, interpretation and validation on separate notebook lines. Explain the result once to a student with the calculation table and once to a parent with the real decision and original units. Add a counterexample in which the same estimate comes from rounding, censoring, mixture structure, dependence or contamination. State what extra graph or measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence 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: Breakdown

Begin with this mechanism: Assess worst-case replacement contamination. Recreate the concrete case: The breakdown point is 50%. Write observations, full distance rows, inner medians, outer medians, consistency constants, finite-sample factors and scale units with units, indices and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Structured contamination can still mislead. Complete the transfer step: Use sensitivity. Change exactly one observation, tie, missing-value rule, pair count, inner-median convention, selection index, consistency constant, finite-sample correction, target probability, beta shape, CDF boundary, tail value, jackknife deletion or confidence multiplier. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, ordering, statistic, correction, uncertainty, magnitude, interpretation and validation on separate notebook lines. Explain the result once to a student with the calculation table and once to a parent with the real decision and original units. Add a counterexample in which the same estimate comes from rounding, censoring, mixture structure, dependence or contamination. State what extra graph or measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 14: Efficiency

Begin with this mechanism: Compare normal-model sampling variance. Recreate the concrete case: Asymptotic Gaussian efficiency is about 58%. Write observations, full distance rows, inner medians, outer medians, consistency constants, finite-sample factors and scale units with units, indices and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: This is not universal. Complete the transfer step: State model. Change exactly one observation, tie, missing-value rule, pair count, inner-median convention, selection index, consistency constant, finite-sample correction, target probability, beta shape, CDF boundary, tail value, jackknife deletion or confidence multiplier. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, ordering, statistic, correction, uncertainty, magnitude, interpretation and validation on separate notebook lines. Explain the result once to a student with the calculation table and once to a parent with the real decision and original units. Add a counterexample in which the same estimate comes from rounding, censoring, mixture structure, dependence or contamination. State what extra graph or measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 15: Local diagnostic

Begin with this mechanism: Plot each inner median by observation. Recreate the concrete case: The value 8 has local radius 5. Write observations, full distance rows, inner medians, outer medians, consistency constants, finite-sample factors and scale units with units, indices and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: A large radius can be genuine subgroup structure. Complete the transfer step: Investigate context. Change exactly one observation, tie, missing-value rule, pair count, inner-median convention, selection index, consistency constant, finite-sample correction, target probability, beta shape, CDF boundary, tail value, jackknife deletion or confidence multiplier. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, ordering, statistic, correction, uncertainty, magnitude, interpretation and validation on separate notebook lines. Explain the result once to a student with the calculation table and once to a parent with the real decision and original units. Add a counterexample in which the same estimate comes from rounding, censoring, mixture structure, dependence or contamination. State what extra graph or measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 16: Claim

Begin with this mechanism: State median-of-local-distances scale evidence. Recreate the concrete case: Typical local separation is summarized. Write observations, full distance rows, inner medians, outer medians, consistency constants, finite-sample factors and scale units with units, indices and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Sn is not an outlier classifier by itself. Complete the transfer step: Keep scope. Change exactly one observation, tie, missing-value rule, pair count, inner-median convention, selection index, consistency constant, finite-sample correction, target probability, beta shape, CDF boundary, tail value, jackknife deletion or confidence multiplier. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, ordering, statistic, correction, uncertainty, magnitude, interpretation and validation on separate notebook lines. Explain the result once to a student with the calculation table and once to a parent with the real decision and original units. Add a counterexample in which the same estimate comes from rounding, censoring, mixture structure, dependence or contamination. State what extra graph or measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.