Chapter map

1. 1. Sample
2. 2. Order
3. 3. Mean
4. 4. Scale
5. 5. Standardize
6. 6. Plotting positions
7. 7. Normal quantiles
8. 8. Signed gaps
9. 9. Absolute gaps
10. 10. T1
11. 11. Squared gaps
12. 12. T2
13. 13. Monte Carlo
14. 14. Version choice
15. 15. Sensitivity
16. 16. Claim

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

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1. Sample

The central mathematical idea is concrete. Freeze iid observations and the missing-data rule.

A concrete situation

Use 1,2,3,4,8.

What the mathematics does not license

Dependence changes the null distribution. Keeping this boundary clear makes the result easier to audit and transfer.

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.

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2. Order

Here the geometry becomes operational. Sort from smallest to largest.

A concrete situation

The order remains 1,2,3,4,8.

What the mathematics does not license

Sorting without retaining IDs weakens traceability. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Keep mapping. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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3. Mean

This step deserves a careful pause. Calculate the sample mean.

A concrete situation

Mean=3.6.

What the mathematics does not license

Rounding here propagates through every score. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Keep precision. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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4. Scale

The useful question is not only how to calculate, but what the calculation assumes. Use s²=n⁻¹Σ(x−x̄)².

A concrete situation

s²=5.84 and s≈2.4166.

What the mathematics does not license

Using n−1 defines a different statistic. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Match formula. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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5. Standardize

A small example makes the mechanism visible. Compute y_i=(x_(i)−x̄)/s.

A concrete situation

Values run from -1.0759 to 1.8207.

What the mathematics does not license

Standardize after ordering or before ordering consistently. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Show 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.

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6. Plotting positions

This is where modelling choices begin to affect evidence. Set p_i=i/(n+1).

A concrete situation

Positions are 1/6 through 5/6.

What the mathematics does not license

Other plotting positions change the test. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Name convention. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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7. Normal quantiles

The central mathematical idea is concrete. Compute q_i=Φ⁻¹(p_i).

A concrete situation

Endpoints are about ±0.9674.

What the mathematics does not license

Calculator tail precision matters. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Keep digits. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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8. Signed gaps

Here the geometry becomes operational. Subtract q_i from y_i.

A concrete situation

The upper gap is about 0.8533.

What the mathematics does not license

Signs reveal direction but can cancel. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Retain signs. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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9. Absolute gaps

This step deserves a careful pause. Take |y_i−q_i|.

A concrete situation

All five contribute linearly.

What the mathematics does not license

Absolute gaps lose direction. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Pair with plot. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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10. T1

The useful question is not only how to calculate, but what the calculation assumes. Average the absolute gaps.

A concrete situation

T1≈0.3413.

What the mathematics does not license

T1 is not a p-value. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Calibrate. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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11. Squared gaps

A small example makes the mechanism visible. Square each difference.

A concrete situation

The upper gap contributes about 0.7281.

What the mathematics does not license

One tail can dominate. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Show shares. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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12. T2

This is where modelling choices begin to affect evidence. Average the squared gaps.

A concrete situation

T2≈0.1851.

What the mathematics does not license

T2 cannot be compared with a generic variance table. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Simulate 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.

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13. Monte Carlo

The central mathematical idea is concrete. Repeat normal sampling, fitting and scoring at the same n.

A concrete situation

Count simulated statistics at least as large.

What the mathematics does not license

Reusing fixed parameters understates uncertainty. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Re-estimate each run. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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14. Version choice

Here the geometry becomes operational. Predefine T1, T2 or a multiplicity plan.

A concrete situation

Use both as planned diagnostics if desired.

What the mathematics does not license

Choosing after seeing results inflates error. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

State plan. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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15. Sensitivity

This step deserves a careful pause. Recompute after a documented measurement check.

A concrete situation

The value 8 drives the upper-tail gap.

What the mathematics does not license

Deleting an inconvenient value is invalid. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Use 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.

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16. Claim

The useful question is not only how to calculate, but what the calculation assumes. State order-statistic discrepancy evidence.

A concrete situation

Normal quantile agreement is strong or weak.

What the mathematics does not license

Normality is never proved. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Keep 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.

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A seven-day practice lab

1. Day 1: define the decision, observation unit, X variables and Y or reference condition.
2. Day 2: reproduce the smallest worked example and check every dimension, unit and sign.
3. Day 3: build a transparent baseline and write what it assumes.
4. Day 4: fit the main method inside a training-only pipeline; record every parameter.
5. Day 5: challenge one assumption with a deliberately difficult dataset or operating regime.
6. Day 6: compare validation metrics with the practical workload created by errors.
7. Day 7: present one supported conclusion, one unresolved uncertainty and one safe next experiment.

Questions for a student, parent or mentor

• Which part of the model is a mathematical definition, and which part is a choice?
• What small calculation would catch a sign, scale or dimension error?
• What evidence comes from training, validation and the untouched test?
• What benign condition could look like a fault or prediction failure?
• How would the answer change for another response, asset, school project or time period?
• What should a human review before acting on the score?

A method-specific audit before deployment

The Hegazy–Green normality tests turn a normal probability plot into two transparent discrepancy averages. Sort the observations, standardize them with the sample mean and the n-divisor standard deviation, compare each standardized order statistic with the standard-normal quantile at i/(n+1), then average either the absolute gaps or the squared gaps. The first version keeps every gap linear; the second gives large disagreements more weight.

For 1,2,3,4,8, the mean is 3.6 and the n-divisor standard deviation is about 2.4166. The ordered standardized values are approximately -1.0759,-0.6621,-0.2483,0.1655 and 1.8207. The comparison quantiles at i/6 are about -0.9674,-0.4307,0,0.4307 and 0.9674. The five signed gaps therefore end with a conspicuous 0.8533 in the upper tail.

Averaging absolute gaps gives T1≈0.3413. Averaging squared gaps gives T2≈0.1851. These are statistics, not probabilities. Because location and scale were estimated from the same sample, a generic table for independent standard-normal observations is inappropriate. The original work used approximations and simulation; modern implementations commonly use Monte Carlo calibration that repeats the complete standardization step.

The two versions answer related but different questions. T1 gives proportional influence to each quantile gap, while T2 magnifies the largest gaps. Agreement can strengthen the diagnostic story, but selecting whichever version rejects after viewing the result inflates error. A responsible analysis chooses one primary statistic in advance or treats both as a planned diagnostic family.

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Frequently asked questions

What does T1 average?

Absolute differences between ordered standardized observations and expected normal quantiles.

What does T2 average?

Squared versions of the same quantile differences.

Why i/(n+1)?

It supplies interior plotting positions for the ordered observations.

Why estimate mean and scale again in simulations?

Their estimation changes the null distribution of the discrepancy statistics.

Which statistic is more tail-sensitive?

T2 magnifies a large individual gap more strongly than T1.

What should be retained?

The ordered values, fitted mean and scale, quantiles, every gap, statistic version and calibration seed.

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Primary source and fact-check note

Y. A. S. Hegazy and J. R. Green, “Some New Goodness-of-Fit Tests Using Order Statistics,” Journal of the Royal Statistical Society Series C: Applied Statistics 24(3), 299–308 (1975), DOI 10.2307/2347090. 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

• Zhang–Wu likelihood-ratio normality tests
• Hosking L-moments for normality
• How Mathematics Works

Keep the options open. Mathematics education supports clearer questions, transferable problem-solving skills and informed choices across science, engineering, computing, AI, statistics, design and many other careers. It does not guarantee intelligence, income, admission or employment; those outcomes depend on far more than one subject or model.

Extended laboratory: topic-specific validation and responsible use

Design audit

Freeze observations, chronology, target residuals, dependence and inclusion rules.

Transformation audit

Reproduce ordering, plotting positions, normal scores, fitted-null probabilities or standardized moments.

Statistic audit

Show every gap, projection, component, dependence factor, penalty and coefficient.

Calibration audit

Match sample size, parameter estimation, dependence, selection and statistic version.

Graph audit

Preserve Q–Q views, score paths, ACF displays, influence plots and component maps.

Magnitude audit

Explain consequences in original units and decision context.

Sensitivity audit

Use only prespecified validation, model and measurement checks.

Final claim

Keep the conclusion inside the measured population, time window and model.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Sample

Begin with this mechanism: Freeze iid observations and the missing-data rule. Recreate the concrete case: Use 1,2,3,4,8. Write ordered observations, means, variance divisors, standard scores, plotting positions, normal quantiles, signed gaps, absolute gaps, squared gaps and simulated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Dependence changes the null distribution. Complete the transfer step: Audit design. Change exactly one observation, ordering rule, scale divisor, plotting position, normal score, basis function, nuisance projection, maximum dimension, penalty, lag autocorrelation, dependence factor, rejection direction, finite-sample correction or Monte Carlo 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 once to a student with the component table and once to a parent with the real decision and original units. Add a counterexample in which the same statistic comes from rounding, censoring, mixture structure, dependence, conditional volatility 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: Order

Begin with this mechanism: Sort from smallest to largest. Recreate the concrete case: The order remains 1,2,3,4,8. Write ordered observations, means, variance divisors, standard scores, plotting positions, normal quantiles, signed gaps, absolute gaps, squared gaps and simulated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Sorting without retaining IDs weakens traceability. Complete the transfer step: Keep mapping. Change exactly one observation, ordering rule, scale divisor, plotting position, normal score, basis function, nuisance projection, maximum dimension, penalty, lag autocorrelation, dependence factor, rejection direction, finite-sample correction or Monte Carlo 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 once to a student with the component table and once to a parent with the real decision and original units. Add a counterexample in which the same statistic comes from rounding, censoring, mixture structure, dependence, conditional volatility 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: Mean

Begin with this mechanism: Calculate the sample mean. Recreate the concrete case: Mean=3.6. Write ordered observations, means, variance divisors, standard scores, plotting positions, normal quantiles, signed gaps, absolute gaps, squared gaps and simulated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Rounding here propagates through every score. Complete the transfer step: Keep precision. Change exactly one observation, ordering rule, scale divisor, plotting position, normal score, basis function, nuisance projection, maximum dimension, penalty, lag autocorrelation, dependence factor, rejection direction, finite-sample correction or Monte Carlo 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 once to a student with the component table and once to a parent with the real decision and original units. Add a counterexample in which the same statistic comes from rounding, censoring, mixture structure, dependence, conditional volatility 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: Scale

Begin with this mechanism: Use s²=n⁻¹Σ(x−x̄)². Recreate the concrete case: s²=5.84 and s≈2.4166. Write ordered observations, means, variance divisors, standard scores, plotting positions, normal quantiles, signed gaps, absolute gaps, squared gaps and simulated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Using n−1 defines a different statistic. Complete the transfer step: Match formula. Change exactly one observation, ordering rule, scale divisor, plotting position, normal score, basis function, nuisance projection, maximum dimension, penalty, lag autocorrelation, dependence factor, rejection direction, finite-sample correction or Monte Carlo 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 once to a student with the component table and once to a parent with the real decision and original units. Add a counterexample in which the same statistic comes from rounding, censoring, mixture structure, dependence, conditional volatility 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 5: Standardize

Begin with this mechanism: Compute y_i=(x_(i)−x̄)/s. Recreate the concrete case: Values run from -1.0759 to 1.8207. Write ordered observations, means, variance divisors, standard scores, plotting positions, normal quantiles, signed gaps, absolute gaps, squared gaps and simulated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Standardize after ordering or before ordering consistently. Complete the transfer step: Show vector. Change exactly one observation, ordering rule, scale divisor, plotting position, normal score, basis function, nuisance projection, maximum dimension, penalty, lag autocorrelation, dependence factor, rejection direction, finite-sample correction or Monte Carlo 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 once to a student with the component table and once to a parent with the real decision and original units. Add a counterexample in which the same statistic comes from rounding, censoring, mixture structure, dependence, conditional volatility 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: Plotting positions

Begin with this mechanism: Set p_i=i/(n+1). Recreate the concrete case: Positions are 1/6 through 5/6. Write ordered observations, means, variance divisors, standard scores, plotting positions, normal quantiles, signed gaps, absolute gaps, squared gaps and simulated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Other plotting positions change the test. Complete the transfer step: Name convention. Change exactly one observation, ordering rule, scale divisor, plotting position, normal score, basis function, nuisance projection, maximum dimension, penalty, lag autocorrelation, dependence factor, rejection direction, finite-sample correction or Monte Carlo 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 once to a student with the component table and once to a parent with the real decision and original units. Add a counterexample in which the same statistic comes from rounding, censoring, mixture structure, dependence, conditional volatility 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: Normal quantiles

Begin with this mechanism: Compute q_i=Φ⁻¹(p_i). Recreate the concrete case: Endpoints are about ±0.9674. Write ordered observations, means, variance divisors, standard scores, plotting positions, normal quantiles, signed gaps, absolute gaps, squared gaps and simulated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Calculator tail precision matters. Complete the transfer step: Keep digits. Change exactly one observation, ordering rule, scale divisor, plotting position, normal score, basis function, nuisance projection, maximum dimension, penalty, lag autocorrelation, dependence factor, rejection direction, finite-sample correction or Monte Carlo 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 once to a student with the component table and once to a parent with the real decision and original units. Add a counterexample in which the same statistic comes from rounding, censoring, mixture structure, dependence, conditional volatility 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: Signed gaps

Begin with this mechanism: Subtract q_i from y_i. Recreate the concrete case: The upper gap is about 0.8533. Write ordered observations, means, variance divisors, standard scores, plotting positions, normal quantiles, signed gaps, absolute gaps, squared gaps and simulated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Signs reveal direction but can cancel. Complete the transfer step: Retain signs. Change exactly one observation, ordering rule, scale divisor, plotting position, normal score, basis function, nuisance projection, maximum dimension, penalty, lag autocorrelation, dependence factor, rejection direction, finite-sample correction or Monte Carlo 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 once to a student with the component table and once to a parent with the real decision and original units. Add a counterexample in which the same statistic comes from rounding, censoring, mixture structure, dependence, conditional volatility 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 9: Absolute gaps

Begin with this mechanism: Take |y_i−q_i|. Recreate the concrete case: All five contribute linearly. Write ordered observations, means, variance divisors, standard scores, plotting positions, normal quantiles, signed gaps, absolute gaps, squared gaps and simulated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Absolute gaps lose direction. Complete the transfer step: Pair with plot. Change exactly one observation, ordering rule, scale divisor, plotting position, normal score, basis function, nuisance projection, maximum dimension, penalty, lag autocorrelation, dependence factor, rejection direction, finite-sample correction or Monte Carlo 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 once to a student with the component table and once to a parent with the real decision and original units. Add a counterexample in which the same statistic comes from rounding, censoring, mixture structure, dependence, conditional volatility 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: T1

Begin with this mechanism: Average the absolute gaps. Recreate the concrete case: T1≈0.3413. Write ordered observations, means, variance divisors, standard scores, plotting positions, normal quantiles, signed gaps, absolute gaps, squared gaps and simulated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: T1 is not a p-value. Complete the transfer step: Calibrate. Change exactly one observation, ordering rule, scale divisor, plotting position, normal score, basis function, nuisance projection, maximum dimension, penalty, lag autocorrelation, dependence factor, rejection direction, finite-sample correction or Monte Carlo 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 once to a student with the component table and once to a parent with the real decision and original units. Add a counterexample in which the same statistic comes from rounding, censoring, mixture structure, dependence, conditional volatility 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: Squared gaps

Begin with this mechanism: Square each difference. Recreate the concrete case: The upper gap contributes about 0.7281. Write ordered observations, means, variance divisors, standard scores, plotting positions, normal quantiles, signed gaps, absolute gaps, squared gaps and simulated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: One tail can dominate. Complete the transfer step: Show shares. Change exactly one observation, ordering rule, scale divisor, plotting position, normal score, basis function, nuisance projection, maximum dimension, penalty, lag autocorrelation, dependence factor, rejection direction, finite-sample correction or Monte Carlo 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 once to a student with the component table and once to a parent with the real decision and original units. Add a counterexample in which the same statistic comes from rounding, censoring, mixture structure, dependence, conditional volatility 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: T2

Begin with this mechanism: Average the squared gaps. Recreate the concrete case: T2≈0.1851. Write ordered observations, means, variance divisors, standard scores, plotting positions, normal quantiles, signed gaps, absolute gaps, squared gaps and simulated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: T2 cannot be compared with a generic variance table. Complete the transfer step: Simulate null. Change exactly one observation, ordering rule, scale divisor, plotting position, normal score, basis function, nuisance projection, maximum dimension, penalty, lag autocorrelation, dependence factor, rejection direction, finite-sample correction or Monte Carlo 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 once to a student with the component table and once to a parent with the real decision and original units. Add a counterexample in which the same statistic comes from rounding, censoring, mixture structure, dependence, conditional volatility 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 13: Monte Carlo

Begin with this mechanism: Repeat normal sampling, fitting and scoring at the same n. Recreate the concrete case: Count simulated statistics at least as large. Write ordered observations, means, variance divisors, standard scores, plotting positions, normal quantiles, signed gaps, absolute gaps, squared gaps and simulated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Reusing fixed parameters understates uncertainty. Complete the transfer step: Re-estimate each run. Change exactly one observation, ordering rule, scale divisor, plotting position, normal score, basis function, nuisance projection, maximum dimension, penalty, lag autocorrelation, dependence factor, rejection direction, finite-sample correction or Monte Carlo 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 once to a student with the component table and once to a parent with the real decision and original units. Add a counterexample in which the same statistic comes from rounding, censoring, mixture structure, dependence, conditional volatility 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: Version choice

Begin with this mechanism: Predefine T1, T2 or a multiplicity plan. Recreate the concrete case: Use both as planned diagnostics if desired. Write ordered observations, means, variance divisors, standard scores, plotting positions, normal quantiles, signed gaps, absolute gaps, squared gaps and simulated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Choosing after seeing results inflates error. Complete the transfer step: State plan. Change exactly one observation, ordering rule, scale divisor, plotting position, normal score, basis function, nuisance projection, maximum dimension, penalty, lag autocorrelation, dependence factor, rejection direction, finite-sample correction or Monte Carlo 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 once to a student with the component table and once to a parent with the real decision and original units. Add a counterexample in which the same statistic comes from rounding, censoring, mixture structure, dependence, conditional volatility 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: Sensitivity

Begin with this mechanism: Recompute after a documented measurement check. Recreate the concrete case: The value 8 drives the upper-tail gap. Write ordered observations, means, variance divisors, standard scores, plotting positions, normal quantiles, signed gaps, absolute gaps, squared gaps and simulated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Deleting an inconvenient value is invalid. Complete the transfer step: Use provenance. Change exactly one observation, ordering rule, scale divisor, plotting position, normal score, basis function, nuisance projection, maximum dimension, penalty, lag autocorrelation, dependence factor, rejection direction, finite-sample correction or Monte Carlo 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 once to a student with the component table and once to a parent with the real decision and original units. Add a counterexample in which the same statistic comes from rounding, censoring, mixture structure, dependence, conditional volatility 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 order-statistic discrepancy evidence. Recreate the concrete case: Normal quantile agreement is strong or weak. Write ordered observations, means, variance divisors, standard scores, plotting positions, normal quantiles, signed gaps, absolute gaps, squared gaps and simulated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Normality is never proved. Complete the transfer step: Keep context. Change exactly one observation, ordering rule, scale divisor, plotting position, normal score, basis function, nuisance projection, maximum dimension, penalty, lag autocorrelation, dependence factor, rejection direction, finite-sample correction or Monte Carlo 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 once to a student with the component table and once to a parent with the real decision and original units. Add a counterexample in which the same statistic comes from rounding, censoring, mixture structure, dependence, conditional volatility 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.