Chapter map

1. 1. Data order
2. 2. Lower endpoint
3. 3. Interior ranks
4. 4. Upper endpoint
5. 5. Normal scores
6. 6. Centre scores
7. 7. Data deviations
8. 8. Score deviations
9. 9. Cross-products
10. 10. Denominator
11. 11. PPCC
12. 12. Critical value
13. 13. Residual curve
14. 14. Ties
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. Data order

The useful question is not only how to calculate, but what the calculation assumes. Sort the complete sample.

A concrete situation

Use 1,2,3,4,5.

What the mathematics does not license

Sorting can detach case IDs. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Keep a rank-to-case map. 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. Lower endpoint

A small example makes the mechanism visible. Compute 1−0.5^(1/n).

A concrete situation

m1≈.12945.

What the mathematics does not license

Using 1/(n+1) changes the test. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

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

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3. Interior ranks

This is where modelling choices begin to affect evidence. Use (i−.3175)/(n+.365).

A concrete situation

m2≈.31361 and m3=.5.

What the mathematics does not license

Off-by-one errors break symmetry. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Check mirrored ranks. 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. Upper endpoint

The central mathematical idea is concrete. Compute 0.5^(1/n).

A concrete situation

m5≈.87055.

What the mathematics does not license

Rounding endpoints changes tail scores. The same reasoning helps students move from an exercise to a real dataset.

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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5. Normal scores

Here the geometry becomes operational. Apply inverse standard-normal CDF.

A concrete situation

About −1.129,−.486,0,.486,1.129.

What the mathematics does not license

Extreme probabilities need stable code. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Verify quantiles. 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. Centre scores

This step deserves a careful pause. Check the score mean.

A concrete situation

Symmetric scores average zero.

What the mathematics does not license

Asymmetry signals coding error. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Audit 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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7. Data deviations

The useful question is not only how to calculate, but what the calculation assumes. Subtract the ordered-data mean.

A concrete situation

For 1–5 the deviations are −2,−1,0,1,2.

What the mathematics does not license

Changing units should not change r. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Rescale once. 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. Score deviations

A small example makes the mechanism visible. Subtract the score mean.

A concrete situation

They equal the symmetric score vector.

What the mathematics does not license

A different plotting convention changes them. That distinction matters because a correct formula can still answer the wrong operational question.

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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9. Cross-products

This is where modelling choices begin to affect evidence. Multiply paired deviations.

A concrete situation

Positive terms support line fit.

What the mathematics does not license

One endpoint may dominate. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Show contributions. 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. Denominator

The central mathematical idea is concrete. Use both sums of squares.

A concrete situation

Correlation is dimensionless.

What the mathematics does not license

Zero sample spread is undefined. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Check variance. 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. PPCC

Here the geometry becomes operational. Divide covariance sum by root products.

A concrete situation

r≈.99835.

What the mathematics does not license

Near one is not proof. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Calibrate by 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.

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12. Critical value

This step deserves a careful pause. Use Filliben tables or faithful simulation.

A concrete situation

Reject for sufficiently small r.

What the mathematics does not license

A generic Pearson table is wrong. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Match statistic. 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. Residual curve

The useful question is not only how to calculate, but what the calculation assumes. Inspect vertical Q–Q departures.

A concrete situation

Symmetric data show tiny residuals here.

What the mathematics does not license

Correlation can hide patterned curvature. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Plot residuals. 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. Ties

A small example makes the mechanism visible. Record repeated measurements.

A concrete situation

Ties create flat steps.

What the mathematics does not license

Jitter invents information. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Model resolution. 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 is where modelling choices begin to affect evidence. Recalculate after verifying a suspect endpoint.

A concrete situation

Only validated corrections belong in data.

What the mathematics does not license

Deleting to improve fit biases inference. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Predefine 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.

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16. Claim

The central mathematical idea is concrete. State probability-plot evidence.

A concrete situation

The ordered pattern is or is not compatible with fitted normality.

What the mathematics does not license

The cause is not identified. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Link to 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

Filliben’s probability-plot correlation coefficient turns a visual normal Q–Q plot into a number. It orders the data, assigns each rank a median plotting position, converts those probabilities to standard-normal quantiles and calculates the ordinary correlation between the two ordered sequences. A nearly straight probability plot gives a correlation near one; systematic curvature or a displaced endpoint lowers it.

For 1,2,3,4,5, Filliben positions are approximately .12945,.31361,.50000,.68639,.87055. Their normal quantiles are about -1.129,-.486,0,.486,1.129. The correlation with the ordered observations is about .99835. This describes strong line fit in a tiny arithmetic example; it is not a universal acceptance rule.

The endpoint formula differs from the interior formula: m1=1-0.5^(1/n), mn=0.5^(1/n), and mi=(i-0.3175)/(n+0.365) inside. Substituting another plotting-position convention changes the quantiles and therefore changes the statistic. The convention is part of the test, not decorative graph formatting.

A high correlation does not prove normality, and a low value does not identify the cause. Rounded measurements, mixtures, censoring, dependence, skewness and tail contamination can all bend a probability plot. Preserve the ordered table and residual pattern so the statistic remains explainable.

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Frequently asked questions

What does the Filliben test measure?

It measures the linear association between ordered observations and Filliben normal-score quantiles.

What are the endpoint plotting positions?

They are 1−0.5^(1/n) and 0.5^(1/n).

What formula is used inside the endpoints?

For ranks 2 through n−1 it uses (i−0.3175)/(n+0.365).

Which direction rejects normality?

Unusually low correlation values reject; critical values depend on n.

Is it identical to Shapiro–Francia?

No. Both use normal scores, but their weights, statistic and calibration differ.

What should be reported?

The ordered table, plotting positions, normal quantiles, correlation, critical-value method and Q–Q residual pattern.

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Primary source and fact-check note

James J. Filliben, “The Probability Plot Correlation Coefficient Test for Normality,” Technometrics 17(1), 111–117 (1975), DOI 10.1080/00401706.1975.10489279. 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

• Shapiro–Francia normal-order correlation test
• Zhang–Wu ordered-CDF tests
• 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 the target observations or residuals, independence assumptions and inclusion rules.

Transformation audit

Reproduce ordering, centring, scaling, plotting positions or polynomial recurrence.

Statistic audit

Show every rank score, moment coordinate, component and coefficient.

Calibration audit

Match sample size, estimated parameters, dependence and rejection direction.

Graph audit

Preserve Q–Q plots, contour maps, component plots and influence displays.

Magnitude audit

Explain practical consequences in original units, not only p-values.

Sensitivity audit

Use only prespecified formula, resolution and influence checks.

Final claim

Keep the conclusion inside the measured population and fitted model.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Data order

Begin with this mechanism: Sort the complete sample. Recreate the concrete case: Use 1,2,3,4,5. Write ordered observations, ranks, plotting positions, normal quantiles, deviations, cross-products, sums of squares, correlations, critical values and Q–Q residuals with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Sorting can detach case IDs. Complete the transfer step: Keep a rank-to-case map. Change exactly one observation, plotting position, quantile convention, moment divisor, polynomial order, transformation coefficient, contour, covariance entry, lag bandwidth, rejection direction 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 twice: first to a student using the visible ordered values or component table, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from rounding, censoring, mixture structure, dependence, skewness or tail contamination. State what extra graph, design record or held-out observation would distinguish the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 2: Lower endpoint

Begin with this mechanism: Compute 1−0.5^(1/n). Recreate the concrete case: m1≈.12945. Write ordered observations, ranks, plotting positions, normal quantiles, deviations, cross-products, sums of squares, correlations, critical values and Q–Q residuals with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Using 1/(n+1) changes the test. Complete the transfer step: Show formula. Change exactly one observation, plotting position, quantile convention, moment divisor, polynomial order, transformation coefficient, contour, covariance entry, lag bandwidth, rejection direction 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 twice: first to a student using the visible ordered values or component table, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from rounding, censoring, mixture structure, dependence, skewness or tail contamination. State what extra graph, design record or held-out observation would distinguish the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 3: Interior ranks

Begin with this mechanism: Use (i−.3175)/(n+.365). Recreate the concrete case: m2≈.31361 and m3=.5. Write ordered observations, ranks, plotting positions, normal quantiles, deviations, cross-products, sums of squares, correlations, critical values and Q–Q residuals with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Off-by-one errors break symmetry. Complete the transfer step: Check mirrored ranks. Change exactly one observation, plotting position, quantile convention, moment divisor, polynomial order, transformation coefficient, contour, covariance entry, lag bandwidth, rejection direction 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 twice: first to a student using the visible ordered values or component table, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from rounding, censoring, mixture structure, dependence, skewness or tail contamination. State what extra graph, design record or held-out observation would distinguish the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 4: Upper endpoint

Begin with this mechanism: Compute 0.5^(1/n). Recreate the concrete case: m5≈.87055. Write ordered observations, ranks, plotting positions, normal quantiles, deviations, cross-products, sums of squares, correlations, critical values and Q–Q residuals with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Rounding endpoints changes tail scores. Complete the transfer step: Keep precision. Change exactly one observation, plotting position, quantile convention, moment divisor, polynomial order, transformation coefficient, contour, covariance entry, lag bandwidth, rejection direction 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 twice: first to a student using the visible ordered values or component table, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from rounding, censoring, mixture structure, dependence, skewness or tail contamination. State what extra graph, design record or held-out observation would distinguish the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 5: Normal scores

Begin with this mechanism: Apply inverse standard-normal CDF. Recreate the concrete case: About −1.129,−.486,0,.486,1.129. Write ordered observations, ranks, plotting positions, normal quantiles, deviations, cross-products, sums of squares, correlations, critical values and Q–Q residuals with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Extreme probabilities need stable code. Complete the transfer step: Verify quantiles. Change exactly one observation, plotting position, quantile convention, moment divisor, polynomial order, transformation coefficient, contour, covariance entry, lag bandwidth, rejection direction 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 twice: first to a student using the visible ordered values or component table, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from rounding, censoring, mixture structure, dependence, skewness or tail contamination. State what extra graph, design record or held-out observation would distinguish the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 6: Centre scores

Begin with this mechanism: Check the score mean. Recreate the concrete case: Symmetric scores average zero. Write ordered observations, ranks, plotting positions, normal quantiles, deviations, cross-products, sums of squares, correlations, critical values and Q–Q residuals with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Asymmetry signals coding error. Complete the transfer step: Audit vector. Change exactly one observation, plotting position, quantile convention, moment divisor, polynomial order, transformation coefficient, contour, covariance entry, lag bandwidth, rejection direction 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 twice: first to a student using the visible ordered values or component table, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from rounding, censoring, mixture structure, dependence, skewness or tail contamination. State what extra graph, design record or held-out observation would distinguish the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 7: Data deviations

Begin with this mechanism: Subtract the ordered-data mean. Recreate the concrete case: For 1–5 the deviations are −2,−1,0,1,2. Write ordered observations, ranks, plotting positions, normal quantiles, deviations, cross-products, sums of squares, correlations, critical values and Q–Q residuals with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Changing units should not change r. Complete the transfer step: Rescale once. Change exactly one observation, plotting position, quantile convention, moment divisor, polynomial order, transformation coefficient, contour, covariance entry, lag bandwidth, rejection direction 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 twice: first to a student using the visible ordered values or component table, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from rounding, censoring, mixture structure, dependence, skewness or tail contamination. State what extra graph, design record or held-out observation would distinguish the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 8: Score deviations

Begin with this mechanism: Subtract the score mean. Recreate the concrete case: They equal the symmetric score vector. Write ordered observations, ranks, plotting positions, normal quantiles, deviations, cross-products, sums of squares, correlations, critical values and Q–Q residuals with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: A different plotting convention changes them. Complete the transfer step: Name convention. Change exactly one observation, plotting position, quantile convention, moment divisor, polynomial order, transformation coefficient, contour, covariance entry, lag bandwidth, rejection direction 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 twice: first to a student using the visible ordered values or component table, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from rounding, censoring, mixture structure, dependence, skewness or tail contamination. State what extra graph, design record or held-out observation would distinguish the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 9: Cross-products

Begin with this mechanism: Multiply paired deviations. Recreate the concrete case: Positive terms support line fit. Write ordered observations, ranks, plotting positions, normal quantiles, deviations, cross-products, sums of squares, correlations, critical values and Q–Q residuals with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: One endpoint may dominate. Complete the transfer step: Show contributions. Change exactly one observation, plotting position, quantile convention, moment divisor, polynomial order, transformation coefficient, contour, covariance entry, lag bandwidth, rejection direction 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 twice: first to a student using the visible ordered values or component table, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from rounding, censoring, mixture structure, dependence, skewness or tail contamination. State what extra graph, design record or held-out observation would distinguish the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 10: Denominator

Begin with this mechanism: Use both sums of squares. Recreate the concrete case: Correlation is dimensionless. Write ordered observations, ranks, plotting positions, normal quantiles, deviations, cross-products, sums of squares, correlations, critical values and Q–Q residuals with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Zero sample spread is undefined. Complete the transfer step: Check variance. Change exactly one observation, plotting position, quantile convention, moment divisor, polynomial order, transformation coefficient, contour, covariance entry, lag bandwidth, rejection direction 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 twice: first to a student using the visible ordered values or component table, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from rounding, censoring, mixture structure, dependence, skewness or tail contamination. State what extra graph, design record or held-out observation would distinguish the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 11: PPCC

Begin with this mechanism: Divide covariance sum by root products. Recreate the concrete case: r≈.99835. Write ordered observations, ranks, plotting positions, normal quantiles, deviations, cross-products, sums of squares, correlations, critical values and Q–Q residuals with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Near one is not proof. Complete the transfer step: Calibrate by n. Change exactly one observation, plotting position, quantile convention, moment divisor, polynomial order, transformation coefficient, contour, covariance entry, lag bandwidth, rejection direction 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 twice: first to a student using the visible ordered values or component table, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from rounding, censoring, mixture structure, dependence, skewness or tail contamination. State what extra graph, design record or held-out observation would distinguish the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 12: Critical value

Begin with this mechanism: Use Filliben tables or faithful simulation. Recreate the concrete case: Reject for sufficiently small r. Write ordered observations, ranks, plotting positions, normal quantiles, deviations, cross-products, sums of squares, correlations, critical values and Q–Q residuals with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: A generic Pearson table is wrong. Complete the transfer step: Match statistic. Change exactly one observation, plotting position, quantile convention, moment divisor, polynomial order, transformation coefficient, contour, covariance entry, lag bandwidth, rejection direction 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 twice: first to a student using the visible ordered values or component table, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from rounding, censoring, mixture structure, dependence, skewness or tail contamination. State what extra graph, design record or held-out observation would distinguish the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 13: Residual curve

Begin with this mechanism: Inspect vertical Q–Q departures. Recreate the concrete case: Symmetric data show tiny residuals here. Write ordered observations, ranks, plotting positions, normal quantiles, deviations, cross-products, sums of squares, correlations, critical values and Q–Q residuals with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Correlation can hide patterned curvature. Complete the transfer step: Plot residuals. Change exactly one observation, plotting position, quantile convention, moment divisor, polynomial order, transformation coefficient, contour, covariance entry, lag bandwidth, rejection direction 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 twice: first to a student using the visible ordered values or component table, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from rounding, censoring, mixture structure, dependence, skewness or tail contamination. State what extra graph, design record or held-out observation would distinguish the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 14: Ties

Begin with this mechanism: Record repeated measurements. Recreate the concrete case: Ties create flat steps. Write ordered observations, ranks, plotting positions, normal quantiles, deviations, cross-products, sums of squares, correlations, critical values and Q–Q residuals with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Jitter invents information. Complete the transfer step: Model resolution. Change exactly one observation, plotting position, quantile convention, moment divisor, polynomial order, transformation coefficient, contour, covariance entry, lag bandwidth, rejection direction 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 twice: first to a student using the visible ordered values or component table, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from rounding, censoring, mixture structure, dependence, skewness or tail contamination. State what extra graph, design record or held-out observation would distinguish the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 15: Sensitivity

Begin with this mechanism: Recalculate after verifying a suspect endpoint. Recreate the concrete case: Only validated corrections belong in data. Write ordered observations, ranks, plotting positions, normal quantiles, deviations, cross-products, sums of squares, correlations, critical values and Q–Q residuals with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Deleting to improve fit biases inference. Complete the transfer step: Predefine checks. Change exactly one observation, plotting position, quantile convention, moment divisor, polynomial order, transformation coefficient, contour, covariance entry, lag bandwidth, rejection direction 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 twice: first to a student using the visible ordered values or component table, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from rounding, censoring, mixture structure, dependence, skewness or tail contamination. State what extra graph, design record or held-out observation would distinguish the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 16: Claim

Begin with this mechanism: State probability-plot evidence. Recreate the concrete case: The ordered pattern is or is not compatible with fitted normality. Write ordered observations, ranks, plotting positions, normal quantiles, deviations, cross-products, sums of squares, correlations, critical values and Q–Q residuals with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: The cause is not identified. Complete the transfer step: Link to context. Change exactly one observation, plotting position, quantile convention, moment divisor, polynomial order, transformation coefficient, contour, covariance entry, lag bandwidth, rejection direction 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 twice: first to a student using the visible ordered values or component table, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from rounding, censoring, mixture structure, dependence, skewness or tail contamination. State what extra graph, design record or held-out observation would distinguish the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.