WHY MATHEMATICS?

Why Mathematics? | Doornik–Hansen Omnibus Test: Decorrelation, Transformed Skewness, Kurtosis and Multivariate Normality

How correlated variables are rotated into comparable directions, why moment components need finite-sample transformations, and how their contributions become one joint diagnostic.

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

Estimate covariance and decorrelate the variable space
Calculate skewness and kurtosis by transformed component
Apply the published finite-sample normalizations
Sum squared scores and inspect the contributors and geometry

Chapter map

1. Rows
2. Dimension
3. Centre
4. Covariance
5. Eigenvalues
6. Rotation
7. Rescaling
8. Component skew
9. Component kurtosis
10. Finite-n transform
11. Kurtosis transform
12. Sum squares
13. Reference law
14. Diagnostics
15. Geometry
16. Claim

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

1. Rows

This step deserves a careful pause. Define n independent vector observations.

A concrete situation

Each student supplies X and Y once.

What the mathematics does not license

Repeated rows reduce effective n. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

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

2. Dimension

The useful question is not only how to calculate, but what the calculation assumes. Count p variables.

A concrete situation

Here p=2.

What the mathematics does not license

Large p relative to n destabilizes covariance. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

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

3. Centre

A small example makes the mechanism visible. Subtract each coordinate mean.

A concrete situation

The cloud moves to the origin.

What the mathematics does not license

Missingness changes all means. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Freeze cohort. 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. Covariance

This is where modelling choices begin to affect evidence. Calculate the p by p matrix.

A concrete situation

Use [[4,3],[3,4]] illustratively.

What the mathematics does not license

Units and scaling shape the matrix. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Show 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.

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. Eigenvalues

The central mathematical idea is concrete. Find variances along principal directions.

A concrete situation

They are 7 and 1.

What the mathematics does not license

A near-zero value signals collinearity. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Inspect condition. 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. Rotation

Here the geometry becomes operational. Use orthogonal eigenvectors.

A concrete situation

Directions are X+Y and X-Y.

What the mathematics does not license

Sign choices do not change evidence. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Verify reconstruction. 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. Rescaling

This step deserves a careful pause. Divide by square roots of direction variances.

A concrete situation

Use sqrt14 and sqrt2 with the chosen normalized vectors.

What the mathematics does not license

A missing sqrt2 mis-scales components. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Check covariance. 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. Component skew

The useful question is not only how to calculate, but what the calculation assumes. Calculate skewness in each transformed direction.

A concrete situation

Two component values are retained.

What the mathematics does not license

High powers amplify unusual rows. Evidence becomes stronger when the limitation is tested instead of hidden.

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.

9. Component kurtosis

A small example makes the mechanism visible. Calculate kurtosis in each direction.

A concrete situation

Two more component values are retained.

What the mathematics does not license

Kurtosis is not just peakedness. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Use tail-and-centre language. 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. Finite-n transform

This is where modelling choices begin to affect evidence. Apply the published skewness normalization.

A concrete situation

Obtain approximately standard-normal scores.

What the mathematics does not license

Raw moments are not comparable Z values. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Use validated code. 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. Kurtosis transform

The central mathematical idea is concrete. Apply the corresponding kurtosis normalization.

A concrete situation

Keep each score visible.

What the mathematics does not license

A shortcut may mis-size small samples. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Record algorithm. 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. Sum squares

Here the geometry becomes operational. Add all four squared scores.

A concrete situation

Illustrative contributions sum to 4.34.

What the mathematics does not license

Illustrative scores are not fitted data. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Label clearly. 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. Reference law

This step deserves a careful pause. Compare with chi-square on 2p df.

A concrete situation

For p=2, df=4.

What the mathematics does not license

Approximation can be imperfect. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Simulate under fitted 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.

14. Diagnostics

The useful question is not only how to calculate, but what the calculation assumes. Map large components back to observations.

A concrete situation

A contrast direction may drive rejection.

What the mathematics does not license

Post-hoc deletion invalidates inference. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Review 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.

15. Geometry

A small example makes the mechanism visible. Add scatterplots and Mahalanobis radii.

A concrete situation

Normal-looking margins can hide curved dependence.

What the mathematics does not license

Moments do not capture every shape. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

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

16. Claim

This is where modelling choices begin to affect evidence. State joint moment evidence.

A concrete situation

The transformed cloud conflicts with normality.

What the mathematics does not license

No causal claim follows. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Connect to analysis. 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 Doornik–Hansen omnibus test builds a multivariate normality check from transformed skewness and kurtosis components. It first decorrelates the variables, applies finite-sample transformations designed to improve the normal approximation of each component statistic, and adds their squared contributions. With p variables, the reference is approximately chi-square with 2p degrees of freedom.

Suppose two standardized measurements have covariance matrix [[4,3],[3,4]]. Its principal directions have variances 7 and 1. Rotating and rescaling gives components proportional to (X+Y)/sqrt(14) and (X-Y)/sqrt(2), each with variance one. The first represents shared movement; the second represents contrast. Skewness and kurtosis are then assessed on these decorrelated directions.

If the transformed scores are, illustratively, 1.2 and -0.4 for skewness and 0.7 and 1.5 for kurtosis, the omnibus sum is 1.44+0.16+0.49+2.25=4.34. The arithmetic illustrates the sum of squares only; real scores must be produced by the published transformations, and the reference degrees of freedom are four when p=2.

Decorrelation does not make a small sample large, and an unstable covariance matrix can distort every transformed component. A rejection identifies joint incompatibility with multivariate normality, not a cause. Show component contributions, scatterplots, marginal probability plots and Mahalanobis-distance diagnostics before changing a model.

Frequently asked questions

What does the Doornik–Hansen test combine?

It combines transformed skewness and kurtosis contributions after decorrelating the variables.

Why decorrelate first?

Correlated coordinates contain overlapping information and depend on the chosen axes.

What is the approximate degrees of freedom?

For p variables, the omnibus reference has 2p degrees of freedom.

Is a rotation a deletion of information?

No. An invertible rotation and rescaling re-express the same full-rank data geometry.

What if covariance is nearly singular?

Component estimates become unstable and the ordinary procedure needs caution.

Does rejection reveal a cause?

No. Report component scores and graphs to locate asymmetry, tail shape, clusters or outliers.

Primary source and fact-check note

Jurgen A. Doornik and Henrik Hansen, “An Omnibus Test for Univariate and Multivariate Normality,” Oxford Bulletin of Economics and Statistics 70(s1), 927–939 (2008), DOI 10.1111/j.1468-0084.2008.00537.x. Definitions and core claims in this article were checked against this primary publication. Calculations shown here were independently recomputed. Examples are educational illustrations; they are not reported industrial performance results.

Useful next reading

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

Extended laboratory: topic-specific validation and responsible use

Design audit

Freeze independent units, target variables and inclusion rules.

Transformation audit

Reproduce centring, scaling, ordering or decorrelation.

Statistic audit

Show every moment, spacing, function value or component score.

Calibration audit

Match sample size, dimension, fitted parameters and window or weighting rule.

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 probability plots, ordered spacings and multivariate geometry.

Magnitude audit

Explain the practical consequence in original units.

Sensitivity audit

Use only prespecified influence and approximation checks.

Final claim

Keep inference inside the model and sampling design.

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

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Rows

Begin with this mechanism: Define n independent vector observations. Recreate the concrete case: Each student supplies X and Y once. Write rows, dimensions, means, covariance matrices, eigenvalues, eigenvectors, decorrelated components, skewness scores, kurtosis scores, squared contributions and calibrated omnibus statistics with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Repeated rows reduce effective n. Complete the transfer step: Preserve IDs. Change exactly one observation, row, tie, order statistic, window, plotting convention, frequency, weight, covariance entry, eigenvalue, component score, dimension, null approximation or simulation 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 moments, spacings, frequencies or geometry, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from skewness, heavy tails, mixtures, rounding, censoring or dependence. State what extra graph, design record or held-out observation would distinguish the explanations. Finish by writing one sentence the data support, one sentence they do not support and one useful next measurement.

Workshop 2: Dimension

Begin with this mechanism: Count p variables. Recreate the concrete case: Here p=2. Write rows, dimensions, means, covariance matrices, eigenvalues, eigenvectors, decorrelated components, skewness scores, kurtosis scores, squared contributions and calibrated omnibus statistics with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Large p relative to n destabilizes covariance. Complete the transfer step: Report both. Change exactly one observation, row, tie, order statistic, window, plotting convention, frequency, weight, covariance entry, eigenvalue, component score, dimension, null approximation or simulation 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 moments, spacings, frequencies or geometry, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from skewness, heavy tails, mixtures, rounding, censoring or dependence. State what extra graph, design record or held-out observation would distinguish the explanations. Finish by writing one sentence the data support, one sentence they do not support and one useful next measurement.

Workshop 3: Centre

Begin with this mechanism: Subtract each coordinate mean. Recreate the concrete case: The cloud moves to the origin. Write rows, dimensions, means, covariance matrices, eigenvalues, eigenvectors, decorrelated components, skewness scores, kurtosis scores, squared contributions and calibrated omnibus statistics with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Missingness changes all means. Complete the transfer step: Freeze cohort. Change exactly one observation, row, tie, order statistic, window, plotting convention, frequency, weight, covariance entry, eigenvalue, component score, dimension, null approximation or simulation 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 moments, spacings, frequencies or geometry, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from skewness, heavy tails, mixtures, rounding, censoring or dependence. State what extra graph, design record or held-out observation would distinguish the explanations. Finish by writing one sentence the data support, one sentence they do not support and one useful next measurement.

Workshop 4: Covariance

Begin with this mechanism: Calculate the p by p matrix. Recreate the concrete case: Use [[4,3],[3,4]] illustratively. Write rows, dimensions, means, covariance matrices, eigenvalues, eigenvectors, decorrelated components, skewness scores, kurtosis scores, squared contributions and calibrated omnibus statistics with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Units and scaling shape the matrix. Complete the transfer step: Show units. Change exactly one observation, row, tie, order statistic, window, plotting convention, frequency, weight, covariance entry, eigenvalue, component score, dimension, null approximation or simulation 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 moments, spacings, frequencies or geometry, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from skewness, heavy tails, mixtures, rounding, censoring or dependence. State what extra graph, design record or held-out observation would distinguish the explanations. Finish by writing one sentence the data support, one sentence they do not support and one useful next measurement.

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

Workshop 5: Eigenvalues

Begin with this mechanism: Find variances along principal directions. Recreate the concrete case: They are 7 and 1. Write rows, dimensions, means, covariance matrices, eigenvalues, eigenvectors, decorrelated components, skewness scores, kurtosis scores, squared contributions and calibrated omnibus statistics with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: A near-zero value signals collinearity. Complete the transfer step: Inspect condition. Change exactly one observation, row, tie, order statistic, window, plotting convention, frequency, weight, covariance entry, eigenvalue, component score, dimension, null approximation or simulation 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 moments, spacings, frequencies or geometry, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from skewness, heavy tails, mixtures, rounding, censoring or dependence. State what extra graph, design record or held-out observation would distinguish the explanations. Finish by writing one sentence the data support, one sentence they do not support and one useful next measurement.

Workshop 6: Rotation

Begin with this mechanism: Use orthogonal eigenvectors. Recreate the concrete case: Directions are X+Y and X-Y. Write rows, dimensions, means, covariance matrices, eigenvalues, eigenvectors, decorrelated components, skewness scores, kurtosis scores, squared contributions and calibrated omnibus statistics with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Sign choices do not change evidence. Complete the transfer step: Verify reconstruction. Change exactly one observation, row, tie, order statistic, window, plotting convention, frequency, weight, covariance entry, eigenvalue, component score, dimension, null approximation or simulation 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 moments, spacings, frequencies or geometry, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from skewness, heavy tails, mixtures, rounding, censoring or dependence. State what extra graph, design record or held-out observation would distinguish the explanations. Finish by writing one sentence the data support, one sentence they do not support and one useful next measurement.

Workshop 7: Rescaling

Begin with this mechanism: Divide by square roots of direction variances. Recreate the concrete case: Use sqrt14 and sqrt2 with the chosen normalized vectors. Write rows, dimensions, means, covariance matrices, eigenvalues, eigenvectors, decorrelated components, skewness scores, kurtosis scores, squared contributions and calibrated omnibus statistics with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: A missing sqrt2 mis-scales components. Complete the transfer step: Check covariance. Change exactly one observation, row, tie, order statistic, window, plotting convention, frequency, weight, covariance entry, eigenvalue, component score, dimension, null approximation or simulation 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 moments, spacings, frequencies or geometry, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from skewness, heavy tails, mixtures, rounding, censoring or dependence. State what extra graph, design record or held-out observation would distinguish the explanations. Finish by writing one sentence the data support, one sentence they do not support and one useful next measurement.

Workshop 8: Component skew

Begin with this mechanism: Calculate skewness in each transformed direction. Recreate the concrete case: Two component values are retained. Write rows, dimensions, means, covariance matrices, eigenvalues, eigenvectors, decorrelated components, skewness scores, kurtosis scores, squared contributions and calibrated omnibus statistics with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: High powers amplify unusual rows. Complete the transfer step: Show contributions. Change exactly one observation, row, tie, order statistic, window, plotting convention, frequency, weight, covariance entry, eigenvalue, component score, dimension, null approximation or simulation 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 moments, spacings, frequencies or geometry, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from skewness, heavy tails, mixtures, rounding, censoring or dependence. State what extra graph, design record or held-out observation would distinguish the explanations. Finish by writing one sentence the data support, one sentence they do not support and one useful next measurement.

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

Workshop 9: Component kurtosis

Begin with this mechanism: Calculate kurtosis in each direction. Recreate the concrete case: Two more component values are retained. Write rows, dimensions, means, covariance matrices, eigenvalues, eigenvectors, decorrelated components, skewness scores, kurtosis scores, squared contributions and calibrated omnibus statistics with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Kurtosis is not just peakedness. Complete the transfer step: Use tail-and-centre language. Change exactly one observation, row, tie, order statistic, window, plotting convention, frequency, weight, covariance entry, eigenvalue, component score, dimension, null approximation or simulation 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 moments, spacings, frequencies or geometry, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from skewness, heavy tails, mixtures, rounding, censoring or dependence. State what extra graph, design record or held-out observation would distinguish the explanations. Finish by writing one sentence the data support, one sentence they do not support and one useful next measurement.

Workshop 10: Finite-n transform

Begin with this mechanism: Apply the published skewness normalization. Recreate the concrete case: Obtain approximately standard-normal scores. Write rows, dimensions, means, covariance matrices, eigenvalues, eigenvectors, decorrelated components, skewness scores, kurtosis scores, squared contributions and calibrated omnibus statistics with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Raw moments are not comparable Z values. Complete the transfer step: Use validated code. Change exactly one observation, row, tie, order statistic, window, plotting convention, frequency, weight, covariance entry, eigenvalue, component score, dimension, null approximation or simulation 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 moments, spacings, frequencies or geometry, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from skewness, heavy tails, mixtures, rounding, censoring or dependence. State what extra graph, design record or held-out observation would distinguish the explanations. Finish by writing one sentence the data support, one sentence they do not support and one useful next measurement.

Workshop 11: Kurtosis transform

Begin with this mechanism: Apply the corresponding kurtosis normalization. Recreate the concrete case: Keep each score visible. Write rows, dimensions, means, covariance matrices, eigenvalues, eigenvectors, decorrelated components, skewness scores, kurtosis scores, squared contributions and calibrated omnibus statistics with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: A shortcut may mis-size small samples. Complete the transfer step: Record algorithm. Change exactly one observation, row, tie, order statistic, window, plotting convention, frequency, weight, covariance entry, eigenvalue, component score, dimension, null approximation or simulation 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 moments, spacings, frequencies or geometry, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from skewness, heavy tails, mixtures, rounding, censoring or dependence. State what extra graph, design record or held-out observation would distinguish the explanations. Finish by writing one sentence the data support, one sentence they do not support and one useful next measurement.

Workshop 12: Sum squares

Begin with this mechanism: Add all four squared scores. Recreate the concrete case: Illustrative contributions sum to 4.34. Write rows, dimensions, means, covariance matrices, eigenvalues, eigenvectors, decorrelated components, skewness scores, kurtosis scores, squared contributions and calibrated omnibus statistics with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Illustrative scores are not fitted data. Complete the transfer step: Label clearly. Change exactly one observation, row, tie, order statistic, window, plotting convention, frequency, weight, covariance entry, eigenvalue, component score, dimension, null approximation or simulation 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 moments, spacings, frequencies or geometry, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from skewness, heavy tails, mixtures, rounding, censoring or dependence. State what extra graph, design record or held-out observation would distinguish the explanations. Finish by writing one sentence the data support, one sentence they do not support and one useful next measurement.

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

Workshop 13: Reference law

Begin with this mechanism: Compare with chi-square on 2p df. Recreate the concrete case: For p=2, df=4. Write rows, dimensions, means, covariance matrices, eigenvalues, eigenvectors, decorrelated components, skewness scores, kurtosis scores, squared contributions and calibrated omnibus statistics with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Approximation can be imperfect. Complete the transfer step: Simulate under fitted null. Change exactly one observation, row, tie, order statistic, window, plotting convention, frequency, weight, covariance entry, eigenvalue, component score, dimension, null approximation or simulation 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 moments, spacings, frequencies or geometry, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from skewness, heavy tails, mixtures, rounding, censoring or dependence. State what extra graph, design record or held-out observation would distinguish the explanations. Finish by writing one sentence the data support, one sentence they do not support and one useful next measurement.

Workshop 14: Diagnostics

Begin with this mechanism: Map large components back to observations. Recreate the concrete case: A contrast direction may drive rejection. Write rows, dimensions, means, covariance matrices, eigenvalues, eigenvectors, decorrelated components, skewness scores, kurtosis scores, squared contributions and calibrated omnibus statistics with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Post-hoc deletion invalidates inference. Complete the transfer step: Review provenance. Change exactly one observation, row, tie, order statistic, window, plotting convention, frequency, weight, covariance entry, eigenvalue, component score, dimension, null approximation or simulation 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 moments, spacings, frequencies or geometry, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from skewness, heavy tails, mixtures, rounding, censoring or dependence. State what extra graph, design record or held-out observation would distinguish the explanations. Finish by writing one sentence the data support, one sentence they do not support and one useful next measurement.

Workshop 15: Geometry

Begin with this mechanism: Add scatterplots and Mahalanobis radii. Recreate the concrete case: Normal-looking margins can hide curved dependence. Write rows, dimensions, means, covariance matrices, eigenvalues, eigenvectors, decorrelated components, skewness scores, kurtosis scores, squared contributions and calibrated omnibus statistics with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Moments do not capture every shape. Complete the transfer step: Show pairs. Change exactly one observation, row, tie, order statistic, window, plotting convention, frequency, weight, covariance entry, eigenvalue, component score, dimension, null approximation or simulation 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 moments, spacings, frequencies or geometry, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from skewness, heavy tails, mixtures, rounding, censoring or dependence. State what extra graph, design record or held-out observation would distinguish the explanations. Finish by writing one sentence the data support, one sentence they do not support and one useful next measurement.

Workshop 16: Claim

Begin with this mechanism: State joint moment evidence. Recreate the concrete case: The transformed cloud conflicts with normality. Write rows, dimensions, means, covariance matrices, eigenvalues, eigenvectors, decorrelated components, skewness scores, kurtosis scores, squared contributions and calibrated omnibus statistics with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: No causal claim follows. Complete the transfer step: Connect to analysis. Change exactly one observation, row, tie, order statistic, window, plotting convention, frequency, weight, covariance entry, eigenvalue, component score, dimension, null approximation or simulation 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 moments, spacings, frequencies or geometry, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from skewness, heavy tails, mixtures, rounding, censoring or dependence. State what extra graph, design record or held-out observation would distinguish the explanations. Finish by writing one sentence the data support, one sentence they do not support and one useful next measurement.