Chapter map

1. 1. Data matrix
2. 2. Centre
3. 3. Covariance
4. 4. Inverse geometry
5. 5. Inner product
6. 6. Cube
7. 7. Average pairs
8. 8. Skew calibration
9. 9. Radius
10. 10. Square radius
11. 11. Average radii
12. 12. Normal benchmark
13. 13. Separate decisions
14. 14. Dimension
15. 15. Effect display
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 matrix

Here the geometry becomes operational. Arrange n independent rows and p variables.

A concrete situation

Four rows contain two coordinates.

What the mathematics does not license

Mixing units across rows breaks geometry. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Audit 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.

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2. Centre

This step deserves a careful pause. Subtract the p-dimensional sample mean.

A concrete situation

The symmetric cross has mean zero.

What the mathematics does not license

Missing rows move every centre. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Lock inclusion. 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. Covariance

The useful question is not only how to calculate, but what the calculation assumes. Estimate S from centred cross-products.

A concrete situation

The axes have equal spread in the toy case.

What the mathematics does not license

Divisor convention affects finite-sample values. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

State 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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4. Inverse geometry

A small example makes the mechanism visible. Compute S inverse.

A concrete situation

It rescales correlated axes.

What the mathematics does not license

Near singularity magnifies noise. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Inspect condition number. 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. Inner product

This is where modelling choices begin to affect evidence. Form dij for every ordered pair.

A concrete situation

Opposite points have negative inner products.

What the mathematics does not license

Euclidean dot products ignore covariance. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Use Mahalanobis form. 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. Cube

The central mathematical idea is concrete. Raise every dij to the third power.

A concrete situation

Signs are preserved.

What the mathematics does not license

Large products gain strong influence. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Check influential 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.

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7. Average pairs

Here the geometry becomes operational. Sum n squared cubed terms and divide by n squared.

A concrete situation

Symmetry cancels positive and negative directions.

What the mathematics does not license

Cancellation can hide shell shape. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Plot the cloud. 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. Skew calibration

This step deserves a careful pause. Scale b1,p with its finite-sample or asymptotic law.

A concrete situation

Dimension controls degrees of freedom.

What the mathematics does not license

A copied chi-square rule may omit correction. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Record 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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9. Radius

The useful question is not only how to calculate, but what the calculation assumes. Set i=j to obtain squared Mahalanobis distance.

A concrete situation

Each cross point has the same radius.

What the mathematics does not license

Equal radii do not imply normality. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Inspect distribution. 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. Square radius

A small example makes the mechanism visible. Calculate dii squared.

A concrete situation

Large radii receive extra weight.

What the mathematics does not license

One outlier can 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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11. Average radii

This is where modelling choices begin to affect evidence. Compute b2,p.

A concrete situation

The result describes radial tail shape.

What the mathematics does not license

It does not measure directional skew. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Keep both components. 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. Normal benchmark

The central mathematical idea is concrete. Calculate p(p+2).

A concrete situation

For p=2 the benchmark is 8.

What the mathematics does not license

Sample expectation needs care. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Use stated calibration. 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. Separate decisions

Here the geometry becomes operational. Report skewness and kurtosis evidence separately.

A concrete situation

One can reject while the other does not.

What the mathematics does not license

Combining after inspection inflates error. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

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

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14. Dimension

This step deserves a careful pause. Compare n with p.

A concrete situation

High p consumes covariance information.

What the mathematics does not license

Asymptotics can fail. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Use simulation. 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. Effect display

The useful question is not only how to calculate, but what the calculation assumes. Show scatterplots and radius QQ plots.

A concrete situation

The shell is immediately visible.

What the mathematics does not license

Moment p-values lack magnitude. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Add geometry. 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

A small example makes the mechanism visible. Describe the component that departs.

A concrete situation

The cloud may have unusual radial shape.

What the mathematics does not license

The cause is not identified. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Check process. 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

Mardia’s measures extend familiar ideas of skewness and kurtosis to a vector of variables. First standardize the cloud through its sample covariance geometry. Multivariate skewness averages cubed Mahalanobis inner products across all ordered pairs, while multivariate kurtosis averages squared Mahalanobis radii.

Write dij=(xi-xbar) transpose S inverse (xj-xbar). Then b1,p is the average of dij cubed over all n squared ordered pairs. Write dii for each squared Mahalanobis radius; b2,p is the average of dii squared. Under multivariate normality the population kurtosis benchmark is p(p+2), which equals 8 when p=2.

Consider four symmetric two-dimensional points (1,0),(-1,0),(0,1),(0,-1) centred at zero. Symmetry makes positive and negative inner-product cubes cancel, so the skewness measure can be small. Yet all points lie on one radius after standardization, creating a shell rather than a normal cloud. The kurtosis component and a scatterplot reveal what skewness alone can miss.

The separate components are a strength: one can detect asymmetric joint shape, unusual radial tails, both, or neither. They are not a complete description of every non-normal alternative. Sample covariance must be invertible, finite-sample corrections matter, and high dimension relative to sample size makes estimation unstable. Report both statistics with the precise calibration used.

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Frequently asked questions

What is Mardia multivariate skewness?

It is the average cubed Mahalanobis inner product over all ordered pairs of observations.

What is Mardia multivariate kurtosis?

It is the average squared Mahalanobis radius.

What is the normal kurtosis benchmark?

For dimension p, the population benchmark is p(p+2).

Why are there two components?

Joint asymmetry and radial tail shape are different departures.

What if covariance is singular?

The ordinary inverse-covariance definitions are not available without changing the problem.

Does passing both tests prove joint normality?

No. Other non-normal shapes can escape these two moment summaries.

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Primary source and fact-check note

K. V. Mardia, “Measures of Multivariate Skewness and Kurtosis with Applications,” Biometrika 57(3), 519–530 (1970), DOI 10.1093/biomet/57.3.519. 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

• Henze–Zirkler multivariate normality test
• Royston multivariate normality test
• 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 independent multivariate rows and inclusion rules.

Geometry audit

Reproduce centring, covariance and transformations.

Statistic audit

Show every score, distance or component contribution.

Calibration audit

Match sample size, dimension and dependence treatment.

Graph audit

Preserve probability plots, scatterplots and radius diagnostics.

Magnitude audit

Explain the practical consequence in original units.

Sensitivity audit

Use prespecified alternatives and influence checks.

Final claim

Keep inference inside the model and sampling design.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Data matrix

Begin with this mechanism: Arrange n independent rows and p variables. Recreate the concrete case: Four rows contain two coordinates. Write rows, dimensions, centred vectors, covariance matrices, inverse geometry, ordered-pair inner products, cubed terms, radii, squared radii and component statistics with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Mixing units across rows breaks geometry. Complete the transfer step: Audit IDs. Change exactly one observation, row, coordinate, tie, plotting position, normal score, covariance entry, distance, bandwidth, component statistic, dimension, dependence estimate, 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 plot 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, clusters, 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: Centre

Begin with this mechanism: Subtract the p-dimensional sample mean. Recreate the concrete case: The symmetric cross has mean zero. Write rows, dimensions, centred vectors, covariance matrices, inverse geometry, ordered-pair inner products, cubed terms, radii, squared radii and component statistics with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Missing rows move every centre. Complete the transfer step: Lock inclusion. Change exactly one observation, row, coordinate, tie, plotting position, normal score, covariance entry, distance, bandwidth, component statistic, dimension, dependence estimate, 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 plot 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, clusters, 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: Covariance

Begin with this mechanism: Estimate S from centred cross-products. Recreate the concrete case: The axes have equal spread in the toy case. Write rows, dimensions, centred vectors, covariance matrices, inverse geometry, ordered-pair inner products, cubed terms, radii, squared radii and component statistics with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Divisor convention affects finite-sample values. Complete the transfer step: State convention. Change exactly one observation, row, coordinate, tie, plotting position, normal score, covariance entry, distance, bandwidth, component statistic, dimension, dependence estimate, 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 plot 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, clusters, 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: Inverse geometry

Begin with this mechanism: Compute S inverse. Recreate the concrete case: It rescales correlated axes. Write rows, dimensions, centred vectors, covariance matrices, inverse geometry, ordered-pair inner products, cubed terms, radii, squared radii and component statistics with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Near singularity magnifies noise. Complete the transfer step: Inspect condition number. Change exactly one observation, row, coordinate, tie, plotting position, normal score, covariance entry, distance, bandwidth, component statistic, dimension, dependence estimate, 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 plot 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, clusters, 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 5: Inner product

Begin with this mechanism: Form dij for every ordered pair. Recreate the concrete case: Opposite points have negative inner products. Write rows, dimensions, centred vectors, covariance matrices, inverse geometry, ordered-pair inner products, cubed terms, radii, squared radii and component statistics with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Euclidean dot products ignore covariance. Complete the transfer step: Use Mahalanobis form. Change exactly one observation, row, coordinate, tie, plotting position, normal score, covariance entry, distance, bandwidth, component statistic, dimension, dependence estimate, 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 plot 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, clusters, 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: Cube

Begin with this mechanism: Raise every dij to the third power. Recreate the concrete case: Signs are preserved. Write rows, dimensions, centred vectors, covariance matrices, inverse geometry, ordered-pair inner products, cubed terms, radii, squared radii and component statistics with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Large products gain strong influence. Complete the transfer step: Check influential pairs. Change exactly one observation, row, coordinate, tie, plotting position, normal score, covariance entry, distance, bandwidth, component statistic, dimension, dependence estimate, 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 plot 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, clusters, 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: Average pairs

Begin with this mechanism: Sum n squared cubed terms and divide by n squared. Recreate the concrete case: Symmetry cancels positive and negative directions. Write rows, dimensions, centred vectors, covariance matrices, inverse geometry, ordered-pair inner products, cubed terms, radii, squared radii and component statistics with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Cancellation can hide shell shape. Complete the transfer step: Plot the cloud. Change exactly one observation, row, coordinate, tie, plotting position, normal score, covariance entry, distance, bandwidth, component statistic, dimension, dependence estimate, 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 plot 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, clusters, 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: Skew calibration

Begin with this mechanism: Scale b1,p with its finite-sample or asymptotic law. Recreate the concrete case: Dimension controls degrees of freedom. Write rows, dimensions, centred vectors, covariance matrices, inverse geometry, ordered-pair inner products, cubed terms, radii, squared radii and component statistics with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: A copied chi-square rule may omit correction. Complete the transfer step: Record formula. Change exactly one observation, row, coordinate, tie, plotting position, normal score, covariance entry, distance, bandwidth, component statistic, dimension, dependence estimate, 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 plot 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, clusters, 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 9: Radius

Begin with this mechanism: Set i=j to obtain squared Mahalanobis distance. Recreate the concrete case: Each cross point has the same radius. Write rows, dimensions, centred vectors, covariance matrices, inverse geometry, ordered-pair inner products, cubed terms, radii, squared radii and component statistics with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Equal radii do not imply normality. Complete the transfer step: Inspect distribution. Change exactly one observation, row, coordinate, tie, plotting position, normal score, covariance entry, distance, bandwidth, component statistic, dimension, dependence estimate, 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 plot 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, clusters, 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: Square radius

Begin with this mechanism: Calculate dii squared. Recreate the concrete case: Large radii receive extra weight. Write rows, dimensions, centred vectors, covariance matrices, inverse geometry, ordered-pair inner products, cubed terms, radii, squared radii and component statistics with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: One outlier can dominate. Complete the transfer step: Show contributions. Change exactly one observation, row, coordinate, tie, plotting position, normal score, covariance entry, distance, bandwidth, component statistic, dimension, dependence estimate, 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 plot 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, clusters, 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: Average radii

Begin with this mechanism: Compute b2,p. Recreate the concrete case: The result describes radial tail shape. Write rows, dimensions, centred vectors, covariance matrices, inverse geometry, ordered-pair inner products, cubed terms, radii, squared radii and component statistics with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: It does not measure directional skew. Complete the transfer step: Keep both components. Change exactly one observation, row, coordinate, tie, plotting position, normal score, covariance entry, distance, bandwidth, component statistic, dimension, dependence estimate, 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 plot 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, clusters, 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: Normal benchmark

Begin with this mechanism: Calculate p(p+2). Recreate the concrete case: For p=2 the benchmark is 8. Write rows, dimensions, centred vectors, covariance matrices, inverse geometry, ordered-pair inner products, cubed terms, radii, squared radii and component statistics with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Sample expectation needs care. Complete the transfer step: Use stated calibration. Change exactly one observation, row, coordinate, tie, plotting position, normal score, covariance entry, distance, bandwidth, component statistic, dimension, dependence estimate, 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 plot 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, clusters, 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 13: Separate decisions

Begin with this mechanism: Report skewness and kurtosis evidence separately. Recreate the concrete case: One can reject while the other does not. Write rows, dimensions, centred vectors, covariance matrices, inverse geometry, ordered-pair inner products, cubed terms, radii, squared radii and component statistics with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Combining after inspection inflates error. Complete the transfer step: Preplan rule. Change exactly one observation, row, coordinate, tie, plotting position, normal score, covariance entry, distance, bandwidth, component statistic, dimension, dependence estimate, 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 plot 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, clusters, 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: Dimension

Begin with this mechanism: Compare n with p. Recreate the concrete case: High p consumes covariance information. Write rows, dimensions, centred vectors, covariance matrices, inverse geometry, ordered-pair inner products, cubed terms, radii, squared radii and component statistics with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Asymptotics can fail. Complete the transfer step: Use simulation. Change exactly one observation, row, coordinate, tie, plotting position, normal score, covariance entry, distance, bandwidth, component statistic, dimension, dependence estimate, 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 plot 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, clusters, 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: Effect display

Begin with this mechanism: Show scatterplots and radius QQ plots. Recreate the concrete case: The shell is immediately visible. Write rows, dimensions, centred vectors, covariance matrices, inverse geometry, ordered-pair inner products, cubed terms, radii, squared radii and component statistics with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Moment p-values lack magnitude. Complete the transfer step: Add geometry. Change exactly one observation, row, coordinate, tie, plotting position, normal score, covariance entry, distance, bandwidth, component statistic, dimension, dependence estimate, 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 plot 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, clusters, 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: Describe the component that departs. Recreate the concrete case: The cloud may have unusual radial shape. Write rows, dimensions, centred vectors, covariance matrices, inverse geometry, ordered-pair inner products, cubed terms, radii, squared radii and component statistics 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: Check process. Change exactly one observation, row, coordinate, tie, plotting position, normal score, covariance entry, distance, bandwidth, component statistic, dimension, dependence estimate, 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 plot 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, clusters, 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.