Chapter map

1. 1. Positive class
2. 2. Counts
3. 3. Precision
4. 4. Recall
5. 5. Harmonic mean
6. 6. Count form
7. 7. Beta two
8. 8. Beta half
9. 9. True negatives
10. 10. Threshold
11. 11. Prevalence
12. 12. Macro average
13. 13. Micro average
14. 14. Weighted average
15. 15. Uncertainty
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. Positive class

The useful question is not only how to calculate, but what the calculation assumes. Define which outcome is positive.

A concrete situation

Hold it fixed across comparisons.

What the mathematics does not license

F1 is class-directional. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Name class. 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. Counts

A small example makes the mechanism visible. Use TP=18, FP=6, FN=12.

A concrete situation

Keep TN available even though F1 omits it.

What the mathematics does not license

The score is not the whole matrix. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Show counts. 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. Precision

This is where modelling choices begin to affect evidence. Compute 18/24.

A concrete situation

Precision=.75.

What the mathematics does not license

Undefined if no cases are predicted positive. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

State 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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4. Recall

The central mathematical idea is concrete. Compute 18/30.

A concrete situation

Recall=.60.

What the mathematics does not license

Undefined if no actual positives exist. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

State support. 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. Harmonic mean

Here the geometry becomes operational. Compute 2PR/(P+R).

A concrete situation

F1=.6667.

What the mathematics does not license

The harmonic mean is pulled toward the smaller component. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Show arithmetic. 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. Count form

This step deserves a careful pause. Compute 2TP/(2TP+FP+FN).

A concrete situation

36/54=.6667.

What the mathematics does not license

Equivalent forms require consistent counts. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Cross-check. 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. Beta two

The useful question is not only how to calculate, but what the calculation assumes. Set beta=2.

A concrete situation

Use F2=90/144=.625.

What the mathematics does not license

Greater recall emphasis does not guarantee a larger number. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Interpret inputs. 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. Beta half

A small example makes the mechanism visible. Set beta=.5.

A concrete situation

Use F0.5=22.5/31.5=.7143.

What the mathematics does not license

Precision emphasis suits only some decisions. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Justify beta. 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. True negatives

This is where modelling choices begin to affect evidence. Observe TN does not enter binary F beta.

A concrete situation

Keep TN in the report.

What the mathematics does not license

A model can change on negatives without the same F score response. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

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

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10. Threshold

The central mathematical idea is concrete. Recompute counts across score cutoffs.

A concrete situation

F1 varies with threshold.

What the mathematics does not license

Selecting and testing on the same data is optimistic. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Separate sets. 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. Prevalence

Here the geometry becomes operational. Change the positive rate cautiously.

A concrete situation

Precision and F1 can move.

What the mathematics does not license

Cross-population comparisons need case-mix context. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Report rate. 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. Macro average

This step deserves a careful pause. Average classwise scores equally.

A concrete situation

Each class gets equal weight.

What the mathematics does not license

Rare classes can dominate interpretation. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Name average. 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. Micro average

The useful question is not only how to calculate, but what the calculation assumes. Pool counts before computing.

A concrete situation

Large classes contribute more observations.

What the mathematics does not license

It answers a different question. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Do not mix. 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. Weighted average

A small example makes the mechanism visible. Weight class scores by support.

A concrete situation

Record supports.

What the mathematics does not license

A weighted summary can hide weak minorities. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Show per class. 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. Uncertainty

This is where modelling choices begin to affect evidence. Resample matched predictions and labels.

A concrete situation

Keep threshold selection outside final evaluation.

What the mathematics does not license

One decimal hides instability. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Report interval. 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. Name beta, class, average and threshold.

A concrete situation

Teaching F1=.6667.

What the mathematics does not license

Do not call F1 utility or calibration. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

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

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A seven-day practice lab

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

Questions for a student, parent or mentor

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

A method-specific audit before deployment

The F1 score combines precision and recall through their harmonic mean. It focuses on the positive class and uses true positives, false positives and false negatives; true negatives do not enter its binary formula. F-beta generalises the score so the analyst can place more emphasis on recall or precision, but beta must be chosen for a reason rather than tuned to make a result look favourable.

Use a fixed threshold that gives eighteen true positives, six false positives and twelve false negatives. Precision is 18/(18+6)=.75 and recall is 18/(18+12)=.60. F1 is 2(.75)(.60)/(.75+.60)=.6667, equivalently 2(18)/(36+6+12)=36/54=.6667.

For beta=2, the count formula gives F2=5(18)/(5(18)+6+4(12))=90/144=.625. For beta=.5, F0.5=1.25(18)/(1.25(18)+6+.25(12))=22.5/31.5=.7143. The different results reflect the unequal precision and recall in this example, not a general ranking of the measures.

F scores depend on the positive-class definition, threshold, prevalence and averaging rule. Macro, micro and support-weighted multiclass summaries answer different questions. A responsible analysis reports the full confusion matrix, precision and recall separately, the beta and averaging convention, zero-division handling, threshold selection and uncertainty.

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Frequently asked questions

What is F1?

The harmonic mean of precision and recall for a specified positive class.

Why use a harmonic mean?

It is pulled toward the smaller component and requires both precision and recall to be strong.

What does beta do?

Beta above one places more emphasis on recall; beta below one places more emphasis on precision.

Does F1 use true negatives?

Not in its standard binary formula.

Why state the averaging rule?

Macro, micro and weighted multiclass summaries answer different questions.

What must be reported?

Positive class, counts, precision, recall, beta, averaging, threshold, zero-division rule and uncertainty.

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Primary source and fact-check note

scikit-learn 1.9 documentation, “fbeta_score,” current formula, averaging options and zero-division behaviour; with the information-retrieval lineage reviewed by Hand and Christen (ACM Computing Surveys, 2023). 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

• Precision-recall curves
• Confusion matrices
• ROC AUC and thresholds
• How Mathematics Works: Statistics
• Mathematics World at Bukit Timah Tutor

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: classification evidence and responsible use

Target audit

Freeze actual-label definition, positive class, prediction threshold, rater roles and evaluation population before calculation.

Table audit

Write the complete cross-tabulation with row totals, column totals, N and every denominator before using a summary.

Formula audit

Show every product, sum, harmonic mean, marginal probability, square root, rounding step and edge convention.

Threshold audit

Choose any score cutoff from the intended decision and keep threshold selection separate from final evaluation.

Case-mix audit

Report prevalence, class support, rater marginals, missing cases, weights, clusters and repeated observations.

Comparison audit

Use paired predictions on the same cases and show classwise metrics before ranking models with one scalar.

Uncertainty audit

Use confidence intervals or resampling that respects sampling design, clustering and threshold selection.

Final claim

State the exact count, rate, association or agreement evidence and stop before causal, universal or guaranteed-performance claims.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Positive class

Begin with this mechanism: Define which outcome is positive. Recreate the concrete case: Hold it fixed across comparisons. Write TP, FP, FN, TN, precision, recall, beta, averaging rule, threshold, zero-division rule and uncertainty with identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: F1 is class-directional. Complete the transfer step: Name class. Change exactly one class label, prediction, threshold, cell count, prevalence, beta, averaging rule, zero-division convention, marginal total, rater label, weight or validation split. Explain which mathematical object changes, which metric moves and which claim must weaken. Keep target definition, arithmetic, sampling uncertainty and interpretation on separate notebook lines. Explain once to a student and once to a parent. Add a counterexample involving high accuracy from a majority-class rule, identical F1 with different true-negative counts, an MCC zero denominator, high raw agreement with altered marginals or strong agreement without truth. State the extra evidence needed to distinguish explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 2: Counts

Begin with this mechanism: Use TP=18, FP=6, FN=12. Recreate the concrete case: Keep TN available even though F1 omits it. Write TP, FP, FN, TN, precision, recall, beta, averaging rule, threshold, zero-division rule and uncertainty with identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: The score is not the whole matrix. Complete the transfer step: Show counts. Change exactly one class label, prediction, threshold, cell count, prevalence, beta, averaging rule, zero-division convention, marginal total, rater label, weight or validation split. Explain which mathematical object changes, which metric moves and which claim must weaken. Keep target definition, arithmetic, sampling uncertainty and interpretation on separate notebook lines. Explain once to a student and once to a parent. Add a counterexample involving high accuracy from a majority-class rule, identical F1 with different true-negative counts, an MCC zero denominator, high raw agreement with altered marginals or strong agreement without truth. State the extra evidence needed to distinguish explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 3: Precision

Begin with this mechanism: Compute 18/24. Recreate the concrete case: Precision=.75. Write TP, FP, FN, TN, precision, recall, beta, averaging rule, threshold, zero-division rule and uncertainty with identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Undefined if no cases are predicted positive. Complete the transfer step: State rule. Change exactly one class label, prediction, threshold, cell count, prevalence, beta, averaging rule, zero-division convention, marginal total, rater label, weight or validation split. Explain which mathematical object changes, which metric moves and which claim must weaken. Keep target definition, arithmetic, sampling uncertainty and interpretation on separate notebook lines. Explain once to a student and once to a parent. Add a counterexample involving high accuracy from a majority-class rule, identical F1 with different true-negative counts, an MCC zero denominator, high raw agreement with altered marginals or strong agreement without truth. State the extra evidence needed to distinguish explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 4: Recall

Begin with this mechanism: Compute 18/30. Recreate the concrete case: Recall=.60. Write TP, FP, FN, TN, precision, recall, beta, averaging rule, threshold, zero-division rule and uncertainty with identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Undefined if no actual positives exist. Complete the transfer step: State support. Change exactly one class label, prediction, threshold, cell count, prevalence, beta, averaging rule, zero-division convention, marginal total, rater label, weight or validation split. Explain which mathematical object changes, which metric moves and which claim must weaken. Keep target definition, arithmetic, sampling uncertainty and interpretation on separate notebook lines. Explain once to a student and once to a parent. Add a counterexample involving high accuracy from a majority-class rule, identical F1 with different true-negative counts, an MCC zero denominator, high raw agreement with altered marginals or strong agreement without truth. State the extra evidence needed to distinguish explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 5: Harmonic mean

Begin with this mechanism: Compute 2PR/(P+R). Recreate the concrete case: F1=.6667. Write TP, FP, FN, TN, precision, recall, beta, averaging rule, threshold, zero-division rule and uncertainty with identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: The harmonic mean is pulled toward the smaller component. Complete the transfer step: Show arithmetic. Change exactly one class label, prediction, threshold, cell count, prevalence, beta, averaging rule, zero-division convention, marginal total, rater label, weight or validation split. Explain which mathematical object changes, which metric moves and which claim must weaken. Keep target definition, arithmetic, sampling uncertainty and interpretation on separate notebook lines. Explain once to a student and once to a parent. Add a counterexample involving high accuracy from a majority-class rule, identical F1 with different true-negative counts, an MCC zero denominator, high raw agreement with altered marginals or strong agreement without truth. State the extra evidence needed to distinguish explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 6: Count form

Begin with this mechanism: Compute 2TP/(2TP+FP+FN). Recreate the concrete case: 36/54=.6667. Write TP, FP, FN, TN, precision, recall, beta, averaging rule, threshold, zero-division rule and uncertainty with identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Equivalent forms require consistent counts. Complete the transfer step: Cross-check. Change exactly one class label, prediction, threshold, cell count, prevalence, beta, averaging rule, zero-division convention, marginal total, rater label, weight or validation split. Explain which mathematical object changes, which metric moves and which claim must weaken. Keep target definition, arithmetic, sampling uncertainty and interpretation on separate notebook lines. Explain once to a student and once to a parent. Add a counterexample involving high accuracy from a majority-class rule, identical F1 with different true-negative counts, an MCC zero denominator, high raw agreement with altered marginals or strong agreement without truth. State the extra evidence needed to distinguish explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 7: Beta two

Begin with this mechanism: Set beta=2. Recreate the concrete case: Use F2=90/144=.625. Write TP, FP, FN, TN, precision, recall, beta, averaging rule, threshold, zero-division rule and uncertainty with identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Greater recall emphasis does not guarantee a larger number. Complete the transfer step: Interpret inputs. Change exactly one class label, prediction, threshold, cell count, prevalence, beta, averaging rule, zero-division convention, marginal total, rater label, weight or validation split. Explain which mathematical object changes, which metric moves and which claim must weaken. Keep target definition, arithmetic, sampling uncertainty and interpretation on separate notebook lines. Explain once to a student and once to a parent. Add a counterexample involving high accuracy from a majority-class rule, identical F1 with different true-negative counts, an MCC zero denominator, high raw agreement with altered marginals or strong agreement without truth. State the extra evidence needed to distinguish explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 8: Beta half

Begin with this mechanism: Set beta=.5. Recreate the concrete case: Use F0.5=22.5/31.5=.7143. Write TP, FP, FN, TN, precision, recall, beta, averaging rule, threshold, zero-division rule and uncertainty with identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Precision emphasis suits only some decisions. Complete the transfer step: Justify beta. Change exactly one class label, prediction, threshold, cell count, prevalence, beta, averaging rule, zero-division convention, marginal total, rater label, weight or validation split. Explain which mathematical object changes, which metric moves and which claim must weaken. Keep target definition, arithmetic, sampling uncertainty and interpretation on separate notebook lines. Explain once to a student and once to a parent. Add a counterexample involving high accuracy from a majority-class rule, identical F1 with different true-negative counts, an MCC zero denominator, high raw agreement with altered marginals or strong agreement without truth. State the extra evidence needed to distinguish explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 9: True negatives

Begin with this mechanism: Observe TN does not enter binary F beta. Recreate the concrete case: Keep TN in the report. Write TP, FP, FN, TN, precision, recall, beta, averaging rule, threshold, zero-division rule and uncertainty with identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: A model can change on negatives without the same F score response. Complete the transfer step: Add matrix. Change exactly one class label, prediction, threshold, cell count, prevalence, beta, averaging rule, zero-division convention, marginal total, rater label, weight or validation split. Explain which mathematical object changes, which metric moves and which claim must weaken. Keep target definition, arithmetic, sampling uncertainty and interpretation on separate notebook lines. Explain once to a student and once to a parent. Add a counterexample involving high accuracy from a majority-class rule, identical F1 with different true-negative counts, an MCC zero denominator, high raw agreement with altered marginals or strong agreement without truth. State the extra evidence needed to distinguish explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 10: Threshold

Begin with this mechanism: Recompute counts across score cutoffs. Recreate the concrete case: F1 varies with threshold. Write TP, FP, FN, TN, precision, recall, beta, averaging rule, threshold, zero-division rule and uncertainty with identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Selecting and testing on the same data is optimistic. Complete the transfer step: Separate sets. Change exactly one class label, prediction, threshold, cell count, prevalence, beta, averaging rule, zero-division convention, marginal total, rater label, weight or validation split. Explain which mathematical object changes, which metric moves and which claim must weaken. Keep target definition, arithmetic, sampling uncertainty and interpretation on separate notebook lines. Explain once to a student and once to a parent. Add a counterexample involving high accuracy from a majority-class rule, identical F1 with different true-negative counts, an MCC zero denominator, high raw agreement with altered marginals or strong agreement without truth. State the extra evidence needed to distinguish explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 11: Prevalence

Begin with this mechanism: Change the positive rate cautiously. Recreate the concrete case: Precision and F1 can move. Write TP, FP, FN, TN, precision, recall, beta, averaging rule, threshold, zero-division rule and uncertainty with identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Cross-population comparisons need case-mix context. Complete the transfer step: Report rate. Change exactly one class label, prediction, threshold, cell count, prevalence, beta, averaging rule, zero-division convention, marginal total, rater label, weight or validation split. Explain which mathematical object changes, which metric moves and which claim must weaken. Keep target definition, arithmetic, sampling uncertainty and interpretation on separate notebook lines. Explain once to a student and once to a parent. Add a counterexample involving high accuracy from a majority-class rule, identical F1 with different true-negative counts, an MCC zero denominator, high raw agreement with altered marginals or strong agreement without truth. State the extra evidence needed to distinguish explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 12: Macro average

Begin with this mechanism: Average classwise scores equally. Recreate the concrete case: Each class gets equal weight. Write TP, FP, FN, TN, precision, recall, beta, averaging rule, threshold, zero-division rule and uncertainty with identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Rare classes can dominate interpretation. Complete the transfer step: Name average. Change exactly one class label, prediction, threshold, cell count, prevalence, beta, averaging rule, zero-division convention, marginal total, rater label, weight or validation split. Explain which mathematical object changes, which metric moves and which claim must weaken. Keep target definition, arithmetic, sampling uncertainty and interpretation on separate notebook lines. Explain once to a student and once to a parent. Add a counterexample involving high accuracy from a majority-class rule, identical F1 with different true-negative counts, an MCC zero denominator, high raw agreement with altered marginals or strong agreement without truth. State the extra evidence needed to distinguish explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 13: Micro average

Begin with this mechanism: Pool counts before computing. Recreate the concrete case: Large classes contribute more observations. Write TP, FP, FN, TN, precision, recall, beta, averaging rule, threshold, zero-division rule and uncertainty with identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: It answers a different question. Complete the transfer step: Do not mix. Change exactly one class label, prediction, threshold, cell count, prevalence, beta, averaging rule, zero-division convention, marginal total, rater label, weight or validation split. Explain which mathematical object changes, which metric moves and which claim must weaken. Keep target definition, arithmetic, sampling uncertainty and interpretation on separate notebook lines. Explain once to a student and once to a parent. Add a counterexample involving high accuracy from a majority-class rule, identical F1 with different true-negative counts, an MCC zero denominator, high raw agreement with altered marginals or strong agreement without truth. State the extra evidence needed to distinguish explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 14: Weighted average

Begin with this mechanism: Weight class scores by support. Recreate the concrete case: Record supports. Write TP, FP, FN, TN, precision, recall, beta, averaging rule, threshold, zero-division rule and uncertainty with identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: A weighted summary can hide weak minorities. Complete the transfer step: Show per class. Change exactly one class label, prediction, threshold, cell count, prevalence, beta, averaging rule, zero-division convention, marginal total, rater label, weight or validation split. Explain which mathematical object changes, which metric moves and which claim must weaken. Keep target definition, arithmetic, sampling uncertainty and interpretation on separate notebook lines. Explain once to a student and once to a parent. Add a counterexample involving high accuracy from a majority-class rule, identical F1 with different true-negative counts, an MCC zero denominator, high raw agreement with altered marginals or strong agreement without truth. State the extra evidence needed to distinguish explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 15: Uncertainty

Begin with this mechanism: Resample matched predictions and labels. Recreate the concrete case: Keep threshold selection outside final evaluation. Write TP, FP, FN, TN, precision, recall, beta, averaging rule, threshold, zero-division rule and uncertainty with identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: One decimal hides instability. Complete the transfer step: Report interval. Change exactly one class label, prediction, threshold, cell count, prevalence, beta, averaging rule, zero-division convention, marginal total, rater label, weight or validation split. Explain which mathematical object changes, which metric moves and which claim must weaken. Keep target definition, arithmetic, sampling uncertainty and interpretation on separate notebook lines. Explain once to a student and once to a parent. Add a counterexample involving high accuracy from a majority-class rule, identical F1 with different true-negative counts, an MCC zero denominator, high raw agreement with altered marginals or strong agreement without truth. State the extra evidence needed to distinguish explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 16: Claim

Begin with this mechanism: Name beta, class, average and threshold. Recreate the concrete case: Teaching F1=.6667. Write TP, FP, FN, TN, precision, recall, beta, averaging rule, threshold, zero-division rule and uncertainty with identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Do not call F1 utility or calibration. Complete the transfer step: Bound conclusion. Change exactly one class label, prediction, threshold, cell count, prevalence, beta, averaging rule, zero-division convention, marginal total, rater label, weight or validation split. Explain which mathematical object changes, which metric moves and which claim must weaken. Keep target definition, arithmetic, sampling uncertainty and interpretation on separate notebook lines. Explain once to a student and once to a parent. Add a counterexample involving high accuracy from a majority-class rule, identical F1 with different true-negative counts, an MCC zero denominator, high raw agreement with altered marginals or strong agreement without truth. State the extra evidence needed to distinguish explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.