THE GOLD STANDARD OF JUDGEMENT · SPECIALIST STUDY
Choose a generator only after deciding which transformations should leave judgement unchanged
A quasi-arithmetic mean transforms each input through a continuous strictly monotone generator, averages in transformed space and maps back. Arithmetic, geometric and harmonic means are familiar cases, but the generator encodes how differences are valued. The gold standard tests admissible scale transformations, elicits comparisons, checks invariance and refuses to aggregate labels whose measurement meaning cannot support the formula.
This guide belongs to the Judgement subject guide, the Gold Standard of Education, and the How X Works learning library. Choose a reading route, work the examples, then use the transfer checks to test whether the method survives a changed context.
Choose a reading route
Build the foundations · chapters 1–2
Apply the core method · chapters 3–5
Test evidence and alternatives · chapters 6–7
Handle limits and disagreement · chapters 8–9
Transfer and govern · chapters 10
The whole problem at a glance
| Generator f(x) | Mean produced | Scale boundary |
|---|---|---|
| x | Arithmetic | Translation and positive scaling |
| log x | Geometric | Positive ratio scale |
| 1/x | Harmonic | Positive domain; rate-like uses |
| General monotone f | Quasi-arithmetic | Depends on f and admissible transforms |
A monotone transform preserves order but not every meaningful numerical operation.
CHAPTER 1 OF 10 · Build the foundations
1. Build the quasi-arithmetic mean from its generator
Core idea. For continuous strictly monotone f, the mean is f inverse of the weighted average of f(x_i).
Method. Declare domain, weights, generator, inverse and endpoint behaviour; verify weights are non-negative and sum to one.
Worked example. With f(x)=log x and inputs 1, 4 and 16, the equal-weight mean is exp((0+log4+log16)/3)=4.
Changed-context transfer. Changed-context transfer: apply the same reasoning structure to a school or public-service decision. Preserve the formal relationship, then replace the population, authority, privacy duties, evidence and remedy; never copy the domain conclusion.
Limits. The inverse may be undefined at a boundary, and decreasing generators require careful orientation even though they can still define a mean.
Evidence and review. For this chapter, keep a dated evidence record: the input, rule or assumption, result, accountable owner and a condition that would change the conclusion. Work an ordinary case and a boundary case, preserve rejected alternatives, and require an authorised independent reader to reproduce where judgement entered. Chapter 1 is complete only when the claimed result survives that review.
Misconception to avoid. Any monotone scoring function followed by an average is equivalent. The inverse mapping and domain are essential.
Practical evidence check
- Reproduce the result in transformed and original space.
- Return to chapter contents
CHAPTER 2 OF 10 · Build the foundations
2. Distinguish generator equivalence from different means
Core idea. Generators related by a positive affine transformation af+b produce the same quasi-arithmetic mean.
Method. Normalise a generator for reporting but test whether a proposed change is affine in generator space, not merely visually similar.
Worked example. f(x)=log x and g(x)=3log x+7 yield the same geometric mean because averaging cancels the affine change before inversion.
Changed-context transfer. Changed-context transfer: apply the same reasoning structure to a school or public-service decision. Preserve the formal relationship, then replace the population, authority, privacy duties, evidence and remedy; never copy the domain conclusion.
Limits. Non-affine changes generally alter compensation and ranking, even if both generators are increasing.
Evidence and review. For this chapter, keep a dated evidence record: the input, rule or assumption, result, accountable owner and a condition that would change the conclusion. Work an ordinary case and a boundary case, preserve rejected alternatives, and require an authorised independent reader to reproduce where judgement entered. Chapter 2 is complete only when the claimed result survives that review.
Misconception to avoid. Every different formula names a different mean. Affine generator variants are representationally equivalent.
Practical evidence check
- Compare outputs for ten vectors under f and af+b.
- Return to chapter contents
CHAPTER 3 OF 10 · Apply the core method
3. Read arithmetic, geometric and harmonic means as cases
Core idea. Familiar means correspond to different generators and therefore different assumptions about increments, ratios and rates.
Method. Match the data scale and decision meaning before selecting a family; test units and zero or negative values explicitly.
Worked example. For 1, 4 and 16, arithmetic gives 7, geometric gives 4 and harmonic is about 2.29, revealing progressively less compensation by the large value.
Changed-context transfer. Changed-context transfer: apply the same reasoning structure to a school or public-service decision. Preserve the formal relationship, then replace the population, authority, privacy duties, evidence and remedy; never copy the domain conclusion.
Limits. The geometric and harmonic means require positive values; shifting a scale to avoid zero changes their meaning.
Evidence and review. For this chapter, keep a dated evidence record: the input, rule or assumption, result, accountable owner and a condition that would change the conclusion. Work an ordinary case and a boundary case, preserve rejected alternatives, and require an authorised independent reader to reproduce where judgement entered. Chapter 3 is complete only when the claimed result survives that review.
Misconception to avoid. One familiar mean is objectively correct. Each answers a different structural question.
Practical evidence check
- Explain which pairwise change the chosen mean treats as equivalent.
- Return to chapter contents
CHAPTER 4 OF 10 · Apply the core method
4. Elicit a generator through meaningful indifferences
Core idea. Generator curvature can be inferred from consistent judgements about mixtures or certainty equivalents, but elicitation embeds framing and noise.
Method. Present repeated balanced comparisons, randomise order, estimate f under monotonicity constraints and hold out cases for validation.
Worked example. A decision-maker is indifferent between the pair (2,8) and the certain value 4, suggesting ratio-sensitive rather than arithmetic compensation.
Changed-context transfer. Changed-context transfer: apply the same reasoning structure to a school or public-service decision. Preserve the formal relationship, then replace the population, authority, privacy duties, evidence and remedy; never copy the domain conclusion.
Limits. Finite comparisons identify the generator only up to equivalence and may conflict across contexts.
Evidence and review. For this chapter, keep a dated evidence record: the input, rule or assumption, result, accountable owner and a condition that would change the conclusion. Work an ordinary case and a boundary case, preserve rejected alternatives, and require an authorised independent reader to reproduce where judgement entered. Chapter 4 is complete only when the claimed result survives that review.
Misconception to avoid. Fitting a smooth curve discovers true preferences. It estimates a model under elicitation conditions.
Practical evidence check
- Repeat one comparison in a new framing and report inconsistency.
- Return to chapter contents
CHAPTER 5 OF 10 · Apply the core method
5. Test translation and affine invariance
Core idea. A judgement rule should be invariant or equivariant under transformations that represent harmless changes of measurement units.
Method. List admissible transforms for each scale, transform all inputs, aggregate, back-transform and compare with the original decision.
Worked example. Converting Celsius to Fahrenheit is positive affine. An arithmetic mean transforms consistently; a geometric mean of Celsius values does not have the same unit-invariant meaning.
Changed-context transfer. Changed-context transfer: apply the same reasoning structure to a school or public-service decision. Preserve the formal relationship, then replace the population, authority, privacy duties, evidence and remedy; never copy the domain conclusion.
Limits. Not every application requires full affine invariance; demanding it can exclude useful ratio-scale models.
Evidence and review. For this chapter, keep a dated evidence record: the input, rule or assumption, result, accountable owner and a condition that would change the conclusion. Work an ordinary case and a boundary case, preserve rejected alternatives, and require an authorised independent reader to reproduce where judgement entered. Chapter 5 is complete only when the claimed result survives that review.
Misconception to avoid. Monotonicity guarantees unit invariance. It preserves order, not numerical aggregation.
Practical evidence check
- Run the rule under two legitimate unit systems.
- Return to chapter contents
Worked three-score comparison
| Inputs | Arithmetic | Geometric |
|---|---|---|
| 1, 4, 16 | 7 | 4 |
| 2, 8, 32 | 14 | 8 |
| Scale doubled | Mean doubles | Mean doubles |
The geometric mean requires positive inputs; translation changes its interpretation.
CHAPTER 6 OF 10 · Test evidence and alternatives
6. Use homogeneity only on ratio-admissible data
Core idea. Homogeneity means scaling every input scales the mean. Power means satisfy it on positive domains, but the property is meaningful only when zero and ratios are meaningful.
Method. Identify the measurement scale, justify a true zero, and test multiplication by several positive constants.
Worked example. Doubling all lengths should double a geometric or power mean; adding ten centimetres is not a harmless rescaling in the same way.
Changed-context transfer. Changed-context transfer: apply the same reasoning structure to a school or public-service decision. Preserve the formal relationship, then replace the population, authority, privacy duties, evidence and remedy; never copy the domain conclusion.
Limits. Scores with conventional zero or bounded labels do not inherit ratio meaning from their numbers.
Evidence and review. For this chapter, keep a dated evidence record: the input, rule or assumption, result, accountable owner and a condition that would change the conclusion. Work an ordinary case and a boundary case, preserve rejected alternatives, and require an authorised independent reader to reproduce where judgement entered. Chapter 6 is complete only when the claimed result survives that review.
Misconception to avoid. A homogeneous formula makes any numeric scale a ratio scale. Measurement semantics come first.
Practical evidence check
- Ask whether “twice as much” is meaningful for every input.
- Return to chapter contents
CHAPTER 7 OF 10 · Test evidence and alternatives
7. Respect ordinal and interval scale admissibility
Core idea. Ordinal labels support order, interval scales support positive affine change, and ratio scales support positive multiplication; aggregation claims must match.
Method. Write the admissible transformation group and test whether the ranking or decision survives it. Prefer medians or order methods when only order is defensible.
Worked example. Relabelling satisfaction levels 1–5 as 10,20,30,40,50 preserves order and intervals, but an arbitrary monotone relabelling can reverse a quasi-arithmetic comparison.
Changed-context transfer. Changed-context transfer: apply the same reasoning structure to a school or public-service decision. Preserve the formal relationship, then replace the population, authority, privacy duties, evidence and remedy; never copy the domain conclusion.
Limits. Real instruments can mix scale properties, and group-level meaning may differ from item-level meaning.
Evidence and review. For this chapter, keep a dated evidence record: the input, rule or assumption, result, accountable owner and a condition that would change the conclusion. Work an ordinary case and a boundary case, preserve rejected alternatives, and require an authorised independent reader to reproduce where judgement entered. Chapter 7 is complete only when the claimed result survives that review.
Misconception to avoid. Numbers permit arithmetic because they are numbers. Numerals can encode labels without cardinal distance.
Practical evidence check
- Apply one admissible and one inadmissible transform and explain the difference.
- Return to chapter contents
CHAPTER 8 OF 10 · Handle limits and disagreement
8. Analyse sensitivity to curvature and weights
Core idea. Generator curvature and weights jointly control compensation, so similar current outputs can diverge at boundaries.
Method. Vary curvature and weights over a justified range, map ranking reversals and separate empirical weights from normative importance.
Worked example. Two candidates tie under the arithmetic mean; a more concave generator favours the balanced profile and reverses the choice.
Changed-context transfer. Changed-context transfer: apply the same reasoning structure to a school or public-service decision. Preserve the formal relationship, then replace the population, authority, privacy duties, evidence and remedy; never copy the domain conclusion.
Limits. Parameter sweeps reveal dependence but cannot select the ethically right curvature or weights.
Evidence and review. For this chapter, keep a dated evidence record: the input, rule or assumption, result, accountable owner and a condition that would change the conclusion. Work an ordinary case and a boundary case, preserve rejected alternatives, and require an authorised independent reader to reproduce where judgement entered. Chapter 8 is complete only when the claimed result survives that review.
Misconception to avoid. Robustness to small input changes proves model choice is robust. The decision may still be fragile to generator family.
Practical evidence check
- Plot every reversal across the approved parameter region.
- Return to chapter contents
Generator audit
| Question | Evidence | Failure |
|---|---|---|
| Monotone? | Derivative or order proof | Reversed ranking |
| Domain valid? | Endpoint tests | Undefined transform |
| Invariant? | Admissible scale transforms | Unit-dependent result |
| Elicited? | Choice comparisons | Arbitrary curvature |
A fitted generator can reproduce choices while concealing inconsistency or policy values.
CHAPTER 9 OF 10 · Handle limits and disagreement
9. Keep missingness, bounds and dependence visible
Core idea. The formula assumes available comparable inputs; it does not decide how to impute missing values or handle dependence.
Method. Return unresolved states, use interval propagation where justified, and model correlated criteria before aggregation.
Worked example. One criterion is missing, so a low-high interval is reported instead of renormalising weights silently and changing the decision contract.
Changed-context transfer. Changed-context transfer: apply the same reasoning structure to a school or public-service decision. Preserve the formal relationship, then replace the population, authority, privacy duties, evidence and remedy; never copy the domain conclusion.
Limits. Wide intervals may prevent ranking, which is a legitimate conclusion.
Evidence and review. For this chapter, keep a dated evidence record: the input, rule or assumption, result, accountable owner and a condition that would change the conclusion. Work an ordinary case and a boundary case, preserve rejected alternatives, and require an authorised independent reader to reproduce where judgement entered. Chapter 9 is complete only when the claimed result survives that review.
Misconception to avoid. Renormalising weights is neutral. It changes the relative importance of observed criteria.
Practical evidence check
- Compare worst, best and unresolved treatments for one missing input.
- Return to chapter contents
CHAPTER 10 OF 10 · Transfer and govern
10. Govern the generator as a contestable judgement rule
Core idea. A generator is a policy-bearing model of compensation and scale, not a technical decoration.
Method. Version data definitions, generator, normalisation, weights, elicitation evidence, invariance tests, overrides and appeal outcomes.
Worked example. A school changes a rubric from interval-like marks to ordered mastery bands; the old quasi-arithmetic rule is retired instead of reusing the same numbers.
Changed-context transfer. Changed-context transfer: apply the same reasoning structure to a school or public-service decision. Preserve the formal relationship, then replace the population, authority, privacy duties, evidence and remedy; never copy the domain conclusion.
Limits. Valid mathematics cannot supply lawful authority, fair inputs or a remedy path.
Evidence and review. For this chapter, keep a dated evidence record: the input, rule or assumption, result, accountable owner and a condition that would change the conclusion. Work an ordinary case and a boundary case, preserve rejected alternatives, and require an authorised independent reader to reproduce where judgement entered. Chapter 10 is complete only when the claimed result survives that review.
Misconception to avoid. Publishing the formula transfers accountability to the model. Decision owners remain responsible.
Practical evidence check
- Audit one decision from raw measurement through scale choice, generator and correction.
- Return to chapter contents
Build the specialist decision record
For “The Gold Standard of Judgement | Quasi-Arithmetic Means, Generator Elicitation, Affine Invariance and Scale Admissibility”, begin with one bounded decision rather than an abstract claim. Name the affected people, data or institutions; the time window; the authority for acting; and the conclusion the method is allowed to support.
Preserve source material before interpretation. Record definitions, exclusions, transformations and the person responsible for each judgement. A reviewer should be able to separate an observed fact from an inferred result, a policy choice and a presentation shortcut.
Work one ordinary case in full. Show the input, every material rule or calculation, the output and the action that follows. Then write what the same evidence would mean under the simplest credible alternative method.
Keep an explicit unresolved state. Missing, protected, conflicting or unevaluated information must not become zero, false, consent or absence merely because a form expects one value.
Finish the record with a claim ledger. For every sentence that could drive action, identify its supporting source, inferential step, confidence or legal status, intended reader and expiry condition. This keeps descriptive evidence, formal consequence, professional judgement and institutional authority from collapsing into one impressive-looking conclusion.
Run the boundary-case audit
Select a case near the method’s domain edge: a missing input, a disputed definition, an endpoint, a sparse group, a resource limit or a legal deadline. Boundary cases reveal the contract that routine examples can conceal.
Change one assumption at a time and retain every output. Mark what stays stable, what reverses and what becomes undefined. If several assumptions move together, report that dependence instead of attributing the change to one convenient cause.
Search deliberately for a counterexample to the strongest claim. A method that works on the published illustration can still fail under a permitted input, an alternative grouping, a changed unit or an inaccessible evidence pathway.
Translate the counterexample into plain language: who would receive a different decision, what evidence would be hidden, and which safeguard or narrower claim repairs the problem.
Include an adversarial but plausible case. Let a missing group, ambiguous identifier, extreme weight, conflicting rule or urgent deadline interact with normal system incentives. The aim is not theatrical failure; it is to learn whether ordinary pressure makes the method conceal uncertainty, shift burden or reward a misleading shortcut.
Test independent reproduction
Give a reviewer the bounded inputs, public method contract and permitted access, but not the desired answer. Ask them to reproduce the result and identify every place where judgement, local knowledge or unavailable evidence was required.
Compare the reviewer’s intermediate states, not only the final number or label. Matching outputs can hide different denominators, inference paths, missing-data treatments or authorities that will diverge on the next case.
Where reproduction fails, classify the cause: ambiguous definition, undocumented transformation, incompatible implementation, inaccessible source, calculation error or legitimate alternative interpretation. Each class needs a different remedy.
Retain the disagreement and resolution. A corrected record should show what changed, why it changed and which prior outputs or decisions must be revisited.
Run the reproduction on a changed but structurally similar case after the first discrepancy is repaired. A method is not transferable merely because its authors can explain the original example. The new case tests whether the public contract carries the reasoning or whether hidden expertise is still doing essential work.
Design correction, challenge and remedy
Assign a visible route for correcting inputs, challenging assumptions and reviewing the application of the method. Technical accuracy without a usable remedy leaves the affected person carrying an error they cannot contest.
Separate urgent live-case repair from later system learning. A current decision may need pause, reassignment, fresh evidence or a human review before the organisation finishes its longer investigation.
Protect confidential and privileged material during review, but do not use confidentiality as a reason for unrecorded authority. Layer public explanation, protected oversight and individual reasons according to lawful access.
Verify that correction reaches cached results, derived records, dashboards and future decisions. Closing a complaint is not enough if the old conclusion continues to travel.
Measure remedy from the affected person’s position. A revised database row is useful, but it is incomplete when a hearing, service allocation, published claim or downstream model still relies on the old result. Record notification, restored opportunity, compensation or procedural repair where the governing framework provides it.
Set review and retirement triggers
Review after a change in definitions, population, law, source access, measurement scale, software, vocabulary or incentives. A stable formula can become invalid when the world or the meaning of its inputs changes.
Monitor leading signals as well as final outcomes: coverage gaps, timeouts, overrides, complaints, disagreement, missingness and boundary frequency. These often reveal failure before a headline metric moves.
Predefine thresholds for recalibration, suspension and retirement. Do not force a method to answer after its assumptions fail merely because replacing it is inconvenient.
At retirement, preserve the versioned contract, evidence, known limitations and migration decision. State which historic results remain interpretable and which require warning, restatement or withdrawal.
Schedule one prospective review before confidence fades or controversy forces it. Assign named owners for evidence, implementation and rights impact, and invite at least one reviewer outside the design team. The review should be able to narrow the claim, change the method or stop use without requiring proof of catastrophic failure.
Questions readers ask
What is a quasi-arithmetic mean?
- An average computed after a monotone transformation and then mapped back through the inverse.
When do two generators give the same mean?
- When they differ by a positive affine transformation.
Is the geometric mean quasi-arithmetic?
- Yes, on positive inputs with a logarithmic generator.
Why does scale admissibility matter?
- A harmless change of units should not arbitrarily change judgement.
Can the generator be learned from choices?
- Partly, under consistency and identification assumptions.
Does formal invariance prove fairness?
- No; it addresses transformation behaviour, not authority or impact.
Practise, verify and continue
Choose one consequential case. Reproduce the complete method, work one ordinary and one boundary case, test the strongest rival explanation, and state the evidence that would reverse or suspend the result. Build an evidence pack containing original inputs, definitions, calculations or rule steps, alternatives rejected, uncertainty, reviewer comments and the final action. Give the pack to a reader who did not design the method. Ask that reader to reproduce the result, identify one missing assumption and construct a case that would make the method return unresolved. Revise the public explanation only after the evidence pack and the operational process agree.
Move the method into an education or public-service setting. Keep the reasoning structure while replacing domain authority, population, privacy, evidence and correction routes. Start with a low-stakes historical example, then compare the specialist method with a transparent baseline. Check whether the same people, records and decisions remain inside scope; whether the scale or vocabulary still means the same thing; and whether a person affected by error can understand and challenge it. Pilot under human review, record reversals and burden, and stop the transfer if the formal resemblance hides a different ethical or legal problem.
Related eduKate reading
Uninorms, neutral elements and associativity limits
Sugeno integrals and non-compensatory choice
Bipolar Sugeno integrals and qualitative gains and losses
Choquet integrals and non-additive expectations
Authoritative sources and scope
Pasteczka, Scales of quasi-arithmetic means determined by invariance property
Matkowski and Páles, Characterization of generalized quasi-arithmetic means
Nielsen, Beyond scalar quasi-arithmetic means
Jarczyk, invariance of weighted quasi-arithmetic means
This article is mathematical and decision-design education. Definitions, domains and theorem assumptions must be checked in the cited research. Examples are original teaching cases. A formally valid mean does not establish that inputs share a meaningful scale, that a generator reflects legitimate values, or that a consequential decision is fair or lawful. Preserve invariance tests, sensitivity, reasons, review and correction. Before operational use, publish the domain, admissible transformations, generator equivalence class, weights, endpoint treatment and counterexamples. Re-test after a scale, instrument or policy change. If stakeholders cannot explain what compensation the generator creates in ordinary language, the rule is not ready for a consequential decision.