What did
the number
leave out?

Faster appointments or more choice?

Compare two hypothetical services using access and participant control. Adjust the weights to see which service ranks higher.

Read the opening story

Possible score differences

Every point in the violet band is compatible with the stated evidence and scoring rule.

Violet: all compatible gaps at each weight. Blue bracket: the full range over your allowed weights. These are scenario bounds, not confidence intervals.

Priority given to access 50%Allow disagreement around that priority 50–50% allowed

Compatible gap[-25, 10]

Both rankings remain possible. The evidence and allowed trade-offs do not settle the comparison.

Evidence and scoring rules

The two moves answer different questions. An audit narrows what is compatible. A minimum standard changes what is acceptable.

Assumed 0–100 rubrics. Higher is better within each dimension. The scores describe the services.

Compatible profiles, without assigned probabilities
Design	Access	Participant control
A	75–85	25–45
B	45–65	65–85

Inspect the control evidence more closelyA hypothetical audit narrows A to 30–40 and B to 70–80.Require a control score of at least 50A declared floor, even if another dimension scores well.

No eligibility floor is imposed. Access can compensate for lower participant control in this scoring rule.

Scoring assumptions

For this exercise we assume meaningful differences on two anchored rubrics, an additive score, nonnegative weights, and all combinations within the profile ranges. Weights and profiles vary independently. Those assumptions make the displayed bounds sharp for this rectangular scenario set. Dependencies could narrow the true range; invalid assumptions could change it entirely.

Mathematics and sources

Ordinal recoding

Now use a separate dataset: A received categories 1 and 4; B received 2 and 3. Only their order is established. Change the spacing while keeping every ranking and both endpoints intact.

If a conclusion reverses under an admissible recoding, the original order facts did not determine it.

Blue endpoints belong to A. Violet interior categories belong to B. The connecting lines track identical categories across codings.

Curvature of the recoding 1

A's recoded mean 0.500

B's recoded mean 0.500

The arithmetic means tie with equal category spacing. Order alone does not require that spacing.

The recoding and what survives it

For category s, use f(s) = (sᵠ − 1) / (4ᵠ − 1), with q between 0.25 and 4. Every function shown is strictly increasing, and f(1) = 0, f(4) = 1. The recoded means average f(1), f(4) for A and f(2), f(3) for B.

A outranks B in two of the four cross-group pairings under every recoding. That invariant comparison does not establish equal value. Additional interval-scale or utility assumptions can justify other comparisons; they have to be supplied.

See the worked example

Export the scenario

Download the ranges, weights, coding rule, and calculated bounds as JSON.

Synthetic data.

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Measurement and decision-making

The essay, field guide, and technical companion examine how a service-design ranking depends on its measurements and value judgments.

The technical companion gives proofs and references, followed by research and teaching proposals that have not yet been tested.

Read the technical companion and research agenda