Engineering calculator

Zeta-Score Calculator

Q: Is zeta the same as En?

No. Zeta uses standard uncertainties ulab and uref. En uses already-expanded Ulab and Uref. |En| ≤ 1 is the En pass rule; this page uses |ζ| ≤ 2 / warning / fail.

Q: Is |ζ| ≤ 2 the same as an ILAC G8 spec decision?

No. Zeta compares a laboratory result to an assigned value in a proficiency test. The decision-rule page compares a measured y to specification limits.

Q: Where does this run?

Locally in the browser by default. REST and MCP call the same metrology-engine zeta_score op.

Compute the proficiency-testing zeta-score from a laboratory result, a reference value, and two standard uncertainties. |ζ| ≤ 2 is pass. Runs locally. Not En, not TUR, and not an ILAC G8 spec decision.

Instant result

Result

—

Enter values to calculate.

Inputs—

Mode—

Formula—

Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · Proficiency-testing zeta from ylab, xref, and standard ulab and uref, + O3 mpmath tabulated zeta.

Input interpretation: Enter values to calculate.
Result: —
Verified scope: Proficiency-testing zeta from ylab, xref, and standard ulab and uref, + O3 mpmath tabulated zeta.
Assurance: Engineering
Declared partition coverage: PASS · 5/5 declared partitions (pass, fail, alias, awkward, invalid-domain) · Matrix
Deferred: Not En (expanded U), not TUR, and not an ILAC G8 specification decision.
Numerical scope: O2: proficiency-testing zeta vs a separate-module hypot identity (≤2 ULP). Not En, not TUR, and not an ILAC G8 specification decision. ≤2 ULP vs O3 applies only to the published tabulated zeta vectors (pass, the |ζ|=2 boundary, warning, fail, an alias, a negative result, one awkward pair, and equal standard uncertainties). It is not a whole-domain guarantee and does not cover En, TUR, or an ILAC G8 specification decision.
Known limitations: Core CVP does not include live graph, viewport, or pointer interaction.
Model: Proficiency-testing zeta-score and ISO 13528 |ζ| bands (pass / warning / fail).
Scope: ISO 13528 zeta-score with standard uncertainties ulab and uref
Verification: Engine tested · Source checked · v1.0.1 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · Proficiency-testing zeta from ylab, xref, and standard ulab and uref, + O3 mpmath tabulated zeta.· View Manifest · CVP overview · Specification
Versions: Calculation 1.0.1 · CVP protocol 1.0.0-proposed · Evidence 2026-09-25.o2-o3
Verification revision: 2026-09-25.o2-o3 · 1/1 property · digest 63c3ec401df5
Legacy regression: 11/11 tests · Production surface contract 6/6
Trust layers: Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference: O1 model · O3 expected_values · O3 numerical_behavior · O2 expected_values · O2 numerical_behavior
Interfaces: PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review: Not performed
Named expert review: Not performed
CVP suite: 4/4 golden · 5/5 CVP boundary · 5/5 invalid · 1/1 property · 1/1 metamorphic · 8/8 O3 · 3/3 cross-interface · 6/6 CVP contract · Manifest
Sources: ISO 13528 — Statistical methods for use in proficiency testing by interlaboratory comparison — Zeta-score using standard uncertainties
ISO/IEC 17043 — Conformity assessment — General requirements for the competence of proficiency testing providers — Performance evaluation using zeta
Sources
Evidence: 1 legacy golden · 5 legacy boundary · legacy regression suite · 4/4 oracle-backed golden · 5/5 invalid · Artifact integrity PASS
Semantic contract: PASS

Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

Zeta-scoreζ = (ylab − xref) / √(ulab² + uref²)

Decision|ζ| ≤ 2 → pass; 2 < |ζ| < 3 → warning; |ζ| ≥ 3 → fail

iulab and uref are standard uncertainties. This is not En (already-expanded U). It is not TUR and not an ILAC G8 decision against a specification.

How to use

Enter ylab and xref

Laboratory result and assigned or reference value, in the same unit.

Enter ulab and uref

Standard uncertainties, both greater than zero. Do not enter expanded U here — that is the En page.

Read ζ

|ζ| ≤ 2 is satisfactory, 2 < |ζ| < 3 is questionable, |ζ| ≥ 3 is unsatisfactory under ISO 13528 z-score interpretation.

Example calculations

Common configurations with formula and result.

Pass

ylab=2.5 · xref=0 · ulab=3 · uref=4

ζ=2.5/5

ζ=0.5 · pass

Warning

ylab=12.5 · xref=0 · ulab=3 · uref=4

ζ=12.5/5

ζ=2.5 · warning

Fail

ylab=15 · xref=0 · ulab=3 · uref=4

ζ=15/5

ζ=3 · fail

Zeta-Score calculator specification

Version 1.0.1 · Engine tested

Calculation status

Engine tested 11/11 tests · Production surface contract 6/6
Named expert review Not performed
Calculation version 1.0.1

Review policy · Evidence

Definition: ζ = (ylab − xref) / √(ulab² + uref²). |ζ| ≤ 2 is pass, 2 < |ζ| < 3 is warning, |ζ| ≥ 3 is fail. ylab is the laboratory result, xref the assigned value, and both u values are standard uncertainties.
What it calculates: Proficiency-testing zeta-score and ISO 13528 |ζ| bands (pass / warning / fail).
Inputs: ylab
xref
ulab
uref
Outputs: zeta
abs_zeta
decision
Formula: ζ = (ylab−xref)/√(ulab²+uref²); |ζ|≤2 pass; 2<|ζ|<3 warning; |ζ|≥3 fail
Assumptions: ISO 13528 zeta-score with standard uncertainties ulab and uref
|ζ| ≤ 2 is pass; 2 < |ζ| < 3 is warning; |ζ| ≥ 3 is fail
Not En (expanded U), not TUR, and not an ILAC G8 spec decision
Units: dimensionless ζ
Boundary conditions: missing ylab, xref, ulab, or uref → MISSING_REQUIRED_INPUT
ulab ≤ 0 or uref ≤ 0 → VALUE_MUST_BE_POSITIVE
Example: ylab=2.5 xref=0 ulab=3 uref=4 → ζ=0.5 pass
Validation cases: 2 published on this page · 11/11 tests · Production surface contract 6/6 · View evidence
ylab=2.5 xref=0 ulab=3 uref=4 → ζ=0.5 pass
ylab=2.5 xref=0 ulab=0 uref=4 → VALUE_MUST_BE_POSITIVE
Sources: ISO 13528 — Statistical methods for use in proficiency testing by interlaboratory comparison — Zeta-score using standard uncertainties
Supports: ζ = (ylab−xref)/√(ulab²+uref²); |ζ| ≤ 2 / 2–3 / ≥ 3 as z-score interpretation
ISO/IEC 17043 — Conformity assessment — General requirements for the competence of proficiency testing providers — Performance evaluation using zeta
Supports: Zeta is a proficiency-testing performance statistic, not a specification decision rule
Calculation version: 1.0.1

Background

Interpretation and common distinctions.

Compute the proficiency-testing zeta-score.

Supported and not supported

Supported — ζ from ylab, xref, ulab, uref · |ζ| ≤ 2 / warning / fail · API engineering.conformity.zeta_score

Not supported — En (expanded U), TUR, ILAC G8 specification decision, U = k·u_c as a product, RSS of a measurement model

Agent / API notes

Capability id: engineering.conformity.zeta_score · tool id: zeta-score · pin 1.0.1.

{ "ylab": 2.5, "xref": 0, "ulab": 3, "uref": 4 }

Aliases: y/x for ylab, X/ref for xref, u_lab/u_x for ulab, u_ref/u_X for uref. Expanded Ulab/Uref are not aliases. Errors: MISSING_REQUIRED_INPUT, INVALID_NUMBER, VALUE_MUST_BE_POSITIVE.

Calculator URL stays /calc/engineering/zeta-score. There is no /calc/metrology.

Other calculators in this family: Proficiency-testing z-score, Proficiency-testing z′, Normalized error En, Conformity decision, Test uncertainty ratio TUR, Type A uncertainty, Coverage factor k, Tolerance / Uncertainty Workspace .

Frequently asked questions

Key distinctions behind the calculation.

Is zeta the same as En?