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Engineering calculator

Normalized Error Calculator

Compute the proficiency-testing normalized error En from a laboratory result, a reference value, and two already-expanded uncertainties. |En| ≤ 1 is pass. Runs locally. Not zeta, 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 En from ylab, xref, and already-expanded Ulab and Uref, + O3 mpmath tabulated En.
Input interpretation
Enter values to calculate.
Result
—
Verified scope
Proficiency-testing En from ylab, xref, and already-expanded Ulab and Uref, + O3 mpmath tabulated En.
Assurance
Engineering
Declared partition coverage
PASS · 5/5 declared partitions (pass, fail, alias, awkward, invalid-domain) · Matrix
Deferred
Not zeta (standard uncertainties), not TUR, and not an ILAC G8 specification decision.
Numerical scope
O2: proficiency-testing En vs a separate-module hypot identity (≤2 ULP). Not zeta, not TUR, and not an ILAC G8 specification decision. ≤2 ULP vs O3 applies only to the published tabulated En vectors (pass, the |En|=1 boundary, fail, an alias, a negative result, one awkward pair, equal expanded uncertainties, and a close pair). It is not a whole-domain guarantee and does not cover zeta, TUR, or an ILAC G8 specification decision.
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Proficiency-testing normalized error En and |En|≤1 pass/fail.
Scope
ISO 13528 / ISO/IEC 17043 En with already-expanded Ulab and Uref
Verification
Engine tested · Source checked · v1.0.1 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · Proficiency-testing En from ylab, xref, and already-expanded Ulab and Uref, + O3 mpmath tabulated En.· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.1 · CVP protocol 1.0.0-proposed · Evidence 2026-09-24.o2-o3
Verification revision
2026-09-24.o2-o3 · 1/1 property · digest 7b7a4c158ed2
Legacy regression
9/9 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 · 4/4 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 — Normalized error En using expanded uncertainties
  • ISO/IEC 17043 — Conformity assessment — General requirements for the competence of proficiency testing providers — Performance evaluation using En
Sources
Evidence
1 legacy golden · 4 legacy boundary · legacy regression suite · 4/4 oracle-backed golden · 5/5 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.0.1 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

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

Formulas

Core equations used by this calculator.

Normalized errorEn = (ylab − xref) / √(Ulab² + Uref²)
Decision|En| ≤ 1 → pass
iUlab and Uref are already expanded. This is not the zeta-score (standard u). It is not TUR and not an ILAC G8 decision against a specification.

How to use

1

Enter ylab and xref

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

2

Enter Ulab and Uref

Already-expanded uncertainties, both greater than zero. Expand with Coverage factor if you still have u and k.

3

Read En

|En| ≤ 1 is satisfactory under ISO 13528 / ISO/IEC 17043. The page does not add a warning zone.

Example calculations

Common configurations with formula and result.

ϟ

Pass

ylab=2.5 · xref=0 · Ulab=3 · Uref=4

En=2.5/5
En=0.5 · pass
ϟ

Fail

ylab=7.5 · xref=0 · Ulab=3 · Uref=4

En=7.5/5
En=1.5 · fail

Normalized Error calculator specification

Version 1.0.1 · Engine tested

Calculation status

Review policy · Evidence

Definition
En = (ylab − xref) / √(Ulab² + Uref²). |En| ≤ 1 is pass. ylab is the laboratory result, xref the assigned value, and both U values are already expanded.
What it calculates
Proficiency-testing normalized error En and |En|≤1 pass/fail.
Inputs
  • ylab
  • xref
  • Ulab
  • Uref
Outputs
  • En
  • abs_En
  • decision
Formula
En = (ylab−xref)/√(Ulab²+Uref²); |En|≤1 → pass
Assumptions
  • ISO 13528 / ISO/IEC 17043 En with already-expanded Ulab and Uref
  • |En| ≤ 1 is pass; |En| > 1 is fail
  • Not zeta (standard u), not TUR, and not an ILAC G8 spec decision
Units
  • dimensionless En
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 → En=0.5 pass
Validation cases

2 published on this page · 9/9 tests · Production surface contract 6/6 · View evidence

  • ylab=2.5 xref=0 Ulab=3 Uref=4 → En=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 — Normalized error En using expanded uncertainties
    Supports: En = (ylab−xref)/√(Ulab²+Uref²); |En| ≤ 1 is satisfactory
  • ISO/IEC 17043 — Conformity assessment — General requirements for the competence of proficiency testing providers — Performance evaluation using En
    Supports: En 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 normalized error En.

Supported and not supported

Supported — En from ylab, xref, Ulab, Uref · |En| ≤ 1 pass · API engineering.conformity.normalized_error

Not supported — zeta-score, TUR, ILAC G8 specification decision, U = k·u_c as a product, RSS of a measurement model

Agent / API notes

Capability id: engineering.conformity.normalized_error · tool id: normalized-error · 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_ref. Errors: MISSING_REQUIRED_INPUT, INVALID_NUMBER, VALUE_MUST_BE_POSITIVE.

Calculator URL stays /calc/engineering/normalized-error. There is no /calc/metrology.

Other calculators in this family: Conformity decision, Maximum permissible error MPE, Zeta-score, Proficiency-testing z-score, Proficiency-testing z′, Measurement error, Test uncertainty ratio TUR, Coverage factor k, Uncertainty propagate, Type A uncertainty, Tolerance / Uncertainty Workspace .

DR Conformity decision › MPE Maximum permissible error MPE › ζ Zeta-score › z Proficiency-testing z-score › z′ Proficiency-testing z′ › ε Measurement error › TUR Test uncertainty ratio TUR › k Coverage factor k › u(y) Uncertainty propagate › uA Type A uncertainty › WS Tolerance / Uncertainty Workspace ›

Frequently asked questions

Key distinctions behind the calculation.

Is En the same as a zeta-score?

No. Zeta uses standard uncertainties u_lab and u_ref. En uses already-expanded Ulab and Uref. Use the Zeta-score page for ζ.

Is |En| ≤ 1 the same as an ILAC G8 spec decision?

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

Where does this run?

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