Engineering calculator

Proficiency-Testing z-Score Calculator

Q: Is z the same as zeta?

No. z divides by the scheme SDPA σ_pt. Zeta divides by √(ulab²+uref²), the laboratory and assigned-value standard uncertainties.

Q: Is z the same as En?

No. En uses already-expanded Ulab and Uref, and |En| ≤ 1 is pass. This page uses σ_pt and the ISO 13528 |z| ≤ 2 / warning / fail bands.

Q: Is z the same as z′?

No. z′ divides by √(σ_pt²+u(x_pt)²) when the assigned-value uncertainty is not neglected. This page ignores u(x_pt).

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

No. A z-score 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 z_score op.

Compute the ISO 13528 proficiency-testing z-score from a laboratory result, an assigned value, and the scheme SDPA σ_pt. |z| ≤ 2 is pass. Runs locally. Not zeta, not En, 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 z-score from ylab, xref, and a scheme σ_pt, + O3 mpmath tabulated z-score.

Input interpretation: Enter values to calculate.
Result: —
Verified scope: Proficiency-testing z-score from ylab, xref, and a scheme σ_pt, + O3 mpmath tabulated z-score.
Assurance: Engineering
Declared partition coverage: PASS · 5/5 declared partitions (pass, fail, alias, awkward, invalid-domain) · Matrix
Deferred: Not zeta (laboratory u), not En (expanded U), and not an ILAC G8 specification decision.
Numerical scope: O2: proficiency-testing z-score vs a separate-module division identity (≤2 ULP). Not zeta, not En, and not an ILAC G8 specification decision. ≤2 ULP vs O3 applies only to the published tabulated z-score vectors (pass, the |z|=2 boundary, warning, fail, an alias, a negative result, one awkward pair, and another |z|=2 pair). It is not a whole-domain guarantee and does not cover zeta, En, or an ILAC G8 specification decision.
Known limitations: Core CVP does not include live graph, viewport, or pointer interaction.
Model: Proficiency-testing z-score and ISO 13528 |z| bands (pass / warning / fail).
Scope: ISO 13528 z-score with scheme SDPA σ_pt
Verification: Engine tested · Source checked · v1.0.1 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · Proficiency-testing z-score from ylab, xref, and a scheme σ_pt, + O3 mpmath tabulated z-score.· 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 e0f1dac3e62d
Legacy regression: 13/13 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 · 6/6 CVP boundary · 6/6 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 — z-score z = (x − X) / σ_pt
ISO/IEC 17043 — Conformity assessment — General requirements for the competence of proficiency testing providers — Performance statistics in proficiency testing
Sources
Evidence: 1 legacy golden · 6 legacy boundary · legacy regression suite · 4/4 oracle-backed golden · 6/6 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.

z-scorez = (ylab − xref) / σ_pt

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

iσ_pt is the scheme SDPA. This is not zeta (that uses √(ulab²+uref²)). It is not En (already-expanded U), 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 σ_pt

Scheme standard deviation for proficiency assessment, greater than zero. Do not enter laboratory u or expanded U here — those are zeta and En.

Read z

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

Example calculations

Common configurations with formula and result.

Pass

ylab=2.5 · xref=0 · σ_pt=5

z=2.5/5

z=0.5 · pass

Warning

ylab=12.5 · xref=0 · σ_pt=5

z=12.5/5

z=2.5 · warning

Fail

ylab=15 · xref=0 · σ_pt=5

z=15/5

z=3 · fail

Proficiency-Testing z-Score calculator specification

Version 1.0.1 · Engine tested

Calculation status

Engine tested 13/13 tests · Production surface contract 6/6
Named expert review Not performed
Calculation version 1.0.1

Review policy · Evidence

Definition: z = (ylab − xref) / σ_pt. |z| ≤ 2 is pass, 2 < |z| < 3 is warning, |z| ≥ 3 is fail. ylab is the laboratory result, xref the assigned value, and σ_pt is the scheme standard deviation for proficiency assessment.
What it calculates: Proficiency-testing z-score and ISO 13528 |z| bands (pass / warning / fail).
Inputs: ylab
xref
sigma
Outputs: z
abs_z
decision
Formula: z = (ylab−xref)/σ_pt; |z|≤2 pass; 2<|z|<3 warning; |z|≥3 fail
Assumptions: ISO 13528 z-score with scheme SDPA σ_pt
|z| ≤ 2 is pass; 2 < |z| < 3 is warning; |z| ≥ 3 is fail
Not zeta (lab u), not En (expanded U), not TUR, and not an ILAC G8 spec decision
Units: dimensionless z
Boundary conditions: missing ylab, xref, or sigma → MISSING_REQUIRED_INPUT
sigma ≤ 0 → VALUE_MUST_BE_POSITIVE
ulab / Ulab are not aliases for σ_pt
Example: ylab=2.5 xref=0 sigma=5 → z=0.5 pass
Validation cases: 2 published on this page · 13/13 tests · Production surface contract 6/6 · View evidence
ylab=2.5 xref=0 sigma=5 → z=0.5 pass
ylab=2.5 xref=0 sigma=0 → VALUE_MUST_BE_POSITIVE
Sources: ISO 13528 — Statistical methods for use in proficiency testing by interlaboratory comparison — z-score z = (x − X) / σ_pt
Supports: z uses the standard deviation for proficiency assessment, not laboratory u; |z| ≤ 2 / 2–3 / ≥ 3 interpretation
ISO/IEC 17043 — Conformity assessment — General requirements for the competence of proficiency testing providers — Performance statistics in proficiency testing
Supports: z-score is a PT performance statistic distinct from En and zeta
Calculation version: 1.0.1

Background

Interpretation and common distinctions.

Compute the ISO 13528 proficiency-testing z-score.

Supported and not supported

Supported — z from ylab, xref, and σ_pt · ISO 13528 |z| bands · API engineering.conformity.z_score

Not supported — zeta (lab u), En (expanded U), TUR, ILAC G8 spec decision, computing σ_pt from a PT dataset

Agent / API notes

Capability id: engineering.conformity.z_score · tool id: z-score · pin 1.0.1.

{ "ylab": 2.5, "xref": 0, "sigma": 5 }

Aliases: y / x for ylab; X / ref for xref; sigma_pt / sdpa / s_pt for sigma. ulab and Ulab are not aliases. Errors: MISSING_REQUIRED_INPUT, INVALID_NUMBER, VALUE_MUST_BE_POSITIVE.

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

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

Frequently asked questions

Key distinctions behind the calculation.

Is z the same as zeta?