Engineering calculator

Proficiency-Testing z′ Calculator

Compute the ISO 13528 z′ score from a laboratory result, an assigned value, the scheme SDPA σ_pt, and the standard uncertainty of that assigned value. |z′| ≤ 2 is pass. Runs locally. Not z-score, not zeta, and not En.

Instant result

Result

—

Enter values to calculate.

Inputs—

Mode—

Formula—

Trust summary Engine tested · Source checked · 12/12 tests · Production surface contract 4/4 · v1.0.0

Input interpretation: Enter values to calculate.
Result: —
Model: ISO 13528 z′ score and |z′| bands (pass / warning / fail) when u(x_pt) is not neglected.
Scope: ISO 13528 z′ = (ylab−xref)/√(σ_pt²+u(x_pt)²)
Verification: Engine tested · 12/12 tests · Production surface contract 4/4 · Source checked · v1.0.0
Named expert review: Optional · Not performed
Sources: ISO 13528 — Statistical methods for use in proficiency testing by interlaboratory comparison — z′ score when the uncertainty of the assigned value is not negligible
Sources
Evidence: 7 boundary · 5 property · Production surface contract 4/4 · Artifact integrity PASS
Production: Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (1 capability; 156 remain CURRENT) @ 2026-09-14T03:54:53.650Z
Semantic contract: PASS

Formulas

Core equations used by this calculator.

z′z′ = (ylab − xref) / √(σ_pt² + u(x_pt)²)

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

iσ_pt is the scheme SDPA and u(x_pt) is the standard uncertainty of the assigned value. This is not z (σ_pt alone), not zeta (ulab and uref), and not En (already-expanded U). u(x_pt)=0 is rejected so this page cannot wrap z-score.

How to use

Enter ylab and xref

Laboratory result and assigned value, in the same unit.

Enter σ_pt and u(x_pt)

Scheme SDPA and the standard uncertainty of the assigned value. Both must be greater than zero. Do not enter laboratory u here — that is zeta.

Read z′

|z′| ≤ 2 is satisfactory, 2 < |z′| < 3 is questionable, |z′| ≥ 3 is unsatisfactory under ISO 13528. Use z-score when u(x_pt) is neglected.

Example calculations

Common configurations with formula and result.

Pass (3-4-5)

ylab=5 · xref=0 · σ_pt=3 · u(x_pt)=4

z′=5/5

z′=1 · pass

Warning

ylab=12.5 · xref=0 · σ_pt=3 · u(x_pt)=4

z′=12.5/5

z′=2.5 · warning

Fail

ylab=15 · xref=0 · σ_pt=3 · u(x_pt)=4

z′=15/5

z′=3 · fail

Proficiency-Testing z′ calculator specification

Version 1.0.0 · Engine tested

Calculation status

Engine tested 12/12 tests · Production surface contract 4/4
Named expert review Not performed
Calculation version 1.0.0

Review policy · Evidence

Definition: z′ = (ylab − xref) / √(σ_pt² + u(x_pt)²). |z′| ≤ 2 is pass, 2 < |z′| < 3 is warning, |z′| ≥ 3 is fail. Both σ_pt and u(x_pt) must be greater than zero. This is the ISO 13528 score used when the uncertainty of the assigned value is not neglected.
What it calculates: ISO 13528 z′ score and |z′| bands (pass / warning / fail) when u(x_pt) is not neglected.
Inputs: ylab
xref
sigma
u_xref
Outputs: z_prime
abs_z_prime
decision
denom
Formula: z′=(ylab−xref)/√(σ_pt²+u_X²); |z′|≤2 pass; 2<|z′|<3 warning; |z′|≥3 fail
Assumptions: ISO 13528 z′ = (ylab−xref)/√(σ_pt²+u(x_pt)²)
σ_pt > 0 and u(x_pt) > 0 — u(x_pt)=0 is not a z-score wrap
Not z-score, not zeta, and not En
Units: dimensionless z′
Boundary conditions: missing ylab, xref, sigma, or u_xref → MISSING_REQUIRED_INPUT
σ_pt ≤ 0 or u(x_pt) ≤ 0 → VALUE_MUST_BE_POSITIVE
ulab is not an alias for u_xref
Example: ylab=5 xref=0 sigma=3 u_xref=4 → z′=1 pass
Validation cases: 3 published on this page · 12/12 tests · Production surface contract 4/4 · View evidence
ylab=5 xref=0 sigma=3 u_xref=4 → z′=1 pass
ylab=12.5 xref=0 sigma=3 u_xref=4 → z′=2.5 warning
ylab=5 xref=0 sigma=3 u_xref=0 → VALUE_MUST_BE_POSITIVE
Sources: ISO 13528 — Statistical methods for use in proficiency testing by interlaboratory comparison — z′ score when the uncertainty of the assigned value is not negligible
Supports: z′ = (x − x_pt) / √(σ_pt² + u²(x_pt)). |z′| bands match the z-score bands.
Calculation version: 1.0.0

Background

Interpretation and common distinctions.

Compute the ISO 13528 z′ score when the assigned value has a stated standard uncertainty that is not neglected.

Supported and not supported

Supported — z′ = (ylab−xref)/√(σ_pt²+u_X²) · |z′| bands · API engineering.conformity.z_prime

Not supported — z-score (σ_pt alone), zeta (ulab+uref), En (expanded U), u(x_pt)=0 as a wrap of z, ILAC G8 spec decision

Agent / API notes

Capability id: engineering.conformity.z_prime · tool id: z-prime · pin 1.0.0.

{ "ylab": 5, "xref": 0, "sigma": 3, "u_xref": 4 }

Aliases: y / x for ylab; X / ref for xref; sigma_pt / sdpa / s_pt for sigma; u_xpt / uxpt / u_assigned for u_xref. ulab is not an alias. Errors: MISSING_REQUIRED_INPUT, INVALID_NUMBER, VALUE_MUST_BE_POSITIVE.

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

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

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Frequently asked questions

Key distinctions behind the calculation.

Is z′ the same as z?

No. z divides by σ_pt alone. z′ divides by √(σ_pt²+u(x_pt)²) because the assigned value is not treated as exact. u(x_pt)=0 is rejected here; use the z-score page when that uncertainty is neglected.

Is z′ the same as zeta?

No. Zeta uses the laboratory standard uncertainty ulab together with uref. z′ uses the scheme SDPA σ_pt and u(x_pt); it does not take ulab.

Is z′ the same as En?

No. En uses already-expanded Ulab and Uref, and |En| ≤ 1 is pass. This page uses standard-uncertainty SDPA plus u(x_pt) and the ISO 13528 |z′| ≤ 2 / warning / fail bands.

Where does this run?

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