Engineering calculator

Uncertainty of a Mean from Precision Calculator

Compute the ISO 21748 standard uncertainty of a mean of n≥2 replicates from stated precision components s_r and s_L. Runs locally. Not R = 2.8·s_R, not Type A from a sample, and not U = k·u.

Instant result

Result

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Enter values to calculate.

Inputs—

Mode—

Formula—

Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · ISO 21748 u = √(s_L² + s_r²/n) for a mean of n≥2 replicates from stated s_r and s_L, + O3 mpmath tabulated u.

Input interpretation: Enter values to calculate.
Result: —
Verified scope: ISO 21748 u = √(s_L² + s_r²/n) for a mean of n≥2 replicates from stated s_r and s_L, + O3 mpmath tabulated u.
Assurance: Engineering
Declared partition coverage: PASS · 5/5 declared partitions (both, repeatability-only, laboratory-only, alias, invalid-domain) · Matrix
Deferred: Not R = 2.8·s_R, not Type A from a sample, and not U = k·u.
Numerical scope: O2: ISO 21748 u = √(s_L² + s_r²/n) vs a separate-module identity (≤2 ULP). Not R = 2.8·s_R, not Type A from a sample, and not U = k·u. ≤2 ULP vs O3 applies only to the published tabulated ISO 21748 vectors (both components, repeatability only, laboratory only, an alias, and one awkward mean). It is not a whole-domain guarantee and does not cover R = 2.8·s_R, Type A from a sample, or U = k·u.
Known limitations: Core CVP does not include live graph, viewport, or pointer interaction.
Model: ISO 21748 standard uncertainty of a mean from stated s_r, s_L, and n≥2.
Scope: ISO 21748: u = √(s_L² + s_r²/n) of a mean of n replicates from stated precision components. n≥2.
Verification: Engine tested · Source checked · v1.0.1 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · ISO 21748 u = √(s_L² + s_r²/n) for a mean of n≥2 replicates from stated s_r and s_L, + O3 mpmath tabulated u.· View Manifest · CVP overview · Specification
Versions: Calculation 1.0.1 · CVP protocol 1.0.0-proposed · Evidence 2026-09-23.o2-o3
Verification revision: 2026-09-23.o2-o3 · 1/1 property · digest 0f27e2e3a0f3
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 · 5/5 CVP boundary · 6/6 invalid · 1/1 property · 1/1 metamorphic · 8/8 O3 · 4/4 cross-interface · 6/6 CVP contract · Manifest
Sources: ISO 21748 — Guidance for the use of repeatability, reproducibility and trueness estimates in measurement uncertainty evaluation — Uncertainty of a mean from precision components
ISO 5725-1 — Accuracy (trueness and precision) of measurement methods and results — Repeatability s_r and reproducibility s_R
Sources
Evidence: 1 legacy golden · 5 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.

Uncertainty of the meanu = √(s_L² + s_r²/n)

in must be at least 2 so this cannot wrap s_R. This page takes stated components — not a sample of observations. It does not report R = 2.8·s_R and does not expand U = k·u.

How to use

Enter s_r

Repeatability standard deviation, already computed. Aliases: sr, s_repeatability, repeatability. Must be ≥ 0.

Enter s_L

Between-laboratory or between-condition standard deviation. Aliases: sL, s_lab, s_between. Must be ≥ 0. Zero is valid (then u = s_r/√n).

Enter n

Number of replicates in the mean. Aliases: n_rep, n_mean, replicates. Must be ≥ 2.

Read u

u is the standard uncertainty of the mean. Use Reproducibility when you need R = 2.8·s_R of a single result. Use Type A when you have a sample of observations.

Example calculations

Common configurations with formula and result.

3-4-5 mean of four

s_r=8 · s_L=3 · n=4

u=√(3²+(8/√4)²)=√(9+16)

u=5

No between-lab term

s_r=8 · s_L=0 · n=4

u=8/√4

u=4

n=1 rejected

s_r=8 · s_L=3 · n=1

n≥2

VALUE_OUT_OF_RANGE

Uncertainty of a Mean from Precision calculator specification

Version 1.0.1 · Engine tested

Calculation status

Engine tested 9/9 tests · Production surface contract 6/6
Named expert review Not performed
Calculation version 1.0.1

Review policy · Evidence

Definition: From stated repeatability s_r ≥ 0, between-laboratory s_L ≥ 0, and n ≥ 2 replicates in the mean, u = √(s_L² + s_r²/n).
What it calculates: ISO 21748 standard uncertainty of a mean from stated s_r, s_L, and n≥2.
Inputs: s_r
s_L
n
Outputs: u
s_r
s_L
n
Formula: u=√(s_L²+s_r²/n)
Assumptions: ISO 21748: u = √(s_L² + s_r²/n) of a mean of n replicates from stated precision components. n≥2.
s_r and s_L are ≥ 0. Both zero is valid (u = 0).
Not R = 2.8·s_R, not Type A from a sample, and not U = k·u.
Units: same as s_r / s_L
Boundary conditions: missing s_r, s_L, or n → MISSING_REQUIRED_INPUT
n = 1 → VALUE_OUT_OF_RANGE
n = 0 → VALUE_MUST_BE_POSITIVE
negative s_r or s_L → VALUE_OUT_OF_RANGE
s_L = 0 is valid
Example: s_r=8 s_L=3 n=4 → u=5
Validation cases: 3 published on this page · 9/9 tests · Production surface contract 6/6 · View evidence
s_r=8 s_L=3 n=4 → u=5, s_r=8, s_L=3, n=4
s_r=8 s_L=0 n=4 → u=4
s_r=8 s_L=3 n=1 → VALUE_OUT_OF_RANGE
Sources: ISO 21748 — Guidance for the use of repeatability, reproducibility and trueness estimates in measurement uncertainty evaluation — Uncertainty of a mean from precision components
Supports: u = √(s_L² + s_r²/n) is the standard uncertainty of a mean of n replicates when s_r and s_L are already stated.
ISO 5725-1 — Accuracy (trueness and precision) of measurement methods and results — Repeatability s_r and reproducibility s_R
Supports: s_r and s_L are the same precision components used for R = 2.8·s_R. This page reports u of a mean, not R.
Calculation version: 1.0.1

Background

Interpretation and common distinctions.

Compute the ISO 21748 standard uncertainty of a mean from stated repeatability and between-laboratory components.

Supported and not supported

Supported — u from paired s_r, s_L, and n≥2 · s_L = 0 is valid · API engineering.uncertainty.precision_mean

Not supported — R = 2.8·s_R, Type A from a sample, n=1 wrap of s_R, U = k·u, combined RSS

Agent / API notes

Capability id: engineering.uncertainty.precision_mean · tool id: precision-uncertainty · pin 1.0.1.

{ "s_r": 8, "s_L": 3, "n": 4 }

Aliases: sr / s_repeatability / repeatability for s_r; sL / s_lab / s_between for s_L; n_rep / n_mean / replicates for n. Errors: MISSING_REQUIRED_INPUT, INVALID_NUMBER, VALUE_OUT_OF_RANGE, VALUE_MUST_BE_POSITIVE.

Calculator URL stays /calc/engineering/precision-uncertainty. There is no /calc/metrology.

Other calculators in this family: ISO 5725 reproducibility, Measurement precision, Type A uncertainty, Pooled standard deviation, Welch–Satterthwaite ν_eff, Coverage factor k, Inverse-variance weighted mean, Tolerance / Uncertainty Workspace .

Frequently asked questions

Key distinctions behind the calculation.

Is this the same as reproducibility / R?

No. Reproducibility reports R = 2.8·s_R of a single result. This page takes the same s_r and s_L plus n≥2 and reports the standard uncertainty of the mean.

Is this Type A uncertainty?

No. Type A takes a sample of observations and reports u_A = s/√n. This page takes already-stated s_r, s_L, and n. n=1 is rejected so it cannot wrap s_R.

Is this pooled s from several series?

No. Pooled s_p combines k≥2 sample standard deviations. This page takes one pair (s_r, s_L) plus n and reports u of a mean.

Is this U = k·u?

No. This page reports the standard uncertainty u. Apply a coverage factor on Coverage factor k. Combined RSS is a different contract.

Why n≥2?

With n=1, u = √(s_L²+s_r²) which is s_R. That wrap is rejected. Use Reproducibility for R = 2.8·s_R.

Where does this run?

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