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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
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Source checked · 8/8 tests · Production surface contract 4/4 · v1.0.0
Input interpretation
Enter values to calculate.
Result
—
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 · 8/8 tests · Production surface contract 4/4 · Source checked · v1.0.0
Named expert review
Optional · Not performed
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
5 boundary · 3 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Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (1 capability; 156 remain CURRENT)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.

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

1

Enter s_r

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

2

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).

3

Enter n

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

4

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.

ϟ

3-4-5 mean of four

s_r=8 · s_L=3 · n=4

u=√(3²+(8/√4)²)=√(9+16)
u=5
ϟ

No between-lab term

s_r=8 · s_L=0 · n=4

u=8/√4
u=4
ϟ

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 1.0.0 · Engine tested

Calculation status

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 · 8/8 tests · Production surface contract 4/4 · View evidence

  • s_r=8 s_L=3 n=4 → u=5
  • 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
1.0.0

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 1.0.0.

{ "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 .

R ISO 5725 reproducibility › s Measurement precision › uA Type A uncertainty › s_p Pooled standard deviation › νeff Welch–Satterthwaite ν_eff › k Coverage factor k › x̄w Inverse-variance weighted mean › WS Tolerance / Uncertainty Workspace ›
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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.