Verified calculation capability

Uncertainty of a mean from precision Calculation Capability

ISO 21748 u = √(s_L²+s_r²/n) of a mean of n≥2 replicates from stated precision components. Not R = 2.8·s_R, not Type A from a sample, and not U = k·u. · Engineering · Uncertainty

Capability ID engineering.uncertainty.precision_mean · version 1.0.0

Open calculator REST API MCP Evidence Evidence JSON Review policy

Model

Uncertainty of the mean

Uncertainty of the mean: u = √(s_L² + s_r²/n)

Inputs / Outputs

Canonical symbols for this capability. Agent schemas remain the contract.

Symbol	Quantity	Unit	Role
`s_r`	Repeatability standard deviation. Aliases: sr, s_repeatability, repeatability.	—	input / output
`s_L`	Between-laboratory standard deviation. Aliases: sL, s_lab, s_between.	—	input / output
`n`	Number of replicates in the mean. Aliases: n_rep, n_mean, replicates. Must be ≥ 2 so this cannot wrap s_R.	—	input / output
`u`	Standard uncertainty of the mean √(s_L²+s_r²/n)	—	output

Assumptions / Validity

Declared scope for this capability. Do not use it outside this domain.

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.

Not supported

R = 2.8·s_R
Type A from a sample
n=1 wrap of s_R
U = k·u
combined RSS

Example

One numerical check. The calculator page holds worked steps and FAQ.

s_r=8 s_L=3 n=4 → u=5

Open this example in the calculator

Trust

Review-policy: engine tested — automated tests, sources, version lock, and evidence published. Supplemental to CVP; not Domain Expert Review. See CVP and Review policy.

Tier: Engine tested — catalog token engine_tested
Calculation version: 1.0.0
Evidence tests: 8/8 calculation tests · golden 0/0 · boundary 5/5 · property 3/3
Engine tested: Yes
Source checked: Yes
Supplemental domain review: Not performed
Named expert review: Not performed
Version lock: Supported (409 VERSION_MISMATCH)
Artifact consistency: Build verified · 1.0.0 (workspace artifacts agree)

Production attestation: Embedded snapshot: STALE · Last attested schema matched 1.0.0 snapshot / local build · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · Public/cache ✓ · Origin ✓ · Live production status PASS (0 stale; 164 CURRENT) @ 2026-09-16T06:44:24.909Z
Semantic contract: PASS
Evidence attested at: 2026-09-16T06:44:24.203Z
Last production check: 2026-09-16T06:44:24.203Z
Public / cache: PASS
Origin: PASS

Call this capability

Human page, REST, and MCP share the same engine and calculation_version. Build verified hashes workspace artifacts only. After deploy, attest:production GETs canonical URLs twice — the public/cached view an Agent actually hits, and origin — including every /capabilities/{id} HTML page. Public Capability HTML version must match catalog before production attestation passes. Live Trust rows above fetch /.well-known/calculatorx-production-attestation.json and the output schema const. Build hashes: release integrity. That proves currently published page / engine / schema / evidence / MCP agree — not that a historical version is independently immutable. Reference Calculation Standard — Production surface contract 4/4.

Tool id: precision-uncertainty
Engine: precision-uncertainty
REST: POST /api/v1/calc/precision-uncertainty
MCP: execute_calculation with capability_id: "engineering.uncertainty.precision_mean"
Agent input schema: Canonical (MCP / Agent)
Output schema: JSON Schema (latest)
Versioned input schema: Immutable pin (1.0.0)
Versioned output schema: Immutable pin (1.0.0)

References

Human-readable sources plus CVP reference evidence classes.

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

Try with an agent

Inspect this capability via same-origin MCP, then call execute_calculation.

curl -sS -X POST 'https://www.calculatorx.com/mcp' \
  -H 'Content-Type: application/json' \
  -H 'MCP-Protocol-Version: 2026-07-28' \
  -H 'Mcp-Method: tools/call' \
  -H 'Mcp-Name: inspect_capability' \
  -d '{"jsonrpc":"2.0","id":1,"method":"tools/call","params":{"name":"inspect_capability","arguments":{"capability_id":"engineering.uncertainty.precision_mean"}}}'

Inspect the trust tier and required assurance level before execution. Pin calculation_version (1.0.0) when reproducibility matters. Full protocol notes: MCP docs.