Verified calculation capability

Inverse-variance weighted mean Calculation Capability

GUM H.3 inverse-variance weighted mean of independent results of the same measurand: x_w = Σ(x_i/u_i²)/Σ(1/u_i²), u_w = 1/√Σ(1/u_i²). Not combined RSS, not Type A u_A = s/√n, and not GUM propagate. · Engineering · Uncertainty

Capability ID engineering.uncertainty.weighted_mean · version 1.0.0

Open calculator REST API MCP Evidence Evidence JSON Review policy

Model

Weighted mean

Weighted mean: x_w = Σ (x_i / u_i²) / Σ (1 / u_i²)
Uncertainty: u_w = 1 / √ Σ (1 / u_i²)

Inputs / Outputs

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

Symbol	Quantity	Unit	Role
`x`	Independent results of the same measurand. Aliases: X, values. Share-URL: comma-separated.	—	input / output
`u`	Standard uncertainty of each result, same length as x. Aliases: u_i, ux. Each u_i > 0.	—	input / output
`x_w`	—	—	output
`u_w`	—	—	output
`chi2`	—	—	output
`nu`	—	—	output
`birge`	√(χ²/ν). null when n=1.	—	output
`n`	amount of substance	—	output

Assumptions / Validity

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

Assumptions

JCGM 100:2008 H.3 inverse-variance weighted mean of independent uncorrelated results of the same measurand
Each u_i is a standard uncertainty greater than zero
Not combined RSS, not Type A u_A = s/√n, and not GUM propagate
χ² / Birge are diagnostics, not a pass/fail

Not supported

combined RSS as a product
Type A u_A = s/√n
GUM propagate of a model
U = k·u_w as a product
Birge pass/fail

Example

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

x=[2,4,6,8] u=[1,1,1,1] → x_w=5 u_w=0.5

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: weighted-mean
Engine: weighted-mean
REST: POST /api/v1/calc/weighted-mean
MCP: execute_calculation with capability_id: "engineering.uncertainty.weighted_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

JCGM 100:2008 — Evaluation of measurement data — Guide to the expression of uncertainty in measurement (GUM) — H.3 Measurements sampled from a distribution: weighted mean
JCGM 100:2008 — Combined standard uncertainty (RSS of a measurement model) — 5.1.2 uncorrelated input quantities

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.weighted_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.