CVP 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 1.0.1 CVP VERIFIED CURRENT Engineering

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	—	input / output
`u`	Standard uncertainty of each result, same length as x	—	input / output
`x_w`	—	—	output
`u_w`	—	—	output
`chi2`	—	—	output
`nu`	—	—	output
`birge`	√	—	output
`n`	—	—	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

CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · JCGM 100 H.3 x_w = Σ(x_i/u_i²)/Σ(1/u_i²) and u_w = 1/√Σ(1/u_i²), plus χ² and Birge diagnostics, + O3 mpmath tabulated x_w.. Computational verification is defined by CVP. Supplemental domain review (internal) and named expert review are defined by the Review policy and do not by themselves determine CVP conformance.

CVP: CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · JCGM 100 H.3 x_w = Σ(x_i/u_i²)/Σ(1/u_i²) and u_w = 1/√Σ(1/u_i²), plus χ² and Birge diagnostics, + O3 mpmath tabulated x_w.· View Manifest · CVP overview · Specification
Verified scope: JCGM 100 H.3 x_w = Σ(x_i/u_i²)/Σ(1/u_i²) and u_w = 1/√Σ(1/u_i²), plus χ² and Birge diagnostics, + O3 mpmath tabulated x_w.
Assurance: Engineering
Declared partition coverage: PASS · 5/5 declared partitions (equal-weights, unequal-weights, single-result, alias, invalid-domain)
Deferred: Not combined RSS, not Type A u_A = s/√n, and not expanded U = k·u_w.
Calculation version: 1.0.1
Verification: 9/9 calculation tests · 1/1 legacy golden · 5/5 legacy boundary · 3/3 legacy property
Production surface contract: 6/6
Versions: Calculation 1.0.1 · CVP protocol 1.0.0-proposed · Evidence 2026-09-23.o2-o3
Verification revision: 2026-09-23.o2-o3 · 2/2 property · digest 77bc67046423
Reference: O1 model · O3 expected_values · O3 numerical_behavior · O2 expected_values · O2 numerical_behavior
Numerical scope: O2: JCGM 100 H.3 inverse-variance weighted mean vs separate-module identities (≤2 ULP). Not RSS, not Type A, and not U=k·u_w. ≤2 ULP vs O3 applies only to the published tabulated H.3 vectors (equal weights, unequal weights, a single result, aliases, a signed measurand, doubled uncertainties, and one awkward triple). It is not a whole-domain guarantee and does not cover RSS, Type A, or U=k·u_w.
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.
CVP suite: 5/5 golden · 5/5 CVP boundary · 7/7 invalid · 2/2 property · 2/2 metamorphic · 8/8 O3 · 4/4 cross-interface · 6/6 CVP contract
Supplemental domain review: Not performed
Named expert review: Not performed
Version lock: Supported (409 VERSION_MISMATCH)
Artifact consistency: Build verified · 1.0.1 (workspace artifacts agree)
Review-policy tier: Engine tested — catalog token engine_tested; supplemental; not CVP status

This calculator CURRENT · Public schema 1.0.1 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓

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 6/6.

curl -sS -X POST 'https://www.calculatorx.com/api/v1/calc/weighted-mean' \
  -H 'Content-Type: application/json' \
  -d '{"inputs":{"x":[2,4,6,8],"u":[1,1,1,1]},"calculation_version":"1.0.1"}'

{
  "inputs": {
    "x": [
      2,
      4,
      6,
      8
    ],
    "u": [
      1,
      1,
      1,
      1
    ]
  },
  "calculation_version": "1.0.1"
}

POST /api/v1/calc/weighted-mean. This is a POST calculate endpoint — do not GET it.

Full REST contract: REST API docs.

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 (1.0.1)
Versioned output schema: Immutable pin (1.0.1)

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

Reference evidence classes: O1 model · O3 expected_values · O3 numerical_behavior · O2 expected_values · O2 numerical_behavior

Try with an agent

Inspect, then execute. Same engine as REST and the human calculator.

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"},"_meta":{"io.modelcontextprotocol/protocolVersion":"2026-07-28","io.modelcontextprotocol/clientInfo":{"name":"calculatorx-docs-example","version":"1.0.0"},"io.modelcontextprotocol/clientCapabilities":{}}}}'

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: execute_calculation' \
  -d '{"jsonrpc":"2.0","id":2,"method":"tools/call","params":{"name":"execute_calculation","arguments":{"capability_id":"engineering.uncertainty.weighted_mean","inputs":{"x":[2,4,6,8],"u":[1,1,1,1]},"calculation_version":"1.0.1"},"_meta":{"io.modelcontextprotocol/protocolVersion":"2026-07-28","io.modelcontextprotocol/clientInfo":{"name":"calculatorx-docs-example","version":"1.0.0"},"io.modelcontextprotocol/clientCapabilities":{}}}}'

Pin calculation_version (1.0.1) when the result must stay reproducible.

Inspect the trust tier and required assurance level before execution. Full protocol notes: MCP docs.