Engineering calculator

Inverse-Variance Weighted Mean Calculator

Combine independent results of the same measurand with the GUM H.3 inverse-variance weighted mean. Runs locally. Not combined RSS, not Type A u_A, and not GUM propagate.

Instant result

Result

—

Enter values to calculate.

Inputs—

Mode—

Formula—

Trust summary 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.

Input interpretation: Enter values to calculate.
Result: —
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) · Matrix
Deferred: Not combined RSS, not Type A u_A = s/√n, and not expanded U = k·u_w.
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.
Known limitations: Core CVP does not include live graph, viewport, or pointer interaction.
Model: GUM H.3 inverse-variance weighted mean and its standard uncertainty.
Scope: JCGM 100:2008 H.3 inverse-variance weighted mean of independent uncorrelated results of the same measurand
Verification: Engine tested · Source checked · v1.0.1 · 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
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
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: 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 · Manifest
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
Sources
Evidence: 1 legacy golden · 5 legacy boundary · legacy regression suite · 5/5 oracle-backed golden · 7/7 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.

Weighted meanx_w = Σ (x_i / u_i²) / Σ (1 / u_i²)

Uncertaintyu_w = 1 / √ Σ (1 / u_i²)

iχ² = Σ ((x_i − x_w)/u_i)² is a consistency diagnostic. This is not RSS of a measurement model and not Type A from repeats without individual u_i.

How to use

Enter x_i

Comma-separated independent results of the same quantity. JSON arrays also work.

Enter u_i

Same length. Each standard uncertainty must be greater than zero.

Read x̄_w and u_w

When all u_i are equal, x_w is the arithmetic mean. u_w is not the RSS of the u_i. Combine a measurement model on Uncertainty propagate.

Example calculations

Common configurations with formula and result.

Equal weights

x=[2,4,6,8] · u=[1,1,1,1]

x_w=5; u_w=1/2

x_w=5 · u_w=0.5

Single result

x=[5] · u=[2]

x_w=5; u_w=2

x_w=5 · u_w=2

Inverse-Variance Weighted Mean 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: For independent uncorrelated results x_i of the same measurand with standard uncertainties u_i > 0, x_w = Σ(x_i/u_i²)/Σ(1/u_i²) and u_w = 1/√Σ(1/u_i²).
What it calculates: GUM H.3 inverse-variance weighted mean and its standard uncertainty.
Inputs: x
u
Outputs: x_w
u_w
chi2
nu
birge
n
Formula: x_w=Σ(x_i/u_i²)/Σ(1/u_i²); u_w=1/√Σ(1/u_i²)
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
Units: same as x; u_w same unit as u_i; χ² and Birge dimensionless
Boundary conditions: missing x or u → MISSING_REQUIRED_INPUT
length mismatch → INVALID_INPUT
any u_i ≤ 0 → VALUE_MUST_BE_POSITIVE
Example: x=[2,4,6,8] u=[1,1,1,1] → x_w=5 u_w=0.5
Validation cases: 3 published on this page · 9/9 tests · Production surface contract 6/6 · View evidence
x=[2,4,6,8] u=[1,1,1,1] → x_w=5 u_w=0.5
x=[5] u=[2] → x_w=5 u_w=2
x=[2,4] u=[1,0] → VALUE_MUST_BE_POSITIVE
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
Supports: x_w = Σ(x_i/u²(x_i))/Σ(1/u²(x_i)); u²(x_w) = 1/Σ(1/u²(x_i)) for independent uncorrelated results of the same measurand
JCGM 100:2008 — Combined standard uncertainty (RSS of a measurement model) — 5.1.2 uncorrelated input quantities
Supports: RSS u_c = √Σ (c_i u_i)² is a different contract; this page does not combine a measurement model
Calculation version: 1.0.1

Background

Interpretation and common distinctions.

Compute the GUM H.3 inverse-variance weighted mean of independent results of the same measurand.

Supported and not supported

Supported — x_w and u_w from paired x_i, u_i · χ² diagnostic · API engineering.uncertainty.weighted_mean

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

Agent / API notes

Capability id: engineering.uncertainty.weighted_mean · tool id: weighted-mean · pin 1.0.1.

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

Aliases: X / values for x; u_i / ux for u. Errors: MISSING_REQUIRED_INPUT, INVALID_NUMBER, VALUE_MUST_BE_POSITIVE, INVALID_INPUT.

Calculator URL stays /calc/engineering/weighted-mean. There is no /calc/metrology.

Other calculators in this family: Type A uncertainty, Type B uncertainty, Welch–Satterthwaite ν_eff, Coverage factor k, Uncertainty propagate, Tolerance / Uncertainty Workspace .

Frequently asked questions

Key distinctions behind the calculation.

Is this the same as combined RSS?

No. RSS combines standard uncertainties of a sum or measurement model: u_c = √Σ u_i². This page averages independent results of the same measurand. Equal u_i = 1 on four results gives u_w = 0.5, while RSS of those u_i is 2.

Is this Type A uncertainty?

No. Type A takes repeats without individual u_i and reports u_A = s/√n. This page requires a standard uncertainty on each result.

Is χ² a pass/fail?

No. χ² and the Birge ratio √(χ²/ν) are diagnostics. This page does not issue a consistency verdict.

Where does this run?

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