Engineering calculator

Welch–Satterthwaite Calculator

Compute GUM G.4 effective degrees of freedom ν_eff from standard uncertainty components u_i and their ν_i. Runs locally. Not combined RSS and not expanded U = k·u_c.

Instant result

Result

—

Enter values to calculate.

Inputs—

Mode—

Formula—

Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · JCGM 100 G.4 ν_eff = u_c⁴ / Σ(u_i⁴/ν_i), including infinite ν_i and an optional supplied u_c, + O3 mpmath tabulated ν_eff.

Input interpretation: Enter values to calculate.
Result: —
Verified scope: JCGM 100 G.4 ν_eff = u_c⁴ / Σ(u_i⁴/ν_i), including infinite ν_i and an optional supplied u_c, + O3 mpmath tabulated ν_eff.
Assurance: Engineering
Declared partition coverage: PASS · 6/6 declared partitions (finite-rss, infinite-component, supplied-uc, all-infinite, alias, invalid-domain) · Matrix
Deferred: Not GUM model combination, not expanded U = k·u_c, and not a t-table lookup of k.
Numerical scope: O2: JCGM 100 G.4 ν_eff = u_c⁴ / Σ(u_i⁴/ν_i) vs separate-module identities (≤2 ULP). Not GUM propagate and not U=k·u_c. ≤2 ULP vs O3 applies only to the published tabulated G.4 vectors (equal Type A, one infinite Type B, supplied u_c, all-infinite, aliases, three components, scale, a zero component). It is not a whole-domain guarantee and does not cover GUM propagate or U=k·u_c.
Known limitations: Core CVP does not include live graph, viewport, or pointer interaction.
Model: Welch–Satterthwaite effective degrees of freedom ν_eff from u_i, ν_i, and optional u_c.
Scope: JCGM 100:2008 G.4 Welch–Satterthwaite
Verification: Engine tested · Source checked · v1.0.1 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · JCGM 100 G.4 ν_eff = u_c⁴ / Σ(u_i⁴/ν_i), including infinite ν_i and an optional supplied u_c, + O3 mpmath tabulated ν_eff.· 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 · 3/3 property · digest 4fa90ba29b01
Legacy regression: 10/10 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 · 8/8 invalid · 3/3 property · 2/2 metamorphic · 8/8 O3 · 4/4 cross-interface · 6/6 CVP contract · Manifest
Sources: JCGM 100:2008 (GUM) — Evaluation of measurement data — G.4 Effective degrees of freedom: the Welch–Satterthwaite formula
NIST Technical Note 1297 — Effective degrees of freedom
Sources
Evidence: 1 legacy golden · 5 legacy boundary · legacy regression suite · 5/5 oracle-backed golden · 8/8 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.

Effective degrees of freedomν_eff = u_c⁴ / Σ (u_i⁴ / ν_i)

Combined (when uc omitted)u_c = √(Σ u_i²)

iInfinite ν_i (typical Type B from a bound) contribute 0 to the denominator. This page does not look up k from a t-table and does not compute expanded U.

How to use

Enter u_i

Comma-separated standard uncertainties. Each u_i ≥ 0.

Enter ν_i

Same length. Use inf for a Type B component with infinite degrees of freedom.

Read ν_eff

Combine a measurement model on Uncertainty propagate. Expanded U = k·u_c is that page, not this one.

Example calculations

Common configurations with formula and result.

Two equal Type A

u=[1,1] ν=[8,8]

ν_eff=16

ν_eff=16 · u_c=√2

One infinite Type B

u=[0.1,0.2] ν=[10,inf]

ν_eff=u_c⁴/(0.1⁴/10)

ν_eff=250

Welch–Satterthwaite calculator specification

Version 1.0.1 · Engine tested

Calculation status

Engine tested 10/10 tests · Production surface contract 6/6
Named expert review Not performed
Calculation version 1.0.1

Review policy · Evidence

Definition: For uncorrelated standard uncertainties u_i with degrees of freedom ν_i, ν_eff = u_c⁴ / Σ (u_i⁴ / ν_i), where u_c is the combined standard uncertainty (RSS of u_i unless you enter uc).
What it calculates: Welch–Satterthwaite effective degrees of freedom ν_eff from u_i, ν_i, and optional u_c.
Inputs: u
nu
uc
Outputs: nu_eff
infinite
uc
u
nu
Formula: ν_eff = u_c⁴ / Σ (u_i⁴ / ν_i)
Assumptions: JCGM 100:2008 G.4 Welch–Satterthwaite
If uc omitted, u_c = RSS of uncorrelated u_i
Infinite ν_i contribute 0 to the sum
Not GUM model combination and not expanded U = k·u_c
Units: dimensionless ν_eff; u_i same unit
Boundary conditions: missing u or nu → MISSING_REQUIRED_INPUT
length mismatch → INVALID_INPUT
u_c = 0 → VALUE_MUST_BE_POSITIVE
ν_i ≤ 0 and not inf → VALUE_MUST_BE_POSITIVE
Example: u=[1,1] nu=[8,8] → ν_eff=16
Validation cases: 2 published on this page · 10/10 tests · Production surface contract 6/6 · View evidence
u=1,1 nu=8,8 → ν_eff=16
u=0,0 nu=8,8 → VALUE_MUST_BE_POSITIVE
Sources: JCGM 100:2008 (GUM) — Evaluation of measurement data — G.4 Effective degrees of freedom: the Welch–Satterthwaite formula
Supports: ν_eff = u_c⁴ / Σ (u_i⁴ / ν_i) for the coverage factor of expanded uncertainty
NIST Technical Note 1297 — Effective degrees of freedom
Supports: Welch–Satterthwaite combination of component degrees of freedom
Calculation version: 1.0.1

Background

Interpretation and common distinctions.

Compute Welch–Satterthwaite effective degrees of freedom.

Supported and not supported

Supported — ν_eff from u_i and ν_i · optional uc · infinite Type B dof · API engineering.uncertainty.welch_satterthwaite

Not supported — GUM model combination, expanded U = k·u_c, t-table k lookup, correlated extra terms

Agent / API notes

Capability id: engineering.uncertainty.welch_satterthwaite · tool id: welch-satterthwaite · pin 1.0.1.

{ "u": [1, 1], "nu": [8, 8] }

nu may include inf (JSON null on output). Errors: MISSING_REQUIRED_INPUT, INVALID_NUMBER, INVALID_INPUT, VALUE_MUST_BE_POSITIVE.

Calculator URL stays /calc/engineering/welch-satterthwaite. There is no /calc/metrology.

Other calculators in this family: Type A uncertainty, Pooled standard deviation, Type B uncertainty, Type B degrees of freedom, Inverse-variance weighted mean, Coverage factor k, Uncertainty propagate, Guard band, Tolerance / Uncertainty Workspace .

Frequently asked questions

Key distinctions behind the calculation.

Is this the same as Uncertainty propagate?

No. This page only evaluates Welch–Satterthwaite ν_eff from already-evaluated u_i and ν_i. Uncertainty propagate combines a measurement model into u_c (and optional U = k·u_c). Combined RSS is an engine op, not a calculator page.

Is this pooled s?

No. Pooled s_p combines sample standard deviations. This page takes standard uncertainties u_i and returns ν_eff.

Why can ν_eff be infinite?

If every finite-u component has infinite ν_i, the GUM denominator is 0 and ν_eff is infinite. REST returns nu_eff=null with infinite=true.

Where does this run?

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