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Engineering calculator

Type B Degrees of Freedom Calculator

Compute GUM G.4.2 Type B degrees of freedom from a stated standard uncertainty u and its standard uncertainty σ(u). σ(u)=0 → ν is infinite. Runs locally. Not Type B u_B, not Welch ν_eff, and not 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 · GUM G.4.2 Type B degrees of freedom ν = ½(u/σ(u))², including σ(u)=0 → infinite, + O3 mpmath tabulated ν.
Input interpretation
Enter values to calculate.
Result
—
Verified scope
GUM G.4.2 Type B degrees of freedom ν = ½(u/σ(u))², including σ(u)=0 → infinite, + O3 mpmath tabulated ν.
Assurance
Engineering
Declared partition coverage
PASS · 5/5 declared partitions (textbook-25, infinite, other-finite, alias, invalid-domain) · Matrix
Deferred
Not Type B u_B from a half-width, not Welch ν_eff, not U = k·u_c, and not trapezoidal Type B.
Numerical scope
O2: GUM G.4.2 ν = ½(u/σ(u))² vs separate-module identities (≤2 ULP). Not Type B u_B, not Welch ν_eff, not U=k·u_c. ≤2 ULP vs O3 applies only to the published tabulated G.4.2 vectors (GUM 25%, infinite, 10%, aliases, scale, σ(u)/u=0.5). It is not a whole-domain guarantee and does not cover Welch ν_eff or U=k·u_c.
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
GUM G.4.2 Type B degrees of freedom from u and σ(u).
Scope
JCGM 100:2008 G.4.2: ν = ½ (u/σ(u))²
Verification
Engine tested · Source checked · v1.0.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · GUM G.4.2 Type B degrees of freedom ν = ½(u/σ(u))², including σ(u)=0 → infinite, + O3 mpmath tabulated ν.· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-16.o2-o3
Verification revision
2026-09-16.o2-o3 · 3/3 property · digest 083418691428
Legacy regression
8/8 tests · Production surface contract 4/4
Trust layers
Verification VERIFIED · Production CURRENT · overall VERIFIED
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 · 4/4 CVP boundary · 7/7 invalid · 3/3 property · 2/2 metamorphic · 8/8 O3 · 4/4 cross-interface · 4/4 CVP contract · Manifest
Sources
Sources
Evidence
1 legacy golden · 4 legacy boundary · legacy regression suite · 5/5 oracle-backed golden · 7/7 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.0.0 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

Relative uncertainty of uσ(u)/u
Degrees of freedomν = ½ (u / σ(u))²
iGUM G.4.2: a 25% relative uncertainty of u gives ν = 8. This page does not compute u_B from a half-width a. Combine several ν_i on Welch–Satterthwaite. Apply k on Coverage factor.

How to use

1

Enter u

Already-evaluated Type B standard uncertainty. Must be greater than zero. Aliases: u_i, uB, u_B. Not half-width a.

2

Enter σ(u)

Standard uncertainty of u. Aliases: su, du, delta_u. Zero means u is treated as known exactly (ν → ∞).

3

Read ν

GUM textbook: σ(u)/u = 0.25 → ν = 8. Feed ν into Coverage factor or Welch–Satterthwaite. Do not use this page for U = k·u_c.

Example calculations

Common configurations with formula and result.

ϟ

GUM 25%

u=2 · σ(u)=0.5

σ(u)/u=0.25; ν=½/(0.25)²
ν=8
ϟ

Known exactly

u=1 · σ(u)=0

σ(u)=0 → ν→∞
ν=∞

Type B Degrees of Freedom calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
From a Type B standard uncertainty u > 0 and the standard uncertainty of that estimate σ(u) ≥ 0, ν = ½ (u/σ(u))². If σ(u) = 0, ν is infinite.
What it calculates
GUM G.4.2 Type B degrees of freedom from u and σ(u).
Inputs
  • u
  • sigma_u
Outputs
  • nu
  • rel
  • u
  • sigma_u
  • infinite
Formula
ν=½(u/σ(u))²
Assumptions
  • JCGM 100:2008 G.4.2: ν = ½ (u/σ(u))²
  • σ(u)=0 → ν is infinite (REST returns nu=null, infinite=true)
  • Not Type B u_B from a half-width, not Welch ν_eff, and not U = k·u_c
Units
  • dimensionless ν; u and σ(u) same unit
Boundary conditions
  • missing u or sigma_u → MISSING_REQUIRED_INPUT
  • u ≤ 0 → VALUE_MUST_BE_POSITIVE
  • σ(u) < 0 → VALUE_OUT_OF_RANGE
  • σ(u)=0 is valid (infinite ν)
Example
u=2 sigma_u=0.5 → ν=8 rel=0.25
Validation cases

3 published on this page · 8/8 tests · Production surface contract 4/4 · View evidence

  • u=2 sigma_u=0.5 → ν=8 rel=0.25
  • u=1 sigma_u=0 → nu=null infinite=true
  • u=0 sigma_u=0.5 → VALUE_MUST_BE_POSITIVE
Sources
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Compute GUM G.4.2 Type B degrees of freedom from a stated standard uncertainty and the uncertainty of that estimate.

Supported and not supported

Supported — ν = ½(u/σ(u))² · σ(u)=0 → infinite · API engineering.uncertainty.type_b_nu

Not supported — Type B u_B from half-width a, Welch ν_eff, U = k·u_c, trapezoidal Type B as a new page

Agent / API notes

Capability id: engineering.uncertainty.type_b_nu · tool id: type-b-nu · pin 1.0.0.

{ "u": 2, "sigma_u": 0.5 }

Aliases: u_i / uB / u_B for u; su / du / delta_u for sigma_u. Errors: MISSING_REQUIRED_INPUT, INVALID_NUMBER, VALUE_MUST_BE_POSITIVE, VALUE_OUT_OF_RANGE.

Calculator URL stays /calc/engineering/type-b-degrees-of-freedom. There is no /calc/metrology.

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

uB Type B uncertainty › νeff Welch–Satterthwaite ν_eff › k Coverage factor k › uA Type A uncertainty › u(y) Uncertainty propagate › x̄w Inverse-variance weighted mean › WS Tolerance / Uncertainty Workspace ›
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Frequently asked questions

Key distinctions behind the calculation.

Is this the same as Type B u_B?

No. Type B u_B is a/√3 or a/√6 from a half-width. This page takes an already-stated u and the uncertainty of that u, then reports GUM G.4.2 ν.

Is this Welch–Satterthwaite?

No. Welch combines several u_i and ν_i into ν_eff. This page evaluates one Type B component’s ν_i so you can feed it into Welch or Coverage factor.

Why is ν = 8 when σ(u)/u = 0.25?

That is the GUM G.4.2 textbook case: ν = ½ / (0.25)² = 8.

Does this compute U = k·u_c?

No. Coverage factor looks up k from ν. Expanded U stays on Uncertainty propagate.

Where does this run?

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