Engineering calculator

Coverage Factor Calculator

Compute the two-sided Student-t coverage factor k from degrees of freedom ν and confidence p. Infinite ν uses the normal quantile. Runs locally. 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 · Two-sided k = t_{(1+p)/2}(ν), including ν→∞ → Φ⁻¹((1+p)/2), + O3 mpmath tabulated k.

Input interpretation: Enter values to calculate.
Result: —
Verified scope: Two-sided k = t_{(1+p)/2}(ν), including ν→∞ → Φ⁻¹((1+p)/2), + O3 mpmath tabulated k.
Assurance: Engineering
Declared partition coverage: PASS · 6/6 declared partitions (student-t, infinite, default-p, other-p, alias, invalid-domain) · Matrix
Deferred: Not expanded U = k·u_c, not one-sided k, and not Welch–Satterthwaite ν_eff.
Numerical scope: O2: two-sided k = t_{(1+p)/2}(ν) vs separate-module inverse (|Δk|≤2e-6). Infinite ν uses Φ⁻¹. Not U=k·u_c, not one-sided k. ≤2e-6 vs O3 applies only to the published tabulated k vectors (ν=8/1/2/30, inf, default p, p=0.99, nu_eff alias). It is not a whole-domain guarantee and does not cover U=k·u_c or one-sided k. Infinite-ν comparison is limited by the IUT Hastings erf polyfill when Math.erfc is absent.
Known limitations: Core CVP does not include live graph, viewport, or pointer interaction.
Model: Two-sided coverage factor k from ν and confidence p.
Scope: JCGM 100:2008 G.3 / G.4 two-sided coverage factor
Verification: Engine tested · Source checked · v1.0.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · Two-sided k = t_{(1+p)/2}(ν), including ν→∞ → Φ⁻¹((1+p)/2), + O3 mpmath tabulated k.· 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 22b395d8c5f7
Legacy regression: 10/10 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 · 5/5 CVP boundary · 7/7 invalid · 3/3 property · 2/2 metamorphic · 8/8 O3 · 4/4 cross-interface · 4/4 CVP contract · Manifest
Sources: JCGM 100:2008 (GUM) — Evaluation of measurement data — G.3 / G.4 Coverage factor and effective degrees of freedom
JCGM 100:2008 (GUM) — Evaluation of measurement data — Table G.1 Values of k_p
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.

Student-t (finite ν)k = t_{(1+p)/2}(ν)

Normal (ν → ∞)k = Φ⁻¹((1+p)/2)

iThis page only looks up k. Compute ν_eff on Welch–Satterthwaite. Expanded U = k·u_c is Uncertainty propagate, not this page.

How to use

Enter ν

Degrees of freedom. Use inf when ν → ∞ (typical after all Type B components).

Enter confidence

Two-sided coverage p in (0, 1). Default 0.95.

Read k

Apply k to an already-evaluated u_c on Uncertainty propagate if you need U.

Example calculations

Common configurations with formula and result.

ν=8, 95%

default confidence

k=t_0.975(8)

k≈2.30600

ν=∞, 95%

normal quantile

k=Φ⁻¹(0.975)

k≈1.95996

Coverage Factor calculator specification

Version 1.0.0 · Engine tested

Calculation status

Engine tested 10/10 tests · Production surface contract 4/4
Named expert review Not performed
Calculation version 1.0.0

Review policy · Evidence

Definition: For two-sided coverage probability p, k = t_{(1+p)/2}(ν). If ν is infinite, k is the standard-normal quantile Φ⁻¹((1+p)/2). Default p = 0.95.
What it calculates: Two-sided coverage factor k from ν and confidence p.
Inputs: nu
confidence
Outputs: k
confidence
nu
infinite
k_source
Formula: k = t_{(1+p)/2}(ν); ν→∞ uses Φ⁻¹((1+p)/2)
Assumptions: JCGM 100:2008 G.3 / G.4 two-sided coverage factor
Default confidence p = 0.95
Infinite ν uses the standard-normal quantile
Not expanded U = k·u_c and not GUM model combination
Units: dimensionless k
Boundary conditions: missing nu → MISSING_REQUIRED_INPUT
ν ≤ 0 and not inf → VALUE_MUST_BE_POSITIVE
confidence not in (0, 1) → VALUE_OUT_OF_RANGE
Example: nu=8 confidence=0.95 → k≈2.30600
Validation cases: 2 published on this page · 10/10 tests · Production surface contract 4/4 · View evidence
nu=8 confidence=0.95 → k≈2.30600
nu=0 confidence=0.95 → VALUE_MUST_BE_POSITIVE
Sources: JCGM 100:2008 (GUM) — Evaluation of measurement data — G.3 / G.4 Coverage factor and effective degrees of freedom
Supports: k = t_{(1+p)/2}(ν) for two-sided coverage; infinite ν uses the normal quantile
JCGM 100:2008 (GUM) — Evaluation of measurement data — Table G.1 Values of k_p
Supports: For p=0.95 and ν→∞, k_p = 1.960
Calculation version: 1.0.0

Background

Interpretation and common distinctions.

Compute the two-sided coverage factor k.

Supported and not supported

Supported — Student-t k from ν and confidence · infinite ν → normal quantile · API engineering.uncertainty.coverage_factor

Not supported — expanded U = k·u_c, GUM model combination, Welch–Satterthwaite ν_eff (that is a different page)

Agent / API notes

Capability id: engineering.uncertainty.coverage_factor · tool id: coverage-factor · pin 1.0.0.

{ "nu": 8, "confidence": 0.95 }

nu may be inf (JSON null on output). Default confidence is 0.95. Errors: MISSING_REQUIRED_INPUT, INVALID_NUMBER, VALUE_MUST_BE_POSITIVE, VALUE_OUT_OF_RANGE.

Calculator URL stays /calc/engineering/coverage-factor. There is no /calc/metrology.

Other calculators in this family: Welch–Satterthwaite ν_eff, Uncertainty propagate, Type A uncertainty, Type B uncertainty, Type B degrees of freedom, Guard band, Test uncertainty ratio TUR, Tolerance / Uncertainty Workspace .

}

Frequently asked questions

Key distinctions behind the calculation.

Is this the same as Uncertainty propagate?

No. This page only evaluates k from ν and confidence. Uncertainty propagate can auto-select the same Student-t k when you give confidence together with a measurement model. Expanded U = k·u_c stays on that page.

Why is k≈1.96 when ν is infinite, not 2?

p=0.95 is the two-sided normal quantile Φ⁻¹(0.975)≈1.95996. k=2 is a round number that corresponds to about 95.45% under a normal assumption.

Where does this run?

Locally in the browser by default. REST and MCP call the same metrology-engine coverage_factor op, which uses statistics-engine Student-t.