HomeCalculatorsEngineeringType A Uncertainty Calculator
Engineering calculator

Type A Uncertainty Calculator

Compute Type A standard uncertainty from repeated observations: mean, sample s, and u_A = s/√n. Runs locally. Not GUM combination and not Monte Carlo.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · Type A u_A = s/√n from a sample (Bessel ddof=1 default, population ddof=0) + O3 mpmath tabulated mean/s/u_A. CSV/JSON/array and sample alias.
Input interpretation
Enter values to calculate.
Result
—
Verified scope
Type A u_A = s/√n from a sample (Bessel ddof=1 default, population ddof=0) + O3 mpmath tabulated mean/s/u_A. CSV/JSON/array and sample alias.
Assurance
Engineering
Declared partition coverage
PASS · 7/7 declared partitions (bessel, population, csv, json, sample-alias, constant, invalid-domain) · Matrix
Deferred
Not GUM RSS of several uᵢ, not expanded U=k·u_c, not Monte Carlo, and not a standalone SEM product page.
Numerical scope
O2: mean / sample s / u_A = s/√n vs separate-module identities (≤2 ULP). Not GUM combination, not SEM product page, not Monte Carlo. ≤2 ULP vs O3 applies only to the published tabulated Type A vectors (Bessel, population, scale, constant, two-point, n=5, centered, default ddof). It is not a whole-domain sample-statistics guarantee.
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Type A mean, s, and u_A = s/√n from repeated observations.
Scope
Type A: u_A = s/√n from summarizeSample (ddof=1 default)
Verification
Engine tested · Source checked · v1.0.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · Type A u_A = s/√n from a sample (Bessel ddof=1 default, population ddof=0) + O3 mpmath tabulated mean/s/u_A. CSV/JSON/array and sample alias.· 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 · 2/2 property · digest 091eb0d8d4cd
Legacy regression
7/7 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
6/6 golden · 3/3 CVP boundary · 7/7 invalid · 2/2 property · 2/2 metamorphic · 8/8 O3 · 4/4 cross-interface · 4/4 CVP contract · Manifest
Sources
Sources
Evidence
1 legacy golden · 3 legacy boundary · legacy regression suite · 6/6 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.

Meanx̄ = Σxᵢ / n
Sample ss = √(Σ(xᵢ − x̄)² / (n − ddof))
Type Au_A = s / √n
iBessel ddof=1 by default. This evaluates a sample, not GUM RSS of several uᵢ. Not Monte Carlo.

How to use

1

Enter repeated observations

Comma, space, or semicolon separated. JSON arrays also work. Sample s needs at least two values when ddof=1.

2

Choose ddof

1 (default) is Bessel's correction. 0 is the population form.

3

Read u_A, then s and the mean

u_A is the Type A standard uncertainty of the mean. Combine several u's on Uncertainty propagate.

Example calculations

Common configurations with formula and result.

ϟ

1, 2, 3

ddof=1

s=1; u_A=1/√3
mean=2 · s=1 · u_A≈0.57735
ϟ

Single value

n=1 with ddof=1

sample variance
VALUE_OUT_OF_RANGE

Type A Uncertainty calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
From n repeated observations, mean x̄, sample standard deviation s (ddof=1 default), and Type A standard uncertainty u_A = s/√n.
What it calculates
Type A mean, s, and u_A = s/√n from repeated observations.
Inputs
  • values
  • ddof
Outputs
  • u_A
  • mean
  • std
  • n
Formula
u_A = s/√n
Assumptions
  • Type A: u_A = s/√n from summarizeSample (ddof=1 default)
  • Not GUM combination of several uᵢ
  • Not Monte Carlo
  • Independent repeated observations of one quantity
Units
  • same as the observations
Boundary conditions
  • missing values → MISSING_REQUIRED_INPUT
  • n<2 with ddof=1 → VALUE_OUT_OF_RANGE
  • ddof not 0 or 1 → INVALID_INPUT
Example
values=[1,2,3] ddof=1 → mean=2 std=1 u_A=1/√3
Validation cases

2 published on this page · 7/7 tests · Production surface contract 4/4 · View evidence

  • values=1,2,3 ddof=1 → mean=2 std=1 u_A=1/√3
  • values=1 ddof=1 → VALUE_OUT_OF_RANGE
Sources
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Compute Type A standard uncertainty from repeated observations.

Supported and not supported

Supported — mean, sample s, u_A = s/√n · ddof 0|1 · API engineering.uncertainty.type_a

Not supported — GUM RSS of several uᵢ, Type B rectangular/triangular, Monte Carlo, ANOVA

Agent / API notes

Capability id: engineering.uncertainty.type_a · tool id: type-a · pin 1.0.0.

{ "values": [1, 2, 3], "ddof": 1 }

Share URLs also accept values=1,2,3. Errors: MISSING_REQUIRED_INPUT, INVALID_NUMBER, INVALID_INPUT, VALUE_OUT_OF_RANGE.

Calculator URL stays /calc/engineering/type-a-uncertainty. There is no /calc/metrology.

Other calculators in this family: Uncertainty propagate, Type B uncertainty, Inverse-variance weighted mean, Welch–Satterthwaite ν_eff, Coverage factor k, Guard band, Process capability (Cp/Cpk), Gauge R&R, Measurement bias (trueness), Measurement precision, Uncertainty of a mean from precision, Pooled standard deviation, Repeatability from duplicates, Cochran's C, Tolerance / Uncertainty Workspace .

u(y) Uncertainty propagate › uB Type B uncertainty › x̄w Inverse-variance weighted mean › νeff Welch–Satterthwaite ν_eff › k Coverage factor k › GB Guard band › Cp Process capability (Cp/Cpk) › GRR Gauge R&R › δ Measurement bias (trueness) › s Measurement precision › u Uncertainty of a mean from precision › s_p Pooled standard deviation › s_r Repeatability from duplicates › C Cochran's C › WS Tolerance / Uncertainty Workspace ›
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Frequently asked questions

Key distinctions behind the calculation.

Is this the same as Uncertainty propagate?

No. This page evaluates Type A u_A from a sample. Uncertainty propagate combines already-evaluated standard uncertainties (GUM). The type_a field on that page is only A/B metadata.

Is this ISO 21748 u from stated precision?

No. This page takes a sample of observations. Uncertainty of a mean from precision takes already-stated s_r, s_L, and n≥2 — it does not take values.

Is this pooled s from several series?

No. Type A takes one sample. Pooled standard deviation takes k≥2 already-stated s_i and ν_i.

Why n≥2 when ddof=1?

Sample variance uses n−1. A single observation has no sample s. Use ddof=0 only when you intentionally want the population form.

Where does this run?

Locally in the browser by default. REST and MCP call the same metrology-engine Type A ops (summarizeSample, no copied variance).