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Engineering calculator

Pooled Standard Deviation Calculator

Compute a pooled standard deviation from k≥2 stated sample s_i and their degrees of freedom ν_i. Runs locally. Not Type A from one sample, not Welch–Satterthwaite ν_eff, and not ISO 21748 u of a mean.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · s_p = √(Σ ν_i s_i² / Σ ν_i) for k≥2 stated groups, + O3 mpmath tabulated s_p.
Input interpretation
Enter values to calculate.
Result
—
Verified scope
s_p = √(Σ ν_i s_i² / Σ ν_i) for k≥2 stated groups, + O3 mpmath tabulated s_p.
Assurance
Engineering
Declared partition coverage
PASS · 5/5 declared partitions (two-group, multi-group, sample-size, alias, invalid-domain) · Matrix
Deferred
Not Type A from one sample, not Welch–Satterthwaite ν_eff, and not ISO 21748 u of a mean.
Numerical scope
O2: s_p = √(Σ ν_i s_i² / Σ ν_i) vs a separate-module identity (≤2 ULP). Not Type A from one sample, not Welch–Satterthwaite ν_eff, and not ISO 21748 u of a mean. ≤2 ULP vs O3 applies only to the published tabulated pooled-standard-deviation vectors (two groups, equal s, sample sizes, an alias, a zero component, and one awkward triple). It is not a whole-domain guarantee and does not cover Type A from one sample, Welch–Satterthwaite ν_eff, or ISO 21748 u of a mean.
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Pooled standard deviation s_p and pooled ν from k≥2 stated (s_i, ν_i) groups.
Scope
Pooled standard deviation: s_p = √(Σ ν_i s_i² / Σ ν_i) from k≥2 stated groups. ν_i > 0 and finite.
Verification
Engine tested · Source checked · v1.0.1 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · s_p = √(Σ ν_i s_i² / Σ ν_i) for k≥2 stated groups, + O3 mpmath tabulated s_p.· 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 · 1/1 property · digest 20cd5760d858
Legacy regression
8/8 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
4/4 golden · 4/4 CVP boundary · 8/8 invalid · 1/1 property · 1/1 metamorphic · 8/8 O3 · 4/4 cross-interface · 6/6 CVP contract · Manifest
Sources
Sources
Evidence
1 legacy golden · 4 legacy boundary · legacy regression suite · 4/4 oracle-backed golden · 8/8 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.0.1 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.

Pooled ss_p = √(Σ ν_i s_i² / Σ ν_i)
Pooled νν = Σ ν_i
ik must be at least 2 so this cannot wrap a single sample s. ν_i must be finite and greater than zero. If nu is omitted, sample sizes n_i ≥ 2 give ν_i = n_i−1. This page does not take a list of observations.

How to use

1

Enter s_i

Comma-separated sample standard deviations, already computed. Aliases: s_i, stdevs, std. Each s_i ≥ 0.

2

Enter ν_i

Same length. Each ν_i > 0 and finite. Aliases: nu_i, dof, df. Infinite ν is rejected — that is Welch. Or omit ν and enter n_i ≥ 2.

3

Read s_p

s_p is the pooled standard deviation. Use Type A when you have one sample of observations. Use Welch–Satterthwaite for ν_eff from u_i. Use Uncertainty of a mean from precision for u from s_r, s_L, and n.

Example calculations

Common configurations with formula and result.

ϟ

1 and 7

s=[1,7] · ν=[1,1]

s_p=√((1+49)/2)
s_p=5 · ν=2
ϟ

Equal s

s=[5,5] · ν=[3,8]

s_p=5
s_p=5 · ν=11
ϟ

One group rejected

s=[3] · ν=[8]

k≥2
VALUE_OUT_OF_RANGE

Pooled Standard Deviation calculator specification

Version 1.0.1 · Engine tested

Calculation status

Review policy · Evidence

Definition
From k≥2 stated sample standard deviations s_i ≥ 0 and finite ν_i > 0 of equal length, s_p = √(Σ ν_i s_i² / Σ ν_i). The pooled degrees of freedom are Σ ν_i.
What it calculates
Pooled standard deviation s_p and pooled ν from k≥2 stated (s_i, ν_i) groups.
Inputs
  • s
  • nu
Outputs
  • s_p
  • nu
  • k
  • s
  • nu_i
Formula
s_p=√(Σνs²/Σν)
Assumptions
  • Pooled standard deviation: s_p = √(Σ ν_i s_i² / Σ ν_i) from k≥2 stated groups. ν_i > 0 and finite.
  • s_i ≥ 0. Both zero is valid (s_p = 0).
  • Not Type A from one sample, not Welch–Satterthwaite ν_eff, and not ISO 21748 u of a mean.
Units
  • same as s_i
Boundary conditions
  • missing s or nu → MISSING_REQUIRED_INPUT
  • k = 1 → VALUE_OUT_OF_RANGE
  • length mismatch → INVALID_INPUT
  • negative s_i → VALUE_OUT_OF_RANGE
  • ν_i ≤ 0 → VALUE_MUST_BE_POSITIVE
Example
s=[1,7] nu=[1,1] → s_p=5
Validation cases

3 published on this page · 8/8 tests · Production surface contract 6/6 · View evidence

  • s=[1,7] nu=[1,1] → s_p=5, nu=2, k=2
  • s=[5,5] nu=[3,8] → s_p=5
  • s=[3] nu=[8] → VALUE_OUT_OF_RANGE
Sources
  • JCGM 100:2008 — Evaluation of measurement data (GUM) — 4.2.4 Type A evaluation from several series
    Supports: Independent series of observations of the same measurand may be pooled: s_p² is the ν-weighted mean of the sample variances.
  • ISO 5725-2 — Repeatability and reproducibility of standard measurement methods — Pooled repeatability standard deviation
    Supports: s_p is the usual pooled-within estimate from several groups. This page takes stated s_i and ν_i rather than a collaborative-study table.
Calculation version
1.0.1

Background

Interpretation and common distinctions.

Compute a pooled standard deviation from stated sample s_i and degrees of freedom.

Supported and not supported

Supported — s_p from k≥2 paired s_i, ν_i · s_i = 0 is valid · n_i alias · API engineering.uncertainty.pooled_sd

Not supported — Type A from one sample, Welch ν_eff, ISO 21748 u of a mean, infinite ν_i, combined RSS

Agent / API notes

Capability id: engineering.uncertainty.pooled_sd · tool id: pooled-sd · pin 1.0.1.

{ "s": [1, 7], "nu": [1, 1] }

Aliases: s_i / stdevs / std for s; nu_i / dof / df for nu; n / n_i for sample sizes when nu is omitted. Errors: MISSING_REQUIRED_INPUT, INVALID_NUMBER, INVALID_INPUT, VALUE_OUT_OF_RANGE, VALUE_MUST_BE_POSITIVE.

Calculator URL stays /calc/engineering/pooled-standard-deviation. There is no /calc/metrology.

Other calculators in this family: Type A uncertainty, Welch–Satterthwaite ν_eff, Uncertainty of a mean from precision, Measurement precision, Repeatability from duplicates, Cochran's C, Coverage factor k, Inverse-variance weighted mean, Tolerance / Uncertainty Workspace .

uA Type A uncertainty › νeff Welch–Satterthwaite ν_eff › u Uncertainty of a mean from precision › s Measurement precision › s_r Repeatability from duplicates › C Cochran's C › k Coverage factor k › x̄w Inverse-variance weighted mean › WS Tolerance / Uncertainty Workspace ›

Frequently asked questions

Key distinctions behind the calculation.

Is this Type A uncertainty?

No. Type A takes one sample of observations and reports s and u_A = s/√n. This page takes k≥2 already-stated s_i and ν_i. A single group is rejected.

Is this Welch–Satterthwaite?

No. Welch takes standard uncertainties u_i and returns ν_eff. This page pools sample standard deviations into s_p. Infinite ν_i are rejected.

Is this ISO 21748 u of a mean?

No. Uncertainty of a mean from precision takes one pair s_r, s_L and n≥2. This page pools several series.

Is this Cochran's C?

No. Cochran's C is s_max² / Σ s_i² from the same kind of stated s_i. This page pools them into s_p and needs ν_i.

Can I enter sample sizes n_i instead of ν_i?

Yes, when nu is omitted: ν_i = n_i−1 and each n_i must be ≥ 2. If both are sent, nu is used.

Where does this run?

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