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.
Trust summary Engine tested · Source checked · 8/8 tests · Production surface contract 4/4 · v1.0.0
- Input interpretation
- Enter values to calculate.
- Result
- —
- 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 · 8/8 tests · Production surface contract 4/4 · Source checked · v1.0.0
- Named expert review
- Optional · Not performed
- Sources
- JCGM 100:2008 — Evaluation of measurement data (GUM)
- ISO 5725-2 — Repeatability and reproducibility of standard measurement methods
- Evidence
- 5 boundary · 3 property · Production surface contract 4/4 · Artifact integrity PASS
- Production
- Embedded snapshot: STALE · Last attested schema matched 1.0.0 snapshot / local build · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · Public/cache ✓ · Origin ✓ · Live production status PASS (0 stale; 164 CURRENT) @ 2026-09-16T06:44:24.909Z
- Semantic contract
- PASS
Formulas
Core equations used by this calculator.
How to use
Enter s_i
Comma-separated sample standard deviations, already computed. Aliases: s_i, stdevs, std. Each s_i ≥ 0.
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.
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.
1 and 7
s=[1,7] · ν=[1,1]
Equal s
s=[5,5] · ν=[3,8]
One group rejected
s=[3] · ν=[8]
Pooled Standard Deviation calculator specification
Version 1.0.0 · Engine tested
- Engine tested 8/8 tests · Production surface contract 4/4
- Named expert review Not performed
- Calculation version 1.0.0
- 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 4/4 · View evidence
- s=[1,7] nu=[1,1] → s_p=5 ν=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 seriesSupports: 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 deviationSupports: 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.
- JCGM 100:2008 — Evaluation of measurement data (GUM) — 4.2.4 Type A evaluation from several series
- Calculation version
- 1.0.0
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 1.0.0.
{ "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.
Related tools
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 .
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.