How to Use This Tool
Weight sample variances by their degrees of freedom instead of averaging standard deviations. Calculate pooled sample standard deviation, variance, degrees of freedom and standard error for two independent groups.
The decision this tool supports
A simple average of two standard deviations ignores sample size and averages the wrong scale. This page keeps the decision bounded to pooled standard deviation and the supporting outputs shown beside it. Pooled Standard Deviation does not import an account, infer a market rate, or silently substitute an industry average.
Inputs and units
The Pooled Standard Deviation calculation uses Group 1 sample size, Group 1 sample SD, Group 2 sample size, Group 2 sample SD. Keep all money values in one currency and all time, distance, mass, energy or volume entries in the unit printed beside the field. Mixing Pooled Standard Deviation scopes can produce a plausible number with the wrong meaning.
- Group 1 sample size is entered in n₁.
- Group 1 sample SD is entered in s₁.
- Group 2 sample size is entered in n₂.
- Group 2 sample SD is entered in s₂.
Formula and worked check
Pooled variance = [(n₁−1)s₁² + (n₂−1)s₂²] ÷ (n₁+n₂−2); pooled SD is its square root. Samples n=10, s=2 and n=15, s=3 pool to variance 7.04, standard deviation 2.65, 23 df and standard error 1.08. The Pooled Standard Deviation default is an executable known-answer case, not a benchmark or recommendation. Change one input and verify that the direction of pooled standard deviation still matches the stated relationship.
How to interpret the result
Use pooled SD only when a common within-group variance model is defensible; otherwise retain separate variances or use a Welch method. The additional Pooled Standard Deviation outputs expose the denominator, comparison, capacity or reverse value needed to audit the primary result instead of presenting one unexplained number.
Assumptions
- Inputs are sample standard deviations from independent groups.
- A common within-group variance is the intended model.
- Each sample size is at least two.
Save the Pooled Standard Deviation input values and date with any material decision. A later Pooled Standard Deviation rerun is reproducible only when the same assumptions and units are available.
Limitations and safety boundary
This page does not test equal variance, calculate a t statistic, handle paired data, weights, more than two groups or population SDs. Pooled Standard Deviation is an estimate and cannot replace a contract, local code, licensed professional, calibrated measurement, lender statement or platform report where one governs the decision.
Source and privacy
The Pooled Standard Deviation definition or rule was checked against NIST/SEMATECH e-Handbook — Pooled standard deviations on 2026-08-26. Recheck NIST/SEMATECH e-Handbook — Pooled standard deviations when a specification or policy behind Pooled Standard Deviation can change. Pooled Standard Deviation arithmetic runs in this browser tab; ecech does not receive the values through a calculation API.
Sources & assumptions
Tool Spec v2 · verified 2026-08-26. Platform rules and fees can change; the editable inputs remain authoritative for your account.
Official references
- NIST/SEMATECH e-Handbook — Pooled standard deviations (checked 2026-08-26)
Model assumptions
- Inputs are sample standard deviations from independent groups.
- A common within-group variance is the intended model.
- Each sample size is at least two.
- This page does not test equal variance, calculate a t statistic, handle paired data, weights, more than two groups or population SDs.
