Pooled Standard Deviation Calculator
Combine the spread of two groups into a single estimate — the denominator of a two-sample t-test and of Cohen's d.
The pooled standard deviation is a weighted average of the two variances, weighted by degrees of freedom: s²ₚ = ((n₁−1)s₁² + (n₂−1)s₂²) ÷ (n₁+n₂−2).
How the pooled standard deviation calculator works
The pooled standard deviation is a weighted average of the two variances, weighted by degrees of freedom: s²ₚ = ((n₁−1)s₁² + (n₂−1)s₂²) ÷ (n₁+n₂−2). Take the square root.
It assumes both groups share a common variance. If the two standard deviations differ by more than about a factor of two, that assumption is doing real damage and Welch's test — which does not pool — is the better tool.
Formula: sₚ = √(((n₁−1)s₁² + (n₂−1)s₂²) / (n₁+n₂−2))
Worked examples
| Inputs | Pooled standard deviation | Note |
|---|---|---|
| SDs of 5 and 7 | 5.988984 | pooled ≈ 5.98 |
| Equal SDs | 6 | pooled = 6 exactly |
| Very unequal | 13.5396 | pooling is unsafe here |
FAQFrequently asked questions
What is a pooled standard deviation?
A single spread estimate built from two groups, weighted by their degrees of freedom.
Why weight by n − 1?
Because each group's variance estimate is worth its degrees of freedom. A group of 100 should count far more than a group of 3.
When should I not pool?
When the two standard deviations differ substantially — more than about a factor of two — or when the group sizes are very unequal. Use Welch's t-test instead.
Is it just the average of the two SDs?
No. It averages the *variances*, then takes the root, so it is always at least as large as the plain average of the SDs.
Where is it used?
The denominator of the classic two-sample t-test, and of Cohen's d for effect size.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Australian Curriculum (ACARA) — Mathematics — the terms and methods taught in Australian schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.