Enter each group's sample size and its own standard deviation. Add as many groups as you need — the calculation handles two, three, or any larger number the same way.
| Group size (n) | Group SD (s) | Remove |
|---|---|---|
Enter each group’s sample size and its own sample standard deviation. You need at least two groups, each with n ≥ 2.
Show the working, step by step
The formula
sp = √[ ((n₁−1)s₁² + (n₂−1)s₂² + … + (nₖ−1)sₖ²) / (n₁ + n₂ + … + nₖ − k) ]
For the common two-group case that reduces to the form most textbooks print:
sp = √[ ((n₁−1)s₁² + (n₂−1)s₂²) / (n₁ + n₂ − 2) ]
Each group contributes in proportion to its degrees of freedom, nᵢ − 1, so a group of 50 carries roughly five times the weight of a group of 10. That is deliberate: the larger group's estimate of the spread is the more trustworthy one.
Why variances, not standard deviations
This is the step people get wrong, so it is worth seeing the size of the error. Two groups, each of size 10, with standard deviations of 2 and 4:
| Method | Calculation | Result |
|---|---|---|
| Averaging the SDs (wrong) | (2 + 4) / 2 | 3.000 |
| Pooling the variances (right) | √((9×4 + 9×16) / 18) | 3.162 |
A 5% discrepancy here, and it grows as the groups diverge. The underlying reason is that variance is what behaves additively — the standard deviation is a square root, and square roots do not average.
A worked example
Three batches from a production process:
| Batch | n | s | n − 1 | (n−1)s² |
|---|---|---|---|---|
| A | 10 | 2.4 | 9 | 51.84 |
| B | 15 | 3.1 | 14 | 134.54 |
| C | 12 | 2.8 | 11 | 86.24 |
| Total | 37 | — | 34 | 272.62 |
sp = √(272.62 ÷ 34) = √8.0182 = 2.8317. Those are the values loaded into the calculator above. Note the pooled figure lands between the smallest and largest group SD, and closer to batch B's, because B contributes the most degrees of freedom.
The homogeneity assumption
Pooling only makes sense if the groups genuinely share a common population variance. The pooled number is a single estimate of one underlying spread, so if the groups differ substantially it describes none of them and quietly distorts anything built on it.
Screen for this before pooling. A practical rule is to be cautious once the largest group SD exceeds roughly twice the smallest; Levene's test or the Brown–Forsythe test makes it formal. Where the assumption fails, Welch's t-test keeps the groups' variances separate and is the better default.
What pooled SD feeds into
- Independent-samples t-test —
t = (x̄₁ − x̄₂) / (sp × √(1/n₁ + 1/n₂)). - Cohen's d —
d = (x̄₁ − x̄₂) / sp, the standardised effect size. - ANOVA — the root mean square error is the pooled SD across all groups.
- Method validation — a pooled precision estimate across several runs or analysts.
Related calculators
-
Standard deviation calculator
Get each group’s own SD from its raw values first.
-
Variance calculator
The quantity that actually gets averaged when pooling.
-
Weighted SD
For weighting individual values rather than whole groups.
-
Standard error
What the pooled SD feeds into for a t-test.
Common questions
What is pooled standard deviation?
A single estimate of spread combining two or more groups, formed by averaging their variances weighted by degrees of freedom and then taking the square root:
sp = √[ Σ(nᵢ − 1)sᵢ² / Σ(nᵢ − 1) ]
It assumes every group is drawing from populations with the same true spread, and it gives larger groups more influence over the result.
Why can I not just average the standard deviations?
Because standard deviations do not add or average linearly — variances do. Averaging the SDs of groups with SDs 2 and 4 gives 3; pooling them at equal size gives √((4 + 16)/2) = 3.162. The gap widens as the groups become less alike, and the plain average is always the smaller and wrong answer.
When is pooling not appropriate?
When the groups plainly do not share a common spread. Pooling assumes homogeneity of variance, and if one group's SD is several times another's, the pooled figure describes none of them.
In that case use Welch's t-test, which does not pool, rather than Student's. A common rough screen is to be wary once the largest SD exceeds twice the smallest.
What is pooled SD used for?
Chiefly the independent-samples t-test, where it forms the denominator, and Cohen's d, where the mean difference is divided by it to give a standardised effect size. It also appears in ANOVA as the root mean square error.