Pooled Standard Deviation Calculator
Combine sample standard deviations from two to six independent groups. Each variance is weighted by its degrees of freedom, so unequal sample sizes are handled correctly.
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Use this estimate only when the independent groups can reasonably be treated as having a common population variance.
Calculate pooled standard deviation
Enter each group’s sample size and sample standard deviation. Do not enter standard error.
Pooled standard deviation formula
For k independent samples, calculate sₚ = √[Σ(nᵢ−1)sᵢ² ÷ Σ(nᵢ−1)]. Squaring each sample SD gives its variance. Multiplying by n − 1 restores that sample’s sum of squared deviations. Adding these quantities and dividing by total degrees of freedom gives pooled variance.
When to use pooled standard deviation
Use it to estimate common within-group spread when samples are independent and an equal-variance model is justified. It appears in the classic equal-variance two-sample t test and in one common version of Cohen’s d.
Do not pool automatically. If group variances differ materially, an unequal-variance method such as Welch’s t test is usually more appropriate. Paired observations also need a method based on within-pair differences.
Worked example with unequal sample sizes
Suppose group 1 has n = 10 and s = 5, while group 2 has n = 12 and s = 6. The weighted sum is (9 × 25) + (11 × 36) = 621. Total degrees of freedom are 9 + 11 = 20. Therefore pooled variance is 621 ÷ 20 = 31.05 and pooled SD is √31.05 ≈ 5.5723.
Can groups have different means or sample sizes?
Yes. Pooled SD estimates the common within-group spread, so the formula does not use group means. Unequal sample sizes are expected and handled by n − 1 weights. This is not the same as combining all raw observations into one overall SD; that overall value also reflects differences between group means.
Frequently asked questions
Can I pool more than two standard deviations?
Yes. Add every group’s (n − 1)s² contribution and divide by the sum of all n − 1 degrees of freedom.
Should I average standard deviations directly?
No. Pool variances with degrees-of-freedom weights, then take the square root.
Can groups have unequal sizes?
Yes. Larger samples receive more weight because they contribute more degrees of freedom.
Is pooled SD for paired samples?
Not this formula. Paired data require the standard deviation of within-pair differences or equivalent information.
Related tools and authoritative reference
For observations stored as frequencies, use the grouped-data standard deviation calculator. For a scale-free percentage, see relative standard deviation in Excel.