Pooled Variance Calculator
Combine two samples into a single pooled variance and standard deviation — the weighted estimate used in two-sample t-tests assuming equal variances.
How it works
s²ₚ = [(n₁ − 1)·s₁² + (n₂ − 1)·s₂²] ÷ (n₁ + n₂ − 2)
Pooled variance combines two samples into one variance estimate, weighting each by its degrees of freedom so the larger, more informative sample counts more. It assumes both groups share the same underlying variance — the equal-variance assumption behind the classic two-sample t-test. Its square root, the pooled standard deviation, feeds the standard error used to compare the two group means.
Worked example
With s₁ = 4 over n₁ = 15 and s₂ = 5 over n₂ = 20, the pooled variance is [(14)(16) + (19)(25)] ÷ 33 = (224 + 475) ÷ 33 ≈ 21.18, so the pooled standard deviation is about 4.60 on 33 degrees of freedom.
Sources: Pooled variance — Wikipedia
Frequently asked questions
When should I pool variances?
Pool only when it is reasonable to assume the two groups have equal population variances. If the sample variances differ a lot, the Welch t-test — which does not pool — is safer.
Why weight by degrees of freedom instead of sample size?
Each sample variance is based on n − 1 degrees of freedom, so weighting by n − 1 gives each group influence proportional to the information it actually contributes, producing an unbiased combined estimate.
Can I enter variances instead of standard deviations?
This tool takes standard deviations and squares them internally. If you already have variances, just enter their square roots as the standard deviations.
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