How this instrument works
Pooled standard deviation combines the spread of two separate samples into a single estimate of the common standard deviation they'd share if drawn from populations with equal variance. Rather than simply averaging the two sample standard deviations, it works with variances — squared standard deviations — weighted by each sample's degrees of freedom (n minus 1), so a larger, more reliable sample pulls the pooled estimate more strongly toward its own variance than a smaller sample does.
The formula sums each sample's variance times its degrees of freedom, divides by the total degrees of freedom across both samples, and takes the square root to get back to a standard deviation: sp = sqrt[((n1−1)s1² + (n2−1)s2²) / (n1+n2−2)]. When the two samples are the same size with the same standard deviation, the pooled value equals that shared standard deviation exactly — the formula only does real work when the samples differ in size, spread, or both.
Pooled standard deviation is a building block, not usually the final answer on its own — it's the denominator inside Cohen's d, a standardized effect size, and inside the standard error used by the independent-samples t-test, both of which assume the two groups share a common underlying variance. This calculator reports the pooled value itself so you can see exactly what's feeding into those downstream statistics, or use it directly if that's all you need.
- Enter Sample 1's standard deviation into s1, and its size into n1 (n1 must be at least 2).
- Enter Sample 2's standard deviation into s2, and its size into n2 (n2 must be at least 2).
- Read the pooled standard deviation (sp) — the shared-spread estimate weighted by each sample's degrees of freedom.
- Use sp directly, or as the denominator when computing Cohen's d or a two-sample t-test's standard error by hand.
Worked example — two equal-sized samples
Sample 1 has standard deviation s1 = 10 across n1 = 20 observations. Sample 2 has standard deviation s2 = 10 across n2 = 20 observations. Enter those four numbers into the calculator.
The pooled standard deviation is sp = sqrt[((19×100)+(19×100))/38] = sqrt[3800/38] = sqrt(100) = 10 exactly. Because both samples happen to share the same standard deviation and the same size here, the pooled value comes out equal to that shared value — a useful sanity check before trying an example where the two samples actually differ in size or spread.
Questions
Why not just average the two standard deviations?
A simple average of s1 and s2 ignores sample size and doesn't correctly combine variability in a statistical sense — standard deviations don't add or average linearly the way variances do. Pooling works with variances weighted by degrees of freedom, then converts back to a standard deviation at the very end, which is why the formula looks more involved than a plain average. A simple average can meaningfully overstate or understate the true combined spread, especially when the two sample sizes differ.
What assumption does pooled standard deviation rely on?
It assumes the two populations the samples were drawn from have equal, or at least similar, variance — the technical term is homogeneity of variance. If that assumption is badly violated, pooling the two variances together produces a misleading combined figure, and analyses like Welch's t-test, which doesn't assume equal variances, are used instead of the pooled-variance version.
Where does pooled standard deviation get used?
Its two most common uses are as the denominator of Cohen's d, a standardized effect size expressing the difference between two group means in pooled-standard-deviation units, and inside the standard error formula for the independent-samples (Student's) t-test, which also assumes equal population variances. Both downstream calculations depend directly on getting this pooled figure right.
Does a larger sample size always shrink the pooled standard deviation?
Not directly — sample size changes the weighting, not the magnitude, of each sample's contribution. A larger sample pulls the pooled estimate more strongly toward its own variance, which could push the pooled value up or down depending on whether that sample's variance is bigger or smaller than the other one's. Sample size affects how much each variance counts, not whether the pooled number goes up or down.