Pooled Standard Deviation Calculator
Enter the sample size and standard deviation for each group to get the pooled standard deviation - the weighted average spread across all groups. The result updates instantly as you type, and the Show Your Work panel walks through every arithmetic step with your actual numbers. Supports two to four groups.
Formula
Worked example
Two groups: n1 = 12, s1 = 4.5 and n2 = 15, s2 = 6.2. Numerator: (12-1) x 4.5^2 + (15-1) x 6.2^2 = 11 x 20.25 + 14 x 38.44 = 222.75 + 538.16 = 760.91. Denominator: 11 + 14 = 25. Pooled variance = 760.91 / 25 = 30.4364. Pooled SD = sqrt(30.4364) = 5.5172.
What is the pooled standard deviation?
The pooled standard deviation (Sp) is a single summary of spread computed by combining the standard deviations from two or more independent groups. Rather than averaging the SDs directly, the formula weights each group's squared SD by its degrees of freedom (n - 1), which gives larger groups more influence on the result. The pooled SD is the square root of that weighted average of variances. It appears in two-sample t-tests that assume equal population variances, in one-way ANOVA as the square root of the mean squared error (MSE), and in Cohen's d effect size calculations where a common within-group spread is needed as the denominator.
How to calculate the pooled standard deviation
The formula for k groups is: Sp = sqrt[ sum((n_i - 1) x s_i^2) / sum(n_i - 1) ]. The calculation has four steps: (1) Square each group's sample SD to get its variance. (2) Multiply each variance by its degrees of freedom (n_i - 1). (3) Add those products together (numerator) and add the degrees of freedom together (denominator). (4) Divide and take the square root. For two groups with n1 = 12, s1 = 4.5 and n2 = 15, s2 = 6.2, the pooled variance is (11 x 20.25 + 14 x 38.44) / (11 + 14) = 760.91 / 25 = 30.44, and Sp = sqrt(30.44) = 5.517. The Show Your Work panel above substitutes your exact values into each step.
Pooled SD vs. simple average SD
Averaging the SDs directly (for example, (4.5 + 6.2) / 2 = 5.35) would give equal weight to each group regardless of sample size. The pooled formula instead weights by degrees of freedom, so a group with 100 observations contributes far more to Sp than one with 10. If all groups have the same size, the two approaches produce similar results, but with unequal sample sizes the difference can be meaningful. Always use the weighted pooled formula in t-tests and ANOVA, not the simple average.
When pooling is (and is not) appropriate
Pooling is valid only when the population variances of all groups are plausibly equal - the homogeneity of variance assumption. You can check this formally with Levene's test or Bartlett's test before pooling. As a rough heuristic, if the largest group SD is more than twice the smallest, pooling may produce a distorted estimate. When variances are clearly unequal, use Welch's t-test (which does not pool) or Welch's ANOVA. Pooling across many groups in ANOVA automatically satisfies this check if the overall F-test model is valid, because ANOVA's MSE already assumes homoscedasticity.
When to use the pooled standard deviation
| Situation | Use pooled SD? | Reason |
|---|---|---|
| Two-sample t-test (equal variances) | Yes | Pooled SE gives the correct test statistic |
| One-way ANOVA (k groups) | Yes | MSE is the pooled variance across all groups |
| Two-sample t-test (unequal variances) | No | Use Welch's t-test with separate SDs |
| Cohen's d effect size (equal n) | Yes | Standard pooled SD formula applies |
| Cohen's d effect size (unequal n) | Yes | Weighted pooled SD accounts for group sizes |
| Groups with very different spreads | No | Pooling inflates or deflates the estimate |
| Combining lab replicates | Yes | Pooling replicates improves precision estimates |
The pooled SD is appropriate only when the equal-variance assumption holds. Check this before combining.
Frequently asked questions
Why is the pooled SD not simply the average of the two SDs?
A simple average gives equal weight to each group regardless of size. The pooled formula weights by degrees of freedom (n - 1), so a larger group pulls the result further toward its SD. This weighting makes Sp an unbiased estimator of the common population SD when variances are equal.
What is the difference between pooled standard deviation and pooled variance?
The pooled variance (Sp squared) is the weighted average of the sample variances. The pooled standard deviation is simply the square root of the pooled variance, putting the result back in the original units of the data. Most textbooks report Sp (not Sp squared) for effect sizes and t-tests.
Can I use this calculator for more than two groups?
Yes. The same formula generalizes to any number of groups by summing (n_i - 1) x s_i^2 across all k groups in the numerator and summing (n_i - 1) across all groups in the denominator. This is exactly what one-way ANOVA does when it computes the mean squared error (MSE): MSE = Sp^2.
How is the pooled SD used in a two-sample t-test?
The standard error of the difference between two means is SE = Sp x sqrt(1/n1 + 1/n2). The t-statistic is then (mean1 - mean2) / SE, tested against a t-distribution with (n1 + n2 - 2) degrees of freedom. This is the equal-variance form of the independent-samples t-test.
How does the pooled SD relate to Cohen's d?
Cohen's d is the standardized mean difference: d = (mean1 - mean2) / Sp. When groups have equal sample sizes, this is sometimes computed with the average SD, but the weighted pooled SD is the statistically correct denominator for unequal group sizes. Most power-analysis software uses the pooled SD definition.
What does degrees of freedom mean here?
Each group contributes (n_i - 1) degrees of freedom to the pooled estimate, one for each observation minus the one used to estimate the group mean. Summing across all groups gives the total degrees of freedom for the pooled SD, which equals the total sample size minus the number of groups (N - k).