Enter the sample sizes and standard deviations for up to three groups, and the Pooled Standard Deviation Calculator computes the combined pooled standard deviation and pooled variance. Provide n1, SD1, n2, and SD2 (plus optional n3 and SD3) to measure overall variability across datasets weighted by group size.
Pooled Standard Deviation
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Pooled Variance
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Total Sample Size (N)
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Total Degrees of Freedom
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Number of Groups Combined
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Pooled standard deviation is a weighted average of the standard deviations from two or more groups, where each group's contribution is weighted by its degrees of freedom (n − 1). It gives a single combined measure of variability that is more reliable than any individual group's SD when sample sizes differ.
For two groups, the formula is: SD_pooled = √[((n₁−1)×SD₁² + (n₂−1)×SD₂²) / (n₁ + n₂ − 2)]. For three or more groups, the numerator extends to include each additional group's weighted sum of squares, and the denominator becomes the total degrees of freedom (sum of all n − number of groups).
Use pooled SD when you assume the populations have equal variances (homogeneity of variance) but different sample sizes. It is commonly used in two-sample t-tests, ANOVA, and quality control applications to combine variability estimates across groups into a single, more precise value.
Pooled variance is the weighted average of the individual group variances (SD²), calculated as the sum of weighted squared deviations divided by total degrees of freedom. Pooled standard deviation is simply the square root of the pooled variance, expressed in the same units as the original data.
No. Pooled standard deviation works with unequal sample sizes — in fact, that is one of its main advantages. Groups with larger sample sizes receive more weight (via larger n − 1 degrees of freedom), so the pooled estimate reflects the more reliable groups more heavily.
The key assumption is homogeneity of variance — that the true population variance is the same across all groups being pooled. If variances differ substantially between groups, pooling them can produce a misleading estimate, and you should instead use Welch's correction or report group variances separately.
In an equal-variance two-sample t-test, pooled SD is used in the denominator of the t-statistic. The formula is t = (x̄₁ − x̄₂) / (SD_pooled × √(1/n₁ + 1/n₂)). Using pooled SD rather than separate SDs increases statistical power when the equal-variance assumption holds.
Yes. The pooled standard deviation formula extends naturally to three or more groups by summing (nᵢ − 1) × SDᵢ² for each group in the numerator and dividing by the total degrees of freedom (sum of all nᵢ minus the number of groups). This calculator supports up to three groups.