Pooled Standard Deviation Calculator (two or more groups from n and SD)

Understanding the pooled standard deviation calculator — concept and principle

This calculator combines the standard deviations of two or more groups into one pooled standard deviation, assuming the groups share a common variance. It also shows the pooled variance, degrees of freedom, total sample size, the simple average of the SDs and the ratio of the largest to smallest SD. Use it for two-sample t-tests, ANOVA, Cohen's d and meta-analysis.

Bigger groups count more, and variances, not SDs, are averaged.

How it is calculated

s_p² = Σ(nᵢ − 1)sᵢ² ÷ Σ(nᵢ − 1), and s_p = √(s_p²).

nᵢ is each group's size and sᵢ its sample standard deviation (with n − 1 in the denominator).
For two groups: s_p² = ((n₁ − 1)s₁² + (n₂ − 1)s₂²) ÷ (n₁ + n₂ − 2).
Degrees of freedom = Σ(nᵢ − 1) = total size − number of groups.

Enter one "n, SD" pair per line.

The principle

If several groups have the same true variance, each sample variance is an estimate of it, and combining them gives a better estimate. Variances, not standard deviations, are averaged because variances add properly. Each group is weighted by its degrees of freedom n − 1, which is the amount of information its variance contains: a group of 15 says more about spread than a group of 3. The result is the same as pooling all the squared deviations from each group's own mean and dividing by the total degrees of freedom.

Worked example

Default groups: (12, 4.2), (15, 5.1), (10, 3.8).

(12 − 1) × 4.2² = 11 × 17.64 = 194.04.
(15 − 1) × 5.1² = 14 × 26.01 = 364.14.
(10 − 1) × 3.8² = 9 × 14.44 = 129.96.
Pooled variance = (194.04 + 364.14 + 129.96) ÷ 34 = 688.14 ÷ 34 ≈ 20.2394.
Pooled SD = √20.2394 ≈ 4.49882. Total size 37; simple SD average 4.36667; largest ÷ smallest SD = 5.1 ÷ 3.8 ≈ 1.342.

How to read the result

The pooled SD is the typical spread within groups. It lies between the smallest and largest SD and is pulled toward the larger groups. If the ratio of the largest to smallest SD is small, the equal-variance assumption is reasonable. If it is large, the assumption is doubtful and methods such as Welch's t-test, which do not pool, are safer.

Common mistakes

Averaging the standard deviations directly instead of the variances.
Weighting by n instead of n − 1.
Entering standard errors instead of standard deviations.
Pooling groups with very different spreads.

Keep in mind: Pooling assumes the groups have equal population variances. Check that assumption with the SD ratio or a formal test before using the pooled value in a t-test.

Frequently asked questions

Why is the pooled SD different from the average SD?

It averages variances and weights by group size, while the simple average treats every group equally.

Is pooled SD used in Cohen's d?

Yes. Cohen's d for two independent groups divides the mean difference by the pooled SD.

Can I pool more than two groups?

Yes. The same weighted formula works for any number of groups; it is the within-group variance used in one-way ANOVA.

Sources: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.3 Two-Sample t-Test for Equal Means

Checked 9 Nov 2026. NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.3 Two-Sample t-Test for Equal Means: pooled variance s_p² = ((N₁ − 1)s₁² + (N₂ − 1)s₂²)/(N₁ + N₂ − 2). Extension to k groups with weights nᵢ − 1 is the standard generalisation.

Your numbers