How this calculator works
Each sample variance contributes in proportion to its degrees of freedom n - 1. Dividing their weighted total by the combined degrees of freedom estimates a common within-group population variance. Taking its square root gives pooled SD. The interpretation assumes a defensible common variance across independent groups. Every group needs at least two observations; zero individual SDs are allowed.
Formula
sp² = Σ((n_i - 1)s_i²)/Σ(n_i - 1); sp = sqrt(sp²); df = Σn_i - number of groups.
Worked example
For sizes 5 and 7 with sample SDs 2 and 4, the weighted squared total is 4 × 4 + 6 × 16 = 112. Divide by 10 to get 11.2 and take the square root: 3.346640.
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Frequently asked questions
Is this the SD of all observations joined together?
No. Joined-data variance also includes differences between group means. Pooled within-group variance deliberately excludes those differences.
Can I average the group SDs?
A plain average generally differs. Pool the variances with n - 1 weights before taking a square root.
What if the group variances differ greatly?
The common-variance interpretation may be unsuitable. A procedure allowing unequal variances may better match your analysis.