Pooled Standard Deviation Calculator
Combine the spread of 2–5 independent groups into one pooled SD — the weighted-average variance used by the Student two-sample t-test and ANOVA. Also compare against the naive "lump everything together" SD and see exactly when not to pool.
Pooled SD Calculator
Enter each group's sample size n, sample mean and sample standard deviation s. Groups are weighted by their degrees of freedom (n − 1) — bigger groups pull the pooled value harder.
Results
Each group's contribution
| Group | n | df (n−1) | s² | (n−1)·s² | Weight |
|---|
Naive combined SD vs Pooled SD
The naive SD lumps all observations into one dataset. That adds the between-group spread (group means drifting apart) on top of the within-group spread — so it answers a different question: "how spread are all the numbers together?"
Worked example
Two-sample t-test worked example
Uses your first two groups. Student's t-test divides the mean difference by a standard error built from the pooled SD:
Worked example
When to pool — and when to use Welch instead
Pooling assumes the groups share one true population variance (homoscedasticity). Rule of thumb: if the largest group variance is more than ~4× the smallest, the assumption is broken.
Quick reference
| Situation | Use |
|---|---|
| 2 groups, variances similar | Pooled SD + Student's t |
| 2 groups, variances differ | Welch's t (unpooled) |
| 3+ groups (ANOVA) | Pooled SD (MSE) — check homogeneity first |
| Describing one mixed dataset | Naive combined SD, not pooled |
| Repeated measures / paired data | Paired t-test — pooling doesn't apply |
Frequently asked questions
What is pooled standard deviation?
The pooled standard deviation is a weighted average of several groups' variances, converted back to a standard deviation. It weights each group by its degrees of freedom (n - 1), so larger samples pull harder. Formula: s_pooled = sqrt( sum((n_i - 1) * s_i^2) / sum(n_i - 1) ).
How do you calculate it? Worked example.
Two groups: n1 = 10, s1 = 8 and n2 = 12, s2 = 6. Numerator: 9 * 64 + 11 * 36 = 972. Denominator (df): 9 + 11 = 20. Pooled variance = 972 / 20 = 48.6. Pooled SD = sqrt(48.6) = 6.971370. This calculator's Load worked example button fills these numbers in.
When should I use pooled standard deviation?
Use it when you assume the groups share one true population variance: the Student two-sample t-test and ANOVA are the classic cases. In ANOVA, the pooled variance is the mean squared error (MSE). For three groups (n = 5, 5, 5 with SDs 2, 3, 4), the pooled variance is 9.666667 and the pooled SD is 3.109126 with df = 12.
What is the difference between pooled SD and naive combined SD?
Pooled SD combines only within-group spread. The naive combined SD lumps every observation into one dataset, adding between-group spread (group means drifting apart). Example: two groups of 10, means 0 and 20, SD 2 each. Pooled SD = 2.000000. Naive combined SD = 10.442827 — five times larger, because the between-group sum of squares (2000) swamps the within-group sum of squares (72).
Why do pooled and naive SD differ even when group means are equal?
Their denominators differ. With two groups of 10, mean 5, SDs 2 and 3: naive divides by N - 1 = 19 (2.481511), pooled divides by N - k = 18 (2.549510). When means are equal the difference is only this small denominator effect; when means differ, the between-group spread dominates.
When should I NOT pool the standard deviations?
Do not pool when the equal-variance assumption is broken. Rule of thumb: if the largest group variance is more than about 4 times the smallest, use Welch's t-test (unpooled) instead. The When to Pool tab checks this variance ratio on your data.
What are the degrees of freedom of the pooled SD?
df = sum(n_i - 1) over all groups. For n1 = 10 and n2 = 12, df = 20. For three groups of n = 5, df = 12. These are the degrees of freedom used by the t-test and ANOVA built on the pooled variance.
How does the pooled SD feed into a t-test? Worked example.
t = (mean1 - mean2) / (s_pooled * sqrt(1/n1 + 1/n2)). Example: n1 = 15, mean1 = 105, s1 = 12; n2 = 18, mean2 = 98, s2 = 10. Pooled SD = 10.948560, standard error = 3.827649, t = 1.828799 with df = 31. Welch's version gives t = 1.798093 with df = 27.35 — close, because the variances are similar (ratio 1.44).
Can I pool more than two groups?
Yes — the formula generalizes to any number of groups; this calculator supports 2 to 5. ANOVA does exactly this with k groups, calling the pooled variance the mean squared error.
How is this different from the average-and-standard-deviation calculator?
That tool summarizes ONE dataset: its mean and standard deviation. This tool COMBINES the standard deviations of 2 to 5 independent groups into a single pooled SD for t-tests and ANOVA — a different computation, not an extension.
Free tool by NRTechStudio