About the Pooled Standard Deviation Calculator
This pooled standard deviation calculator combines the spread of two or more groups into a single estimate of the common standard deviation. Enter each group on its own line as "n, mean, SD" (or just "n, SD") from a paper or report, or switch to raw data and paste each group's values on a separate line. It returns the pooled standard deviation, the pooled variance, the total sample size and the degrees of freedom, plus a per-group breakdown showing how much weight each group carries.
Pooled SD is the denominator of Cohen's d and the error term in the two-sample (equal-variance) t-test and one-way ANOVA, so researchers, students and meta-analysts reach for it whenever they need to compare group means on a common scale. With exactly two groups and their means, the calculator also reports Cohen's d and Hedges' g.
The calculation assumes the groups share the same underlying variance; it weights each sample variance by its degrees of freedom (n − 1). If one group's SD is more than about twice another's, consider a method that does not assume equal variances, such as Welch's t-test. Each group needs at least two observations.
With the default inputs, the pooled standard deviation is 8.5158. Change any value above to recalculate instantly.
How to use the pooled standard deviation calculator
- 1Choose summary statistics or raw data.
- 2For summaries, enter one group per line as "n, mean, SD" (or "n, SD").
- 3For raw data, paste each group's values on its own line.
- 4Read the pooled SD, pooled variance and degrees of freedom.
- 5With two groups and means, read Cohen's d and Hedges' g for the effect size.
Formula and method
Each group's sample variance is weighted by its degrees of freedom, n − 1. Adding up these weighted variances gives the pooled sum of squares within groups, and dividing by the total degrees of freedom N − k gives the pooled variance — the same quantity as the mean square error (MSW) in a one-way ANOVA. The pooled standard deviation is its square root.
Pooling is appropriate when the groups are assumed to have a common population variance; larger groups carry more weight because their variance estimates are more precise. Note that pooled SD is not the standard deviation of all the data combined: it ignores differences between group means. For two groups, dividing the difference in means by the pooled SD gives Cohen's d; Hedges' g applies a small-sample correction.
- sₚ
- Pooled standard deviation
- nᵢ, sᵢ
- Size and sample standard deviation of group i
- N
- Total sample size across all groups
- k
- Number of groups
- x̄₁, x̄₂
- Means of groups 1 and 2 (for Cohen’s d)
Worked examples
Three classes’ test scores from summary statistics
Weighted sums of squares: 24 × 8.1² = 1,574.64, 29 × 9.6² = 2,672.64 and 21 × 7.3² = 1,119.09, totalling 5,366.37. Dividing by 77 − 3 = 74 gives a pooled variance of about 72.52, so the pooled SD is √72.52 ≈ 8.52 points.
Two treatment groups with Cohen's d
Pooled variance = (39 × 5.2² + 34 × 6.1²) ÷ 73 = (1,054.56 + 1,265.14) ÷ 73 ≈ 31.78, so sₚ ≈ 5.64. The means differ by 3.5, giving d = 3.5 ÷ 5.64 ≈ 0.62, a medium effect; Hedges’ g ≈ 0.61.
Raw data for two small groups
Group 1 (12, 15, 14, 10, 13) has variance 3.7 and group 2 (18, 16, 20, 17, 19, 21) has variance 3.5. Pooled variance = (4 × 3.7 + 5 × 3.5) ÷ 9 = 32.3 ÷ 9 ≈ 3.589, so sₚ ≈ 1.894. The means are 12.8 and 18.5, so d = −5.7 ÷ 1.894 ≈ −3.01 — group 2 is about three pooled SDs higher.
Frequently asked questions
What is pooled standard deviation?+
It is a weighted average of the variances of two or more groups, square-rooted, that estimates a common standard deviation assumed to be shared by all groups. Each group is weighted by its degrees of freedom, n − 1.
When should I use pooled standard deviation?+
Use it when the groups plausibly have equal population variances: for the classic two-sample t-test, one-way ANOVA, and when computing Cohen's d. If the SDs differ a lot (one more than about twice another), use Welch's t-test instead.
Is pooled SD the same as the SD of all data combined?+
No. The pooled SD only measures spread within groups. The SD of all the data lumped together also includes the differences between group means, so it is usually larger when the means differ.
What is the formula for pooled standard deviation with two groups?+
sₚ = √[((n₁ − 1)s₁² + (n₂ − 1)s₂²) ÷ (n₁ + n₂ − 2)]. With equal group sizes this simplifies to the square root of the average of the two variances.
How does pooled SD relate to Cohen’s d?+
Cohen's d is the difference between two group means divided by the pooled standard deviation. It expresses the gap in standard-deviation units, so d = 0.5 means the means are half a pooled SD apart.