What this calculator does
When two samples are assumed to come from populations with the same variance, their standard deviations are combined into a single pooled estimate, weighted by degrees of freedom. Standard deviations of 9 and 11 from samples of 20 and 25 pool to 10.165.
The weighting happens on variances, not standard deviations, and that distinction produces results people find surprising. Pooling 5 and 20 from equal-sized samples gives 14.577, not the midpoint of 12.5, because the average of 25 and 400 is 212.5 and its square root is 14.577. The larger spread dominates.
The formula
Each sample variance is weighted by its degrees of freedom, n−1, the weighted values are summed and divided by the total degrees of freedom, and the square root gives the pooled standard deviation. The larger sample therefore has more influence. This is the estimate used by the equal-variance two-sample t-test and by Cohen d.
| Term | Meaning |
|---|---|
| Pooled standard deviation | A single spread estimate combining two samples. |
| Degrees of freedom | n−1 for each sample, and n₁+n₂−2 in total, which is the weighting basis. |
| Homogeneity of variance | The assumption that both populations share a variance, which pooling requires. |
| Standard error of the difference | The pooled SD scaled for comparing two means. |
The inputs explained
| Field | What to enter |
|---|---|
| Sample 1 standard deviation | Sample 1 standard deviation. |
| Sample 1 size | Sample 1 size. |
| Sample 2 standard deviation | Sample 2 standard deviation. |
| Sample 2 size | Sample 2 size. |
When to use it
Running a two-sample t-test
The equal-variance form of the test needs a pooled standard deviation as its denominator.
Calculating an effect size
Cohen d divides the mean difference by the pooled standard deviation.
Combining pilot data
Two small studies of the same measure can be pooled for a better spread estimate than either gives alone.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How do the inputs affect the pooled value?
Equal and unequal standard deviations and sample sizes.
| Sample 1 standard deviation | Pooled standard deviation | Pooled variance | Degrees of freedom |
|---|---|---|---|
| s₁ = 5 | 14.577 | 212.500 | 58 |
| s₁ = 10 | 15.811 | 250.000 | 58 |
| s₁ = 15 | 17.678 | 312.500 | 58 |
| s₁ = 20 | 20.000 | 400.000 | 58 |
Questions
Why is the pooled standard deviation not the average of the two?
Because variances are pooled, not standard deviations. Squaring before averaging and taking a square root afterwards gives more weight to the larger value. Pooling 5 and 20 gives 14.577, well above the arithmetic mean of 12.5.
When should I not pool?
When the two variances are genuinely different, particularly if the sample sizes are also unequal. Pooling then gives a misleading standard error and the test can be badly wrong. Use Welch t-test instead, which does not assume equal variances and costs very little when they are equal.
How do I check whether variances are equal enough?
An F-test compares two variances directly, and Levene test is a more robust alternative. A common informal rule accepts pooling when the larger standard deviation is less than about twice the smaller. Many statisticians now simply default to Welch and avoid the question.
Which sample has more influence?
The larger one, because weighting is by degrees of freedom. A sample of 60 contributes roughly three times the weight of a sample of 20, so the pooled estimate sits closer to the larger sample standard deviation.
For the effect size it feeds, see the Cohen’s d calculator. For the test statistic, see the t-statistic calculator.