Pooled Variance Calculator
When comparing the means of two independent groups using a t-test, you may need a pooled estimate of variance if you assume the two populations have equal variances. The pooled variance combines the two sample variances, weighting each by its degrees of freedom, to produce a single, more precise variance estimate. This pooled estimate is used to compute the standard error of the difference between means in the two-sample t-test. Enter the sample sizes and standard deviations for each group below. The calculator outputs the pooled variance, the pooled standard deviation, and the degrees of freedom.
Pooled variance formula
sp² = ((n1-1) s1² + (n2-1) s2²) / (n1 + n2 - 2)
The pooled standard deviation is the square root of the pooled variance. The degrees of freedom equal n1 plus n2 minus 2. This formula gives more weight to the group with the larger sample.
Using pooled variance in a t-test
- The pooled variance is used to estimate the standard error of the difference between two means: SE = sp times the square root of (1/n1 plus 1/n2).
- The t-statistic = (mean1 minus mean2) divided by SE, tested against the t-distribution with (n1 plus n2 minus 2) degrees of freedom.
- Pool only when you have evidence that the two population variances are approximately equal. Use Levene's or Bartlett's test first.
Frequently asked questions
What is pooled variance?
Pooled variance is a weighted average of the variances from two or more groups, where each group's variance is weighted by its degrees of freedom. It assumes the true population variance is the same in both groups and combines them for a more precise estimate.
When is pooled variance used?
Pooled variance is used in the independent-samples t-test (equal variances assumed) and in ANOVA. It provides a single, more stable variance estimate when you can reasonably assume the populations have equal variances.
What is the pooled variance formula?
Pooled variance = ((n1-1) times s1-squared plus (n2-1) times s2-squared) divided by (n1 plus n2 minus 2), where n1 and n2 are the sample sizes and s1 and s2 are the sample standard deviations.
How do I know if I should pool variances?
Use Levene's test or the F-test for equality of variances. If the p-value is greater than 0.05, pooling is typically acceptable. If variances differ substantially (ratio greater than 4:1 or test is significant), use the Welch t-test which does not pool variances.
Does pooled variance require equal sample sizes?
No. The formula weights each group's variance by its degrees of freedom (n minus 1), so groups of different sizes contribute proportionally. Larger samples contribute more to the pooled estimate.
Official sources
- NIST/SEMATECH e-Handbook of Statistical Methods: Two-Sample t-Test.
- NIST/SEMATECH e-Handbook: Engineering Statistics Handbook.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.