Math & Statistics
Pooled Variance Calculator
Calculate each sample's degrees-of-freedom weighted variance contribution, pooled within-group variance and SD, size-weighted mean, raw mean difference, standardized difference, and SD ratio with a complete derivation.
Pooled within-group variance-
Combined within-group degrees of freedom-
Pooled standard deviation-
Size-weighted combined mean-
Sample 2 mean minus sample 1 mean-
Mean difference divided by pooled SD-
Larger SD divided by smaller SD-
Decision view
Degrees-of-freedom weighted variance pool
Degrees-of-freedom weighted variance poolTwo variance contributions are sized by within-group degrees of freedom before combining into pooled variance and pooled SD.
Exact scenario comparisonSample 2 standard deviation changes while all other entered assumptions remain constant.
| Sample 2 standard deviation | Pooled within-group variance | Combined within-group degrees of freedom | Pooled standard deviation | Size-weighted combined mean | Sample 2 mean minus sample 1 mean | Mean difference divided by pooled SD | Larger SD divided by smaller SD |
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How to use Pooled Variance Calculator
- Enter both sample sizes, SDs, and means using compatible units and definitions.
- Confirm each sample size is at least two and each SD is nonnegative.
- Inspect the two weighted variance contributions before reading the pool.
- Use pooled SD for downstream standardization only when a common-variance assumption is defensible.
Calculator guide
Understanding Pooled Variance Calculator
Pooled variance combines two within-group variance estimates using their degrees of freedom. This page separates spread pooling from the group means so between-group location differences are not mistaken for within-group variance.
Calculation method
How the calculation works
Weight each sample variance by its within-group degrees of freedom and divide by total degrees of freedom, keeping means and standardized difference separate. In the Pooled Variance Calculator, the live scenario varies sample 2 standard deviation and tracks pooled standard deviation while the remaining results preserve the reconciliation path. Square each sample SD to obtain variance, multiply by that sample's n minus one degrees of freedom, add the two contributions, and divide by the combined within-group degrees of freedom. Take the square root for pooled SD.
Detailed calculation process
Combine two within-group variance estimates
The default calculation gives sample 2 more variance weight because it has both a larger SD and more degrees of freedom.
General formula: s_p^2 = [((n1 - 1)s1^2) + ((n2 - 1)s2^2)] / (n1 + n2 - 2); s_p = sqrt(s_p^2) Each sample variance is weighted by the independent within-group information available to estimate it. Group means do not enter the pooled variance formula.
What each symbol means
Worked substitution with the default inputs
The defaults combine 60 within-group degrees of freedom into pooled variance 45.0235 and pooled SD 6.70996.
Pooling audit
Trace each group's contribution
The pooled result should preserve how much variance information came from each group.
Worked situations
Practical examples
- Sample 1 contributes 1,037.88 weighted squared units and sample 2 contributes 1,663.53.
- The size-weighted combined mean is approximately 56.194 but does not determine pooled variance.
- A four-unit mean gap becomes 0.596 pooled SD units.
Better inputs
Useful tips
- Compare group SDs and distribution shapes before pooling.
- Keep variance and SD units distinct throughout the calculation.
- Use methods designed for unequal variances when the common-spread assumption is doubtful.
Before relying on the result
Limitations and common mistakes
- Pooling assumes both samples estimate a common within-group variance.
- Dependence or overlapping samples invalidate the independent two-group weighting story.
- Nonrepresentative samples, strong skew, outliers, and measurement differences can undermine downstream inference.
Reference
Key terms
- Pooled variance
- Degrees-of-freedom weighted average of two sample variances.
- Within-group df
- Sample size minus one for each variance estimate.
- Pooled SD
- Square root of pooled variance.
- Standardized difference
- Group mean difference divided by pooled SD.
Important note
Calculated directly from the entered values using the displayed formula and rounding settings.
Frequently asked questions
Why are SDs squared before pooling?
Variances add under this weighting formula; taking the final square root converts the pooled variance back to SD.
Why not average the two SDs?
A simple SD average ignores squared units and the different information carried by each sample size.
Do the group means affect pooled variance?
No. This formula combines within-group spread; means are used separately for location comparisons.
What if the group variances differ substantially?
The arithmetic pool still computes, but equal-variance downstream methods may be inappropriate.