A pooled variance calculator combines the variances of two or more independent samples into a single weighted estimate, assuming that the underlying populations share the same variance (homoscedasticity). The pooled variance weights each group's variance by its degrees of freedom (nᵢ − 1), giving more influence to larger samples. This combined estimate is essential in two-sample t-tests, analysis of variance (ANOVA), and meta-analyses where data from multiple independent groups or studies must be merged into one coherent measure of spread.
Formula
Pooled variance uses degrees-of-freedom weighting to combine group variances.
Pooled Variance (Two Groups)
sp² = [(n₁−1)s₁² + (n₂−1)s₂²] / (n₁ + n₂ − 2)
sp²Pooled variance
n₁, n₂Sample sizes of Group 1 and Group 2
s₁², s₂²Sample variances of Group 1 and Group 2
Pooled Variance (k Groups)
sp² = Σ[(nᵢ−1)sᵢ²] / Σ(nᵢ−1)
kNumber of groups
nᵢSample size of group i
sᵢ²Sample variance of group i
How to Calculate Pooled Variance Step by Step
Calculate each group's variance.Find s² for every independent sample.
Note sample sizes.Record n for each group.
Compute weighted sum of squares.Multiply each variance by (nᵢ − 1) and sum: Σ[(nᵢ − 1)sᵢ²].
Sum the degrees of freedom.Add up all (nᵢ − 1) values: Σ(nᵢ − 1).
Divide.sp² = Σ[(nᵢ − 1)sᵢ²] / Σ(nᵢ − 1).
Practical Pooled Variance Examples
1Comparing Two Teaching Methods
Data:Method A (n=12, s²=18) | Method B (n=15, s²=22)
Combine variances from multiple groups effortlessly:
Enter all raw data values from each group into the calculator's input field.
The calculator computes individual group variances and the pooled variance automatically.
Review the step-by-step breakdown showing each group's contribution to the pooled estimate.
Use the pooled variance in your t-test, ANOVA, or meta-analysis calculations.
Benefits of Using a Pooled Variance Calculator
Automates a multi-step calculation that is tedious and error-prone by hand.
Correctly weights by degrees of freedom — a step many students forget.
Supports more than two groups, unlike many simple online tools.
Provides the pooled standard deviation as well, ready for t-test denominators.
Shows individual group statistics alongside the combined result for full transparency.
Where Pooled Variance Is Commonly Used
Hypothesis Testing
The pooled t-test requires a pooled variance estimate to compare two group means.
ANOVA
The Mean Square Within (MSW) in ANOVA is exactly the pooled variance across all treatment groups.
Meta-Analysis
Combining results from multiple independent studies uses pooled variance for a unified effect-size estimate.
Clinical Trials
Comparing treatment vs control group outcomes under the assumption of equal variance.
Education Research
Evaluating whether different teaching methods produce different average outcomes.
Assumptions and Limitations of Pooled Variance
Pooled variance is powerful but relies on assumptions. Violating them can invalidate your analysis:
Homogeneity of variances required
If group variances differ substantially (heteroscedasticity), the pooled estimate is unreliable. Test with Levene's test or Bartlett's test before pooling.
Independent samples only
Pooled variance assumes groups are independent. If the same subjects appear in multiple groups (repeated measures), use paired analyses instead.
Sensitive to unequal sample sizes
When one group is much larger, it dominates the pooled estimate. This can mask important differences in the smaller group's variability.
Not suitable for very different means
If group means are far apart, the between-group differences can inflate the overall error estimate. ANOVA handles this by separating between- and within-group variance.
Alternative when assumptions fail
Use Welch's t-test instead — it does not assume equal variances and adjusts degrees of freedom accordingly.
プールされた分散は分散の均一性（等分散性）、つまりすべてのグループが同じ分散を持つ母集団に由来していることを前提としています。この仮定に違反すると、プールされた推定値は信頼できなくなります。この仮定をレベンヌ検定、バートレット検定、または分散等価性のF 検定で検定します。分散が等しくない場合は、プールされた t 検定の代わりにウェルチの t 検定を使用します。