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Sample Variance (s²)
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Mean (x̄)
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Std. Deviation
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Count (n)
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Sum (Σx)
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Minimum
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Maximum
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Range
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Sum of Squares
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Coeff. of Variation
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What Is a Pooled Variance Tool?
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.
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Frequently Asked Questions — Pooled Variance Calculator
Everything you need to know about variance.
What is pooled variance?
Pooled variance is a weighted average of the variances from two or more independent samples. It assumes that the populations have equal variances (homoscedasticity). The formula for two groups is: sp² = [(n₁−1)s₁² + (n₂−1)s₂²] / (n₁ + n₂ − 2). It gives more weight to larger samples and is used as the variance estimate in pooled t-tests.
When do you use pooled variance?
Pooled variance is used when: 1. Performing a two-sample t-test (assuming equal variances). 2. Conducting ANOVA to compare means across multiple groups. 3.Combining results from multiple studies (meta-analysis). 4. Estimating a common variance when groups are believed to come from populations with the same spread. Always verify the equal variance assumption first using Levene's test or an F-test.
How to calculate pooled variance step by step?
For two groups: 1. Calculate the variance of each group (s₁² and s₂²). 2. Note the sample sizes (n₁ and n₂). 3. Compute: sp² = [(n₁−1)·s₁² + (n₂−1)·s₂²] / (n₁ + n₂ − 2). Example: Group 1 (n=10, s²=8), Group 2 (n=15, s²=12): sp² = [(9)(8) + (14)(12)] / 23 = (72 + 168) / 23 = 10.43.
What is the difference between pooled variance and combined variance?
Pooled variance assumes equal population variances and weights by degrees of freedom — used in hypothesis testing. Combined variance is a broader term for merging datasets that may account for differences in both means and variances between groups. For combining two datasets into one, you also need to account for the difference between group means, not just within-group variance.
What assumption does pooled variance require?
Pooled variance assumes homogeneity of variances (homoscedasticity) — that all groups come from populations with the same variance. If this assumption is violated, the pooled estimate is unreliable. Test this assumption with Levene's test, Bartlett's test, or the F-test for equality of variances. If variances are unequal, use Welch's t-test instead of the pooled t-test.
Can I pool variances from more than two groups?
Yes. The general formula for k groups is: sp² = Σ[(nᵢ−1)·sᵢ²] / Σ(nᵢ−1). This is a weighted average where each group's variance is weighted by its degrees of freedom (nᵢ−1). This is exactly what ANOVA uses as the Mean Square Within (MSW) — the pooled estimate of variance across all groups.