Pooled Variance Calculator: Compute Combined Sample Variance

The Pooled Variance Calculator computes the pooled variance from two independent samples using their variances and sample sizes. This pooled estimate is the key input for many two-sample t-tests that assume equal variances.

Enter each sample’s variance and size, and the calculator returns the pooled variance, plus the pooled standard deviation for easier interpretation.

What is pooled variance?

Pooled variance is a weighted average of two sample variances. It estimates a single common variance for the population when the equal-variance assumption is reasonable. You use it to improve the stability of a variance estimate when each group has a limited sample size.

For two groups (often called group 1 and group 2), you typically start with:

• s1² and s2²: sample variances from each group
• n1 and n2: sample sizes

Core formula (and what each term means)

When assuming equal variances, the pooled variance is computed as a degrees-of-freedom weighted average of the two sample variances.

Pooled variance:

sp² = ((n1 − 1)·s1² + (n2 − 1)·s2²) / (n1 + n2 − 2)

Where:

• n1 − 1 and n2 − 1 are the degrees of freedom for each sample variance
• n1 + n2 − 2 is the total degrees of freedom
• sp² is the pooled variance estimate

If you also need the pooled standard deviation, take the square root:

sp = √sp²

Why weighting by (n − 1) matters

The factors (n1 − 1) and (n2 − 1) reflect how much information each sample variance provides. Larger samples produce more reliable variance estimates, so their variances receive more weight in the pooled result.

This approach reduces noise compared with simply averaging variances, especially when sample sizes are unequal.

When to use pooled variance (and when not to)

Pooled variance is designed for situations where the two populations have the same variance. A common use case is the classic two-sample t-test with equal variances (often called the “Student’s t-test” or “pooled t-test”).

Do not use pooled variance when variances are clearly different. In that case, a “Welch’s t-test” is usually more appropriate because it does not assume equal variances.

How to interpret the output

• Pooled variance (sp²): variance on the squared scale of your measurement (e.g., if data are in seconds, variance is in seconds²).
• Pooled standard deviation (sp): variance converted back to the original measurement scale (seconds).

Use the pooled standard deviation when you want an intuitive sense of spread in the same units as the raw data.

Worked example: quality control with two machines

Suppose you measure the thickness (in micrometers) of parts produced by two machines. You collect:

• Machine 1: n1 = 12, sample variance s1² = 4.0
• Machine 2: n2 = 10, sample variance s2² = 5.5

Pooled variance:

sp² = ((12 − 1)·4.0 + (10 − 1)·5.5) / (12 + 10 − 2)

sp² = (11·4.0 + 9·5.5) / 20 = (44 + 49.5) / 20 = 4.675

So pooled standard deviation is √4.675 ≈ 2.16 micrometers. This single spread estimate helps you compare the machines using a pooled-variance t-test.

Worked example: compare test scores across two classes

Imagine two classes taking the same exam. You compute variances from each class’s scores:

• Class A: n1 = 20, s1² = 36
• Class B: n2 = 18, s2² = 25

Pooled variance:

sp² = ((19)·36 + (17)·25) / (20 + 18 − 2) = (684 + 425) / 36 = 30.25

Pooled standard deviation is √30.25 = 5.50 points. You can then use sp in the equal-variance two-sample t-test workflow.

Step-by-step: how to use the calculator

1. Enter n1 (sample size for group 1). It must be an integer greater than 1.
2. Enter s1² (sample variance for group 1). It must be a non-negative number.
3. Enter n2 and s2² for group 2.
4. Click Calculate to compute pooled variance sp² and pooled standard deviation sp.
5. If you see an error, check that both sample sizes are > 1 and variances are ≥ 0.

Frequently Asked Questions

What does pooled variance measure?

Pooled variance measures a combined estimate of the variability shared by two groups. It takes a weighted average of each group’s sample variance using degrees of freedom (n − 1). This produces one variance estimate used in equal-variance two-sample t-tests and related confidence calculations.

How is pooled variance different from averaging two variances?

Simple averaging treats both sample variances as equally reliable. Pooled variance weights each variance by (n − 1), so larger samples influence the result more. This weighting reflects statistical information and typically yields a more accurate combined variance estimate under the equal-variance assumption.

Can pooled variance be used with unequal sample sizes?

Yes. Pooled variance is specifically designed to handle unequal sample sizes because it uses degrees-of-freedom weights (n1 − 1) and (n2 − 1). The formula automatically adjusts the contribution from each group, as long as the equal-variance assumption is reasonable.

Why do we divide by (n1 + n2 − 2)?

The denominator (n1 + n2 − 2) equals the total degrees of freedom from both sample variances. Since each sample variance is based on (n − 1) degrees of freedom, pooling must use the combined degrees of freedom to keep the estimator on the correct variance scale.

When should I avoid pooled variance?

Avoid pooled variance when the two populations likely have different variances. If one group shows much higher spread than the other, a Welch’s t-test is usually safer. You can assess variance equality using exploratory plots or formal tests like Levene’s test.

Quick reference table

Bottom line

The Pooled Variance Calculator gives you a single combined variance estimate from two sample variances and their sample sizes. Use it when you can justify equal variances; otherwise, choose a method that does not pool variances.