Pooled Variance Test Calculator

Input mode

Choose quick summary entry or paste values.

Test type

Select the hypothesis direction.

Alpha (α)

Decision threshold for p-values.

Confidence level

Used for the mean-difference interval.

Export file name

Used for CSV and PDF names.

Sample 1

Mean (x̄₁)

Std dev (s₁)

Size (n₁)

Sample 2

Mean (x̄₂)

Std dev (s₂)

Size (n₂)

Results appear above this form after you compute.

Example data table

Group	n	Mean	Std dev	Notes
Sample 1	25	12.40	3.10	Model A response times
Sample 2	30	10.70	2.80	Model B response times

Try these numbers to verify your setup.

Formula used

Pooled variance (equal variance assumption):

s_p² = \( ((n_1-1)s_1^2 + (n_2-1)s_2^2) / (n_1+n_2-2) \)

Standard error of the mean difference:

SE = \( s_p\sqrt{1/n_1 + 1/n_2} \)

Test statistic:

t = \( (\bar{x}_1 - \bar{x}_2) / SE \), with df = \( n_1 + n_2 - 2 \)

Confidence interval for \(\Delta = \bar{x}_1 - \bar{x}_2\):

\( \Delta \pm t^* SE \)

Effect size:

Cohen’s d = \( \Delta / s_p \); Hedges’ g applies a small-sample correction.

The variance ratio check uses the F distribution as an optional diagnostic.

How to use this calculator

Pick an input mode: summary statistics or raw lists.
Provide both samples, ensuring sizes are at least two.
Select the test direction and set alpha and confidence.
Press Compute to see t, p-value, and pooled variance.
Use CSV or PDF export to save the full output.

FAQs

1) What does pooled variance mean?

It combines both sample variances into one estimate. This assumes both groups share a common population variance. The combined estimate improves stability when the assumption is reasonable.

2) When should I use this test?

Use it when comparing two independent means and you can justify equal variances. It is common for controlled experiments and well-matched groups with similar spread.

3) What if variances differ a lot?

Large variance differences can bias the pooled approach. Consider a heteroscedastic alternative like Welch’s t-test. The variance ratio check can flag potential issues quickly.

4) What does the p-value tell me?

It measures how surprising your mean difference is under the null hypothesis. Smaller values suggest stronger evidence against equal means, given the model assumptions you chose.

5) Why choose one-tailed versus two-tailed?

Two-tailed tests detect differences in either direction. One-tailed tests focus on a specific direction and should be selected before seeing data. They can increase power when justified.

6) How is the confidence interval computed?

It uses the pooled standard error and the critical t value. The interval provides a plausible range for the true mean difference. Wider intervals reflect smaller samples or higher variance.

7) What do Cohen’s d and Hedges’ g represent?

They standardize the mean difference using pooled spread. This makes results comparable across scales. Hedges’ g slightly corrects d for small samples and is often preferred.

8) Can I paste raw data instead of summaries?

Yes. Switch to raw input mode and paste values separated by spaces, commas, or new lines. The tool computes mean, sample standard deviation, and then runs the pooled analysis.