Pooled T Test Calculator

Calculator Form

Input Type

Group 1 Label

Group 2 Label

Group 1 Sample Size

Group 1 Mean

Group 1 Standard Deviation

Group 2 Sample Size

Group 2 Mean

Group 2 Standard Deviation

Hypothesized Difference

Alpha

Test Direction

Decimal Places

Group 1 Raw Values

Group 2 Raw Values

Formula Used

Pooled variance: Sp² = ((n1 - 1)s1² + (n2 - 1)s2²) / (n1 + n2 - 2)

Pooled standard deviation: Sp = √Sp²

Standard error: SE = Sp × √(1/n1 + 1/n2)

T statistic: t = ((x̄1 - x̄2) - D0) / SE

Degrees of freedom: df = n1 + n2 - 2

Confidence interval: (x̄1 - x̄2) ± t critical × SE

Cohen d: d = (x̄1 - x̄2) / Sp

Hedges g: g = d × (1 - 3 / (4df - 1))

How to Use This Calculator

Select summary statistics or raw values.
Enter both group labels for clearer reports.
Add sample sizes, means, and standard deviations.
Use raw values when you want automatic means and deviations.
Enter the hypothesized difference. Use zero for most tests.
Select alpha and the test direction.
Press Calculate to view the result above the form.
Use CSV or PDF buttons to save the report.

Example Data Table

Group	Sample Size	Mean	Standard Deviation	Alpha	Tail	Expected Result
Training A	25	84.2	9.1	0.05	Two tailed	t ≈ 2.2207, df = 45
Training B	22	78.5	8.4	0.05	Two tailed	Mean difference = 5.7

Understanding the Pooled T Test

A pooled t test compares two independent means when both groups are treated as having the same population variance. It is useful when samples come from similar processes, labs, classes, or production lines. The method joins both sample variances into one pooled estimate. That pooled value then builds the standard error for the mean difference.

When Equal Variance Matters

The test assumes independent observations, roughly normal data, and similar variances. It can still work well with moderate sample sizes. Yet very different spreads can distort the result. In that case, Welch’s t test is safer. A pooled test is best when the equal variance assumption is planned, justified, or supported by domain knowledge.

What The Calculator Shows

This calculator accepts summary statistics or raw comma separated values. It reports the mean difference, pooled standard deviation, standard error, degrees of freedom, t statistic, p value, critical value, confidence interval, Cohen’s d, and Hedges’ g. These outputs help you judge both statistical evidence and practical size. A tiny p value may show a clear difference. A small effect size may still mean the change is not important in practice.

Interpreting Results Carefully

Start with the hypothesized difference. Many studies use zero, meaning both group means are expected to match. Choose a two tailed test when either direction matters. Choose a one tailed test only when the direction was decided before seeing data. Next, compare the p value with alpha. If p is below alpha, the result is statistically significant. Also review the confidence interval. If it excludes the hypothesized difference, it agrees with the test decision.

Why Reporting Details Helps

Good reports include sample sizes, means, standard deviations, degrees of freedom, test direction, alpha, t value, p value, and confidence interval. Effect sizes add context. They make results easier to compare across studies. Export buttons help preserve records for audits, homework, or research notes. Always pair the calculation with sound study design and honest assumptions.

Common Input Checks

Use positive sample sizes above one. Enter standard deviations, not variances. Keep units consistent for both groups. Remove text labels from raw values. Check outliers before testing. The calculator can guide decisions, but it cannot fix biased data well.

FAQs

What is a pooled t test?

It is an independent samples t test that assumes both groups share the same population variance. It combines both sample variances into one pooled variance before calculating the test statistic.

When should I use this test?

Use it when two groups are independent, data are roughly normal, and equal variance is reasonable. It is common in experiments, classroom comparisons, lab studies, and quality checks.

What is the null hypothesis?

The usual null hypothesis says the difference between the two population means equals the hypothesized difference. Most often, that hypothesized difference is zero.

What does the p value mean?

The p value shows how unusual the observed result is under the null hypothesis. A smaller value gives stronger evidence against the null hypothesis.

What does pooled standard deviation mean?

It is a shared spread estimate made from both group standard deviations. Larger samples receive more weight in this pooled estimate.

Can I enter raw data?

Yes. Choose raw values, then enter numbers separated by commas, spaces, or line breaks. The calculator will compute sample size, mean, and standard deviation.

What is Cohen d?

Cohen d is an effect size. It expresses the mean difference in pooled standard deviation units, making the result easier to compare across studies.

What if variances are not equal?

If group variances differ strongly, use Welch’s t test instead. Welch’s method does not require the equal variance assumption and is often safer for unequal spreads.