Two Sample T Test With Unequal Variance Calculator

Calculator

Input mode

Sample 1 label

Sample 2 label

Sample 1 raw values

Sample 2 raw values

Raw data note

Separate values with commas, spaces, semicolons, or new lines.

Sample 1 size

Sample 1 mean

Sample 1 standard deviation

Sample 2 size

Sample 2 mean

Sample 2 standard deviation

Hypothesized mean difference

Alternative hypothesis

Alpha level

Confidence level percent

Decimal precision

Formula Used

The calculator uses Welch's two sample t test. This test does not assume equal population variances.

Mean difference: d = x̄1 - x̄2

Test statistic: t = ((x̄1 - x̄2) - Δ0) / √((s1² / n1) + (s2² / n2))

Welch degrees of freedom:

df = ((s1² / n1 + s2² / n2)²) / (((s1² / n1)² / (n1 - 1)) + ((s2² / n2)² / (n2 - 1)))

Confidence interval: d ± t critical × standard error

How to Use This Calculator

Select raw data or summary statistics.
Enter both independent sample groups.
Add the hypothesized mean difference. Use 0 for equal means.
Choose the test direction and alpha level.
Set the confidence level for the interval.
Press calculate to see the result above the form.
Use CSV or PDF export for reporting.

Example Data Table

Group	Raw values	n	Mean	Standard deviation
Training A	82, 85, 88, 91, 90, 86	6	87	3.3466
Training B	78, 79, 81, 83, 80, 82	6	80.5	1.8708
Suggested setup	Use raw data mode	Both groups	Two tailed	Alpha 0.05

About This Calculator

This two sample t test calculator helps compare two independent means. It is built for unequal variance cases. The method is often called Welch's t test. It works when sample sizes differ. It also works when standard deviations are not alike.

Why Unequal Variance Matters

A pooled test assumes both groups share one variance. Real data often breaks that rule. Machines may vary. Classes may differ. Medical groups may have different spread. Welch's method keeps each group variance separate. That makes the test more flexible. It also protects decisions when sample balance is poor.

What You Can Enter

You can enter raw sample values. Separate numbers with commas, spaces, or new lines. The calculator finds the mean, variance, and standard deviation. You can also enter summary statistics. Use this option when you already know the sample size, mean, and standard deviation. Both modes use the same final test.

How Results Are Read

The t statistic shows how far the observed mean difference is from the hypothesized difference. The standard error shows expected sampling noise. The degrees of freedom are adjusted by the Welch formula. The p value measures evidence against the null hypothesis. A small p value supports rejecting the null claim at your selected alpha level.

Confidence Interval Meaning

The confidence interval gives a reasonable range for the true mean difference. It is centered on the observed difference. A wider interval means more uncertainty. Large standard deviations make it wider. Small sample sizes also make it wider. If a two sided interval excludes zero, the two sided test often rejects equality.

Good Practice Tips

Use independent observations. Do not mix paired data with this test. Check for extreme outliers before trusting results. Very skewed data can affect small samples. Report the test direction, t value, degrees of freedom, p value, and confidence interval. Also report group means and sample sizes. These details make the conclusion easier to audit.

When To Prefer It

Choose this test when the two groups are unrelated. Choose it when spreads look different. Choose it when group counts are not equal. It is a safer default for many comparisons. It avoids the risky assumption that both populations share one common variance exactly.

FAQs

What is a two sample t test with unequal variance?

It is Welch's t test. It compares two independent sample means without assuming equal variances. It is useful when group spreads or sample sizes differ.

When should I use Welch's t test?

Use it for two unrelated groups when equal variance is doubtful. It is often a safe choice for real data because it handles unequal standard deviations well.

Can I use raw data in this calculator?

Yes. Choose raw data mode. Enter each group in its own box. Use commas, spaces, semicolons, or new lines between values.

Can I use summary statistics instead?

Yes. Choose summary statistics mode. Enter sample size, mean, and standard deviation for each group. The calculator then applies the Welch formulas.

What does the p value mean?

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

What is the null hypothesis?

The null hypothesis states that the true mean difference equals the hypothesized difference. Most tests use zero, meaning both population means are equal.

Why are degrees of freedom not whole numbers?

Welch's formula adjusts degrees of freedom using both variances and sample sizes. That calculation often produces a decimal value, which is expected.

Is this calculator for paired data?

No. This calculator is for independent samples. For before and after data from the same subjects, use a paired t test instead.