Two-Sample t-Test Statistic (Independent Samples) Calculator

Formula first

Overview

Also known as Welch's t-test, this formula is used to compare the means of two independent samples under the assumption of unequal variances. It measures the distance between the observed difference of sample means and the hypothesized population difference in units of standard error. The resulting t-value is then compared against a t-distribution to determine the p-value.

Symbols

Variables

t = t-statistic, $\bar{x}$_1 = Mean of sample 1, $\bar{x}$_2 = Mean of sample 2, $s_1^2$ = Variance of sample 1, $s_2^2$ = Variance of sample 2

Apply it well

When To Use

When to use: Use this test when comparing the means of two independent groups when the population standard deviations are unknown and you cannot assume equal variances.

Why it matters: It is a foundational tool in scientific research and A/B testing, allowing analysts to infer population differences from limited sample data without assuming homogeneity of variance.

Avoid these traps

Common Mistakes

• Assuming equal variances when the sample sizes or distributions differ significantly.
• Failing to confirm that the samples are truly independent (e.g., using it on paired data).
• Using the standard pooled variance formula instead of the unpooled version.

One free problem

Practice Problem

Two groups are tested. Group 1: mean=50, $s^2$=10, n=20. Group 2: mean=45, $s^2$=12, n=25. Assuming the hypothesized difference (mu1-mu2) is 0, what is the t-statistic?

Solve for: $t$

Hint: Calculate the denominator by summing $s{1}^2$/n1 and $s{2}^2$/n2, then take the square root of the result.

The full worked solution stays in the interactive walkthrough.

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Related Formulas

References

Sources

1. Rice, J. A. (2006). Mathematical Statistics and Data Analysis.
2. Welch, B. L. (1947). The generalization of 'Student's' problem when several different population variances are involved.
3. Welch, B. L. (1947). 'The generalization of 'Student's' problem when several different population variances are involved'.