t-Test Online: One-Sample, Welch Two-Sample and Paired
What is the One-Sample t-Test?
The one-sample t-test checks whether a sample mean differs from a known target value such as a specification center or standard value, testing H0: μ = μ0. The statistic is t = (x̄ − μ0)/(s/√n), which follows a t-distribution with n − 1 degrees of freedom, where s is the sample standard deviation and n the sample size. The tool outputs the t-value, degrees of freedom, p-value and a confidence interval for the mean difference. If the interval contains the target value, there is insufficient evidence that the mean deviates from it.
When to Use It
Use the one-sample version to compare a process mean with a target or standard value, and the two-sample version to compare two independent populations such as two processes, two suppliers or two shifts. Use the paired version for before-and-after measurements or matched pairs, where eliminating between-subject variation makes the test more sensitive. The Welch version is the default choice because it does not assume equal variances.
How to Use It (Step by Step)
Choose the test type (one-sample, two-sample or paired) and paste the data, or enter summary statistics. For paired data, make sure the two measurements of each pair are on the same row so differences are computed correctly. The tool returns the t-value, degrees of freedom, p-value and confidence interval; p < 0.05 rejects the null hypothesis.
Key Formulas / Example
One-sample t = (x̄ − μ0)/(s/√n). The Welch two-sample t = (x̄1 − x̄2)/√(s1²/n1 + s2²/n2) uses Satterthwaite-corrected degrees of freedom. The paired t = d̄/(sd/√n) is computed on the differences d. Check normality and outliers before testing, and interpret significance together with the size of the mean difference.