🧪 t-Test Calculator

Run one-sample, two-sample (pooled or Welch), and paired t-tests. Enter your summary statistics (or paired datasets), choose the alternative hypothesis, and get the test statistic, degrees of freedom, p-value, effect size, and plain-language conclusions.

1. Select t-Test Type

One-sample t-test Two-sample pooled Two-sample Welch Paired t-test

Tail Direction

Two-tailed (≠) Left-tailed (<) Right-tailed (>)

Significance Level (α)

Custom α must lie between 0.001 and 0.25.

2. Provide Sample Information

Sample Mean (\\(\bar{x}\\))

Hypothesised Mean (\\(\mu_0\\))

Sample Standard Deviation (\\(s\\))

Sample Size (\\(n\\))

Formula Reference

One-sample

\\[ t = \dfrac{\bar{x} - \mu_0}{s / \sqrt{n}}, \qquad \text{df} = n - 1 \\]

Pooled two-sample

\\[ t = \dfrac{(\bar{x}_1 - \bar{x}_2) - \Delta_0}{s_p \sqrt{\tfrac{1}{n_1} + \tfrac{1}{n_2}}}, \quad s_p^2 = \dfrac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2} \\]

Welch two-sample

\\[ t = \dfrac{(\bar{x}_1 - \bar{x}_2) - \Delta_0}{\sqrt{\tfrac{s_1^2}{n_1} + \tfrac{s_2^2}{n_2}}}, \quad \nu = \dfrac{(s_1^2/n_1 + s_2^2/n_2)^2}{\tfrac{(s_1^2/n_1)^2}{n_1 - 1} + \tfrac{(s_2^2/n_2)^2}{n_2 - 1}} \\]

Paired

\\[ t = \dfrac{\bar{d} - \mu_d}{s_d / \sqrt{n}}, \qquad \text{df} = n - 1 \\]

How to Use This Calculator

Select the t-test that matches your design (one sample, two independent samples, or paired observations).
Enter the summary statistics (or paste paired raw data) requested for that test.
Choose the alternative hypothesis direction and significance level \\(\alpha\\).
Click “Run t-Test” to compute the test statistic, degrees of freedom, p-value, and effect size.
Use the decision panel and step-by-step notes to interpret and report the result confidently.

References

Gossett, W. S. (1908). “The probable error of a mean.” Biometrika, 6(1), 1–25.
Montgomery, D. C., & Runger, G. C. (2014). Applied Statistics and Probability for Engineers (6th ed.). Wiley.
Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury.
Welch, B. L. (1947). “The generalization of Student's problem when several different population variances are involved.” Biometrika, 34(1-2), 28–35.

Disclaimer

Verify assumptions (independent observations, approximate normality for small samples, variance equality for the pooled test) before relying on these inferences. Complement numerical output with diagnostic plots where possible.