Calculator Inputs
Choose the test design, fill the sample statistics, then calculate the t statistic, p-value, effect size, and interval.
Formula Used
One-sample t-test
t = (x̄ − μ0) / (s / √n)
Degrees of freedom equal n − 1. Use this test when one sample mean is compared against a target mean.
Independent equal-variance t-test
t = [(x̄₁ − x̄₂) − μ0] / [sp √(1/n₁ + 1/n₂)]
sp is the pooled standard deviation. Degrees of freedom equal n₁ + n₂ − 2.
Welch t-test
t = [(x̄₁ − x̄₂) − μ0] / √(s₁²/n₁ + s₂²/n₂)
Degrees of freedom use the Welch–Satterthwaite approximation, which is better when the group variances differ.
Paired t-test
t = (d̄ − μ0) / (sd / √n)
The paired test works on within-pair differences, so the standard deviation belongs to those differences rather than the original samples.
How to Use This Calculator
- Select the t-test design matching your study: one-sample, paired, independent equal-variance, or Welch.
- Choose whether your alternative hypothesis is two-tailed, greater than, or less than.
- Enter the hypothesized mean or mean difference and your significance level.
- Fill the sample means, standard deviations, and sizes. For paired tests, use the mean and standard deviation of differences.
- Press Calculate T-Test to show the result panel above the form.
- Use the CSV or PDF buttons to save the calculated metrics and interpretation.
Example Data Table
| Scenario | Test type | Mean 1 | SD 1 | n 1 | Mean 2 / μ₀ | SD 2 | n 2 | α |
|---|---|---|---|---|---|---|---|---|
| Exam score vs benchmark | One-sample | 78.4 | 8.2 | 30 | 75.0 | — | — | 0.05 |
| Method A vs Method B | Independent equal | 12.4 | 2.8 | 25 | 10.9 | 3.1 | 24 | 0.05 |
| Before-and-after improvement | Paired | 3.6 | 1.7 | 18 | 0.0 | — | — | 0.01 |
Frequently Asked Questions
1. What does a Student t-test measure?
A Student t-test checks whether a sample mean, or the difference between means, is large enough relative to variability to suggest a real effect instead of random sampling noise.
2. When should I use a one-sample t-test?
Use a one-sample t-test when you have one sample and want to compare its mean against a known target, standard, benchmark, or claimed population mean.
3. What is the difference between paired and independent tests?
Paired tests use matched observations or before-and-after data. Independent tests compare two unrelated groups, such as two different classes, treatments, or production methods.
4. When is Welch’s t-test better?
Welch’s t-test is safer when group standard deviations or sample sizes differ noticeably. It adjusts the degrees of freedom instead of assuming equal population variances.
5. What does the p-value tell me?
The p-value shows how unusual the observed t statistic would be if the null hypothesis were true. Smaller values provide stronger evidence against the null.
6. Why is the confidence interval useful?
A confidence interval gives a plausible range for the true mean difference. It helps you judge practical magnitude, uncertainty, and whether zero remains credible.
7. What assumptions should I check?
Check approximate normality, independence of observations, and reasonable measurement quality. For equal-variance tests, also verify that both groups have similar variance patterns.
8. Does this calculator use raw data?
This page uses summary statistics: means, standard deviations, and sample sizes. That makes it fast for reports, classroom work, and quick decision support.