One-Sample t-Test Calculator (t Statistic & p-Value)
By Santaji GadepatilinUpdated November 7, 2026
A one-sample t-test checks whether the mean of a sample differs from a hypothesised population mean when the population standard deviation is unknown. This one-sample t-test calculator turns the sample mean, standard deviation and sample size into the t statistic, degrees of freedom, two-tailed and one-tailed p-values and Cohen's d.
How to use the one-sample t-test calculator
Enter the sample mean, the hypothesised mean μ₀ you are testing against, the sample standard deviation and the sample size. The calculator returns the standard error, the t statistic, the degrees of freedom and three p-values. Compare the p-value with your significance level α (usually 0.05): if p < α, reject the null hypothesis that the population mean equals μ₀. If you have raw data, get the mean and standard deviation first with the standard deviation calculator.
One-sample t-test formula
t = (x̄ − μ₀) ÷ (s / √n) · df = n − 1
p (two-tailed) = 2 × P(Tdf ≥ |t|) · Cohen’s d = (x̄ − μ₀) ÷ s
Worked example
A filling machine should deliver 100 g. A sample of n = 25 packs has a mean of 103.2 g and a standard deviation of 8.5 g. Is the mean different from 100 g?
- Standard error = 8.5 ÷ √25 = 1.7000 g
- t = (103.2 − 100) ÷ 1.7000 = 1.8824, with 24 degrees of freedom
- Two-tailed p-value = 0.0720; right-tailed p = 0.0360
- Cohen’s d = 3.2 ÷ 8.5 = 0.376 (a small-to-medium effect)
At α = 0.05 the two-tailed test does not reject μ = 100 g (p = 0.072 > 0.05), but a one-tailed test of “mean above 100 g” would (p = 0.036). This is why the direction must be chosen before looking at the data.
Critical values of t
| df | one-tailed 0.05 / two-tailed 0.10 | two-tailed 0.05 | two-tailed 0.01 |
|---|---|---|---|
| 1 | 6.314 | 12.706 | 63.657 |
| 2 | 2.920 | 4.303 | 9.925 |
| 3 | 2.353 | 3.182 | 5.841 |
| 4 | 2.132 | 2.776 | 4.604 |
| 5 | 2.015 | 2.571 | 4.032 |
| 6 | 1.943 | 2.447 | 3.707 |
| 8 | 1.860 | 2.306 | 3.355 |
| 10 | 1.812 | 2.228 | 3.169 |
| 12 | 1.782 | 2.179 | 3.055 |
| 15 | 1.753 | 2.131 | 2.947 |
| 20 | 1.725 | 2.086 | 2.845 |
| 24 | 1.711 | 2.064 | 2.797 |
| 30 | 1.697 | 2.042 | 2.750 |
| 40 | 1.684 | 2.021 | 2.704 |
| 60 | 1.671 | 2.000 | 2.660 |
| 120 | 1.658 | 1.980 | 2.617 |
| ∞ (normal) | 1.645 | 1.960 | 2.576 |
Reject the null hypothesis when |t| exceeds the critical value for your df and α. Values computed from the Student t distribution.
Practical notes
- Statistical vs practical significance — with a large n even a tiny difference gives a small p-value. Report the effect size (Cohen’s d ≈ 0.2 small, 0.5 medium, 0.8 large) and a confidence interval alongside p.
- Paired data — for before/after measurements on the same items, compute the differences and run a one-sample t-test on them with μ₀ = 0. That is exactly a paired t-test.
- Outliers — one extreme value inflates s and can hide a real difference; check the data with a box-plot or quartiles first.
Frequently asked questions
What is the formula for a one-sample t-test?
t = (x̄ − μ₀) ÷ (s ÷ √n), with n − 1 degrees of freedom. The p-value is the probability of a t value at least this extreme if the true mean were μ₀.
When should I use a t-test instead of a z-test?
Use the t-test when the population standard deviation is unknown and you estimate it from the sample — which is almost always. For large samples (n above about 30) the two give nearly the same answer.
What are the assumptions of a one-sample t-test?
The observations are independent and come from a roughly normal population. The test is fairly robust to non-normality for n of 30 or more, but small samples with strong skew or outliers call for a non-parametric test such as the Wilcoxon signed-rank test.
Should I use the one-tailed or two-tailed p-value?
Use the two-tailed p-value unless you decided before seeing the data that only one direction matters. A one-tailed p-value is half the two-tailed one when the effect is in the predicted direction.