Student t Probability Calculator
The Student t-distribution is used for inference about a mean when the population standard deviation is unknown and the sample is small. This calculator takes a t statistic and its degrees of freedom and returns the cumulative probability to the left, the one-tailed p-value, and the two-tailed p-value. The computation uses the regularized incomplete beta function, which gives exact tail areas for any degrees of freedom. As the degrees of freedom grow, the results converge to those of the standard normal distribution, as expected.
Student t-distribution formula
Let x = df / (df + t^2)
Two-tailed tail area = I(df/2, 1/2, x), the regularized incomplete beta
CDF F(t) = 1 - 0.5 * tail area when t > 0, else 0.5 * tail area
One-tailed p-value = 1 - F(t) for t > 0
Two-tailed p-value = 2 * (1 - F(|t|))
Degrees of freedom must be positive; the t statistic may be any real number.
t-test context
- One-sample or paired t-test degrees of freedom: sample size minus one.
- Two-sample equal-variance t-test degrees of freedom: combined size minus two.
- Use a one-tailed p-value for a directional hypothesis, two-tailed otherwise.
- A t of 2.776 at 4 degrees of freedom gives a two-tailed p of about 0.05.
- As degrees of freedom rise, the t-distribution approaches the standard normal.
Student t probability: frequently asked questions
What does this Student t calculator return?
Enter a t statistic and the degrees of freedom. It returns the cumulative probability to the left of the statistic, the one-tailed p-value, and the two-tailed p-value, which are the quantities most often needed when interpreting a t-test.
How is the t-distribution probability computed?
The calculator uses the regularized incomplete beta function. With x = df / (df + t squared), the two-tailed tail area is the incomplete beta I(df/2, 1/2, x). The cumulative probability is derived from this, and one-tailed and two-tailed p-values follow directly.
What is the difference between one-tailed and two-tailed p-values?
A one-tailed test asks whether the statistic is extreme in a single direction, so it uses the area in one tail. A two-tailed test asks whether it is extreme in either direction and uses both tails, so the two-tailed p-value is twice the smaller one-tailed area. Choose the one that matches your hypothesis.
How do I find degrees of freedom for a t-test?
For a one-sample or paired t-test, degrees of freedom equal the sample size minus one. For a two-sample t-test with equal variances, it is the combined sample size minus two. Welch's unequal-variance test uses a separate formula; enter whichever value your test produced.
Does the t-distribution approach the normal distribution?
Yes. As degrees of freedom increase, the Student t-distribution converges to the standard normal distribution. For large degrees of freedom, say above 100, the t and normal p-values are nearly identical, which is why z-tests are used for large samples.
Official sources
- U.S. NIST/SEMATECH e-Handbook of Statistical Methods: t-Distribution.
- NIST Digital Library of Mathematical Functions: Incomplete Beta and Gamma Functions.
Reviewed by the CalculatorHub team, edited by James Graham, 17 June 2026. See our methodology.