Student's t-Distribution Calculator (p-Value from t)

How it works

P(T ≤ t) = 1 − ½·I_{ν/(ν+t²)}(ν/2, ½) for t ≥ 0 (regularized incomplete beta) · two-tailed p = 2·min(P(T ≤ t), P(T ≥ t)) · f(t) = Γ((ν+1)/2) / (√(νπ)·Γ(ν/2)) · (1 + t²/ν)^(−(ν+1)/2)

Student's t distribution describes a sample mean standardized with the sample standard deviation. It is bell-shaped like the normal curve but with heavier tails, and it approaches the normal as the degrees of freedom ν grow. The p-value is the probability of a t at least as extreme as yours if the null hypothesis is true: one tail for a directional test, both tails for a two-sided test. The probabilities are computed exactly from the incomplete beta function.

Worked example

For t = 2.1 with 15 degrees of freedom, the area above 2.1 is 0.026528, so the two-tailed p-value is 0.053055. That is just above 0.05, so the result is not significant at the 5% level two-sided, although a right-tailed test (p = 0.026528) would be.

Sources: Student's t-distribution — Wikipedia · NIST/SEMATECH e-Handbook — t Distribution

Frequently asked questions