From a t statistic to its p-value
Every t-test ends in a t statistic and a degrees-of-freedom count. This tool converts the pair into the p-value: the area in the tails of that t distribution at least as far from zero as your statistic. You need two things: the t value itself, and the df that came with it. The same t means different things at different df, because the t distribution's tails are fat when df is small and slim toward the normal as df grows.
t = 2.306, df = 8. The two-tailed p-value is 0.0500. Not a coincidence: 2.306 is exactly the critical value for α = 0.05 at df 8, the boundary of the rejection region. A t of 2.679 at df 7 (the bakery example from the one-sample t-test page) gives p = 0.0316.
Finding your df
- One-sample and paired t-tests: n − 1
- Two-sample pooled t-test: n₁ + n₂ − 2
- Welch's t-test: a decimal df from the Welch–Satterthwaite formula (your software or our two-sample page computes it)
- Correlation significance test: n − 2
Reading a t statistic on the normal (z) curve instead of its t curve understates the p-value: at df 7, badly. The fatter tails are the price of estimating the standard deviation from a small sample, and they are not optional.
Frequently asked questions
What p-value does t = 2 give?
It depends on df. Two-tailed: about 0.116 at df 5, about 0.073 at df 10, about 0.055 at df 25, and 0.0455 in the normal limit. Small samples forgive less.
My t is negative. What do I enter?
Enter it as it is. For a two-tailed p the sign is irrelevant; for one-tailed tests the sign decides which tail holds the area.