P-value calculator.

What a p-value is

The p-value is the probability of the test statistic being at least as extreme as the one observed, given that the null hypothesis is true. For a z-score: two-tailed p = 2 × P(Z > |z|); right-tailed p = P(Z > z); left-tailed p = P(Z < z). For a t-score the same tail areas are taken from Student's t distribution with the given degrees of freedom. Compare the p-value with your significance level α: if p < α, reject the null hypothesis.

The critical value shown is the statistic at which the p-value equals α for the chosen tail: z_1−α/2 (or t_{1−α/2, df}) for a two-tailed test and z_1−α for a one-tailed test (the magnitude is shown; for a left-tailed test the cut-off is its negative).

Worked example

z = 2.16, two-tailed: P(Z > 2.16) = 0.01539, so p = 0.0308. At α = 0.05 the critical value is 1.960, and 2.16 is beyond it, so the result is significant. For t = 2.5 with 10 degrees of freedom the two-tailed p-value is 0.0314 and the 5% critical value is 2.228. The same 2.5 as a z-score would give 0.0124: the t distribution has heavier tails, so with few degrees of freedom the same statistic is less convincing.

z or t?

A z-score is used when the standard deviation is known or the sample is large enough that the normal approximation is accepted, as in the two-proportion z-test. When the standard deviation is estimated from a small sample, the statistic follows a t distribution. As the degrees of freedom grow the t distribution approaches the normal.

Sources, opened 8 October 2026: NIST/SEMATECH e-Handbook, 7.1.3.1 Critical values and p values (definition of the p-value, comparing the p-value with α, common α of 0.1, 0.05 and 0.01); NIST/SEMATECH e-Handbook, 1.3.6.7.1 Cumulative Distribution Function of the Standard Normal Distribution (z critical values 1.645 and 1.960); NIST/SEMATECH e-Handbook, 1.3.6.7.2 Critical Values of the Student's t Distribution (the 2.228 critical value for 10 degrees of freedom, and the infinity row that matches the normal critical values).