Suppose we have a family of hypothesis tests of a null hypothesis that let us test the hypothesis at any significance level p between 0 and 100% we choose. The P value of the null hypothesis given the data is the smallest significance level p for which any of the tests would have rejected the null hypothesis.

For example, let X be a test statistic, and for p between 0 and 100%, let xp be the smallest number such that, under the null hypothesis,

P( X ≤ x ) ≥ p.

Then for any p between 0 and 100%, the rule

reject the null hypothesis if X < xp

tests the null hypothesis at significance level p. If we observed X = x, the P-value of the null hypothesis given the data would be the smallest p such that x < xp.