P-value Calculator
The p-value is the probability of getting a test result at least as extreme as the one you observed, assuming the null hypothesis is true. It is the core output of most statistical hypothesis tests. A small p-value indicates that your result would be rare under the null hypothesis, giving you grounds to reject it. The conventional threshold is 0.05: results with p below 0.05 are called statistically significant at the 5% level. Results with p below 0.01 are significant at the 1% level. The choice between a one-tailed and a two-tailed test should be made before you collect data. A two-tailed test asks whether the result is extreme in either direction; a one-tailed test asks whether it is extreme in a specific direction. For the same test statistic, the one-tailed p-value is exactly half the two-tailed p-value. This calculator supports two types of test statistic. The z-score mode uses the standard normal distribution and is appropriate when you know the population standard deviation or have a large sample. The t-score mode uses the t-distribution with your specified degrees of freedom, which is appropriate for smaller samples or when the population SD is unknown. The z = 1.96 default corresponds to a two-tailed p-value of 0.0500, which is the exact boundary of the 5% significance level. Enter your test statistic, choose the test type and tail option, and add degrees of freedom if you are using a t-test.
How it works
For the z-test, the calculator uses the complementary error function (erfc) to compute the tail area of the standard normal distribution. The one-tailed p-value is the area in one tail beyond |z|; the two-tailed value doubles it. For the t-test, the calculator uses an accurate numerical approximation of the regularised incomplete beta function to compute tail probabilities for the t-distribution with the specified degrees of freedom. Results are significant at 0.05 if p is below 0.05, and significant at 0.01 if p is below 0.01.
Worked example
With a z-score of 1.96 and a two-tailed test, the area in both tails of the standard normal distribution equals exactly 0.0500. This is the boundary of the 5% significance level: a z-score of 1.96 gives a p-value of exactly 0.05 two-tailed, so results with |z| greater than 1.96 are significant at the 5% level. These match the default values pre-filled above.
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