P-Value Calculator

Look up the area in the tail of the standard normal distribution beyond your z-score. For a one-tailed test the p-value is the area in one tail; for a two-tailed test it is twice the area in the smaller tail. For example, z = 1.96 gives a two-tailed p-value of about 0.05.

Use the Student-t distribution with the right degrees of freedom (df) instead of the normal distribution. The p-value is the tail area beyond your t-statistic. As df grows large the t-distribution approaches the normal distribution, so the two p-values converge.

A one-tailed test looks for an effect in a single direction and uses the area in one tail. A two-tailed test looks for an effect in either direction and doubles the tail area. The two-tailed p-value is therefore twice the one-tailed p-value for the same test statistic.

The p-value is the probability of observing a test statistic at least as extreme as the one you got, assuming the null hypothesis is true. A small p-value (commonly p < 0.05) suggests the data are unlikely under the null hypothesis, so you reject it.

0.05 is a common convention, not a law. The right threshold (alpha) depends on the field and the cost of a false positive. A p-value just below 0.05 is weak evidence; many fields use 0.01 or smaller, and you should choose alpha before running the test.

The normal probabilities use the Abramowitz-Stegun error-function approximation (absolute error under 1.5e-7), and the t-distribution uses the regularized incomplete beta function computed by a continued fraction. Results match standard statistical tables to the decimals shown.