P-Value Calculator

The p-value is the probability of seeing a test statistic at least as extreme as the one you observed, assuming the null hypothesis is true. A small p-value means your data would be surprising under the null, which is taken as evidence against it. It is not the probability that the null is true.

Use Z when you have a large sample and a known population standard deviation. Use t when you estimated the standard deviation from a small sample, the usual case. Use chi-square for tests of independence or goodness-of-fit on counts. Use F for ANOVA and for comparing two variances.

Two-tailed tests count departures in either direction. Use a one-tailed test only when you specified the direction before collecting data. For symmetric Z and t distributions, the one-tailed p-value is half the two-tailed value only when the observed statistic points in the predicted direction.

The critical value is the test statistic that corresponds exactly to your chosen alpha. If your observed statistic is more extreme than the critical value, the p-value is below alpha and the result is statistically significant.

Very small p-values like 3.45e-5 are hard to read with leading zeros and lose precision when rounded to six decimals. The scientific form keeps the order of magnitude visible so you can compare results across orders of magnitude.