P-Value Calculator

A p-value is the probability of observing a test statistic at least as extreme as the one your data produced, assuming the null hypothesis is exactly true. Fisher (1925) introduced the p-value as a continuous index of evidence against the null; Neyman and Pearson (1933) embedded it in a decision rule by pre-specifying a significance level α (commonly 0.05) and rejecting the null whenever p falls below it. Modern reporting standards including APA, CONSORT, PRISMA, and ICH-E9 use the Neyman-Pearson decision rule for confirmatory inference but require the exact p-value, not just a significant-versus-not label, so readers can judge the strength of evidence themselves.

The arithmetic is mechanical once you know the sampling distribution of your test statistic. For a z-test the statistic is approximately standard normal under the null, so the two-tailed p-value is twice the upper tail of the standard normal at the absolute value of z. For a t-test the statistic follows Student's t distribution with df degrees of freedom, computed using the regularized incomplete beta function. For chi-square goodness-of-fit and independence tests, the statistic follows a chi-square distribution with df = categories minus 1 or (r minus 1) multiplied by (c minus 1), and the test is upper-tailed by construction. For F-tests (ANOVA, regression overall F, variance ratios) the statistic follows an F distribution with numerator and denominator df, and the test is also upper-tailed.

Pearson correlations have a built-in conversion that lets you reuse the t machinery. Given a correlation r based on n paired observations, the test statistic t equals r multiplied by the square root of (n minus 2) divided by (1 minus r squared), with df equal to n minus 2. The calculator's correlation tab applies this conversion automatically and returns the p-value along with the underlying t. For Spearman's rho, the same conversion is a serviceable approximation for n above roughly 30; for smaller samples or for explicit non-parametric inference, exact permutation tests are preferred.

Three reporting reminders matter. First, always pair the p-value with an effect size and confidence interval. The American Statistical Association statement (Wasserstein and Lazar 2016) and the Wilkinson Task Force (1999) both argue that p-values alone are an impoverished summary of evidence; the effect size says how large the difference is, and the confidence interval says how precise that estimate is. The Cochrane Handbook treats the effect size with CI as the minimum reportable unit. Second, default to two-tailed tests unless the alternative hypothesis is directional and pre-specified; switching from two-tailed to one-tailed after seeing the data is a well-documented form of p-hacking that doubles the false-positive rate. Third, never interpret p > 0.05 as evidence that the null is true; absence of evidence is not evidence of absence (Altman and Bland 1995). Report the p-value, the effect size, and the precision, and let readers reach their own conclusions.

When the test statistic comes from a model with assumptions that are not satisfied, the nominal p-value is wrong. For chi-square tests with sparse cells (expected counts below 5), use Fisher's exact test instead. For t-tests with violated normality and small n, bootstrap or permutation tests give more accurate p-values. For meta-analytic synthesis where individual study p-values are not directly poolable, convert each study to an effect size with CI using our p-value to confidence interval converter (the inverse operation of this calculator) before pooling. For full study design and analysis from a PhD statistician, our statistical analysis service handles assumption checks, robust alternatives, and APA-formatted reporting end-to-end.