Find the probability of observing a test statistic at least as extreme under the null hypothesis.
Overview
A p-value measures evidence against a null hypothesis, assuming that hypothesis is true. Small p-values mean the observed statistic would be unusual under the null, not that the null is “impossible.”
Calculation requires the correct sampling distribution: z for large normal means, t for small samples with estimated variance, chi-square for variance or count models, and F for comparing variances or nested regression models.
One-tailed p-values sum one tail; two-tailed double the appropriate tail for symmetric tests. Always state which tail definition you used so results compare fairly across studies.
p-values do not measure effect size or practical importance. Combine with confidence intervals, plots, and domain context. Thresholds like 0.05 are conventions, not physical laws.
Steps
Select z, t, chi-square, or F as appropriate.
Input the calculated z, t, etc.
Required for t, chi-square, and F.
One-tailed or two-tailed.
Integrate tail probability of distribution.
Compare with alpha and report effect size.
Math
p = P(|T| ≥ |t_obs| | H₀) under chosen distribution
Tail area under the null sampling distribution beyond the observed statistic, doubled for two-tailed symmetric tests.
Worked cases
p ≈ 2×(1 − Φ(2.1))
p ≈ 0.0357.
Upper tail of t₁₀ distribution
p smaller than two-tailed counterpart.
Two-tailed area beyond ±1.2
p > 0.05 typical threshold.
Why use this
Questions