P-Value Calculator

The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one calculated from your data — assuming the null hypothesis is true. Small p-values cast doubt on the null; large p-values fail to provide evidence against it. A common misconception is that the p-value is the probability the null hypothesis is true; it is not. It measures how surprising your data would be if the null were true.

For a z-test with standard normal distribution, the left-tailed p-value is P(Z ≤ z) = Φ(z). A right-tailed p-value is P(Z ≥ z) = 1 − Φ(z). A two-tailed p-value doubles the upper-tail area: p = 2 × P(Z ≥ |z|) = 2 × (1 − Φ(|z|)). Example: z = 1.96 in a two-tailed test gives p ≈ 0.05 because each tail holds 2.5%. z = 2.58 yields p ≈ 0.01 two-tailed.

When population standard deviation is unknown and sample size is small, the t distribution replaces z. Enter degrees of freedom df = n − 1 for a one-sample t-test. t distributions have heavier tails, so p-values are larger than z would give for the same numeric statistic — you need a more extreme t to achieve the same significance. As df → ∞, t → z.

Compare p to a pre-chosen significance level α (often 0.05). If p < α, reject the null at that level; if p ≥ α, fail to reject. This calculator checks α = 0.01, 0.05, and 0.10 automatically. A result significant at 0.05 may not be significant at 0.01 — always report both p and α. Multiple testing without correction inflates false positives; Bonferroni and FDR methods address this in advanced work.

P-values connect to confidence intervals and z-scores: a two-sided 95% CI excludes values whose z would give p < 0.05. Use the z-score calculator to standardize test statistics and the confidence interval calculator for interval-based inference. The statistics calculator helps compute sample mean and SD needed to form t statistics.

Effect size matters alongside p: a tiny difference can yield p < 0.05 with a huge sample, while a meaningful difference may be non-significant with n = 10. Report test statistic, df, p-value, and practical significance. This tool supports z and t distributions with left, right, and two-tailed tail selections and shows cumulative probability steps for transparency.

Common benchmarks to memorize: |z| = 1.96 → p ≈ 0.05 two-tailed; |z| = 2.58 → p ≈ 0.01 two-tailed; |z| = 1.645 → p ≈ 0.10 two-tailed. Always state your tail choice before comparing to α — a left-tailed test at z = −1.645 gives p = 0.05, but the same z in a two-tailed test does not.