Calculate p-values from z, t, chi-square, or F statistics. Supports two-tailed, left-tail, and right-tail tests with significance at multiple alpha levels.
The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. It's one of the most reported numbers in hypothesis testing because it gives a compact summary of how surprising the observed result is under H₀.
This calculator converts any test statistic (z, t, chi-square, or F) into a p-value. Select your distribution, enter the statistic and degrees of freedom, and get left-tail, right-tail, and two-tailed p-values along with a significance check at common alpha levels.
That makes it useful for homework, software verification, and quick review of published results when you want the numeric p-value rather than a table lookup.
Statistical tables are limited to a fixed set of values, and software often reports only one tail. This calculator keeps the distribution choice, tail choice, and alpha level together so you can see the p-value in the same form you need for the decision step.
For Normal (Z) distribution: P(Z > z) = 1 − Φ(z) (right-tail) P(Z < z) = Φ(z) (left-tail) P(|Z| > |z|) = 2 × [1 − Φ(|z|)] (two-tailed) For Student's t: Uses regularized incomplete beta function with ν degrees of freedom For Chi-Square: Uses regularized lower incomplete gamma function with k degrees of freedom For F distribution: Uses regularized incomplete beta function with d₁ and d₂ degrees of freedom A result is significant if p-value < α
Result: p = 0.0500
A z-score of 1.96 gives a two-tailed p-value of exactly 0.05, which is the boundary of significance at α = 0.05. This is why z = 1.96 is the famous critical value for 95% confidence intervals.
The p-value is perhaps the most misunderstood concept in statistics. It is NOT the probability that the null hypothesis is true. It is NOT the probability of getting the result by chance. It IS the probability of seeing data at least as extreme as observed, given that H₀ is true. The distinction matters: a p-value of 0.03 doesn't mean there's a 3% chance the result is due to chance.
A statistically significant result can be practically meaningless, and vice versa. With 100,000 observations, a correlation of r = 0.01 might be significant (p < 0.05) despite being negligibly small. Always report effect sizes (Cohen's d, r², odds ratios) alongside p-values. The p-value tells you if an effect exists; effect size tells you if it matters.
The reproducibility crisis in psychology, medicine, and other fields has been partly attributed to p-value misuse: p-hacking (running many tests until p < 0.05), HARKing (hypothesizing after results are known), and the publication bias favoring significant results. The American Statistical Association's 2016 statement and subsequent calls for reform emphasize reporting exact p-values, using confidence intervals, and moving beyond binary significance decisions.
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The p-value is the probability of observing a test statistic as extreme as (or more extreme than) the actual result, assuming the null hypothesis is true. It's NOT the probability that H₀ is true, nor the probability that H₁ is false.
A one-tailed (directional) test checks for an effect in only one direction. A two-tailed test checks for an effect in either direction. Two-tailed p-values are double the one-tailed value for symmetric distributions. Use two-tailed unless you have a strong prior reason to test only one direction.
It means the p-value is less than the chosen significance level (α, typically 0.05). It indicates the observed result would be unlikely under the null hypothesis, providing evidence against H₀. It does not mean the result is practically important.
Z: for large samples (n > 30) with known population variance. t: for small samples or unknown variance. Chi-square: for categorical data tests or variance tests. F: for comparing variances or ANOVA results.
In theory, no — there's always some probability of any outcome. In practice, very small p-values are reported as "< 0.0001" or in scientific notation. The calculator shows exact computed values.
R.A. Fisher suggested 0.05 as a convenient threshold in the 1920s — it's a convention, not a fundamental level. Different fields use different thresholds: particle physics requires 5σ (p ≈ 3×10⁻⁷), and genomics often uses 5×10⁻⁸ for genome-wide significance.