P-value calculator

How to use it

Choose the distribution your test statistic follows, enter the statistic and its degrees of freedom, and pick the tail that matches your alternative hypothesis. The tool gives the p-value, both tail areas and the critical values at your significance level.

Formulas

• z: left tail Φ(z).
• t with ν df: P(|T| > |t|) = I_{ν/(ν + t²)}(ν/2, ½), the regularized incomplete beta function; each tail is half of that on its side.
• Chi-square with k df: right tail Q(k/2, x/2), the regularized upper incomplete gamma function.
• F with d₁ and d₂ df: left tail I_{d₁x/(d₁x + d₂)}(d₁/2, d₂/2).
• Two-tailed p = 2 × the smaller tail (capped at 1). Critical values invert the same functions numerically.

Which tail?

t and z tests: two-tailed unless you stated a direction before seeing the data. Chi-square tests of independence or fit and ANOVA F tests are right-tailed: large values are evidence against the null hypothesis.

Assumptions and limits

• Degrees of freedom can be fractional (Welch’s t-test) and up to 10 million.
• All arithmetic runs in your browser in double precision. Sums are compensated and variances use the corrected two-pass method; tail probabilities come from the regularized incomplete beta and gamma functions and keep about 12 significant digits even for p-values as small as 1e-300. At 10 million degrees of freedom the t tail is accurate to about 5e-10 relative.
• A p-value is the probability of a result at least this extreme if the null hypothesis were true. It is not the probability that the null hypothesis is true, and it says nothing about the size of an effect.

Worked example

t = 2.1 with 15 degrees of freedom, two-tailed: each tail beyond ±2.1 holds 0.02653, so p = 0.05306. The 5% two-tailed critical value is ±2.131, so the result is not significant at α = 0.05.

References

• NIST/SEMATECH e-Handbook of Statistical Methods: critical values of the t distribution
• NIST/SEMATECH e-Handbook of Statistical Methods: critical values of the chi-square distribution
• NIST/SEMATECH e-Handbook of Statistical Methods: critical values of the F distribution
• R documentation: the t distribution
• R documentation: the chi-squared distribution
• R documentation: the F distribution
• ASA statement on p-values (Wasserstein and Lazar 2016)