P-value and critical value calculator
Convert a z, t, chi-square or F statistic into an exact p-value and critical value.
A clear answer. Here’s how.
Computes exact tail areas from the regularised incomplete gamma (chi-square, normal) and incomplete beta (t, F) functions. Two-sided p = 2 × the smaller tail for z and t. Critical values invert the same functions: z and t by quantile search, chi-square and F by bisection on the cumulative probability. For chi-square and F the right tail is the usual test; the two-sided value doubles the smaller tail. Degrees of freedom may be fractional, for example a Welch t test.
See a worked example
Worked example
A fixed demonstration, separate from the values you enter above.
- Test statistic
- t (Student)
- Test statistic value
- 2.228139
- Degrees of freedom (t, χ²) or numerator df (F)
- 10
- Denominator degrees of freedom (F only)
- 12
- Tail
- Two-sided
- Significance level α
- 0.05
Example result
- P-value (two-sided)
- 0.05
- P(statistic ≤ 2.228139)
- 0.975
- P(statistic ≥ 2.228139)
- 0.025
- Critical values at α = 0.05
- -2.228139 and 2.228139
- Decision at α = 0.05
- p is below α: statistically significant
Exact tail areas come from the regularised incomplete gamma and beta functions. A p-value is the chance of a statistic at least this extreme if the null hypothesis were true; it does not give the probability that the hypothesis is true or the size of an effect. For chi-square and F the right tail is the usual test; the two-sided value doubles the smaller tail, which suits variance checks.