Critical Value Calculator

How it works

Right-tailed: c = F⁻¹(1 − α) · left-tailed: c = F⁻¹(α) · two-tailed: F⁻¹(α/2) and F⁻¹(1 − α/2), where F⁻¹ is the inverse CDF (quantile) of the Z, t, χ² or F distribution

A critical value is the cut-off of the test statistic’s distribution beyond which you reject the null hypothesis at significance level α. The area in the rejection region equals α: all of it in one tail for a one-tailed test, split α/2 in each tail for a two-tailed test. Z and t are symmetric, so a two-tailed test uses ±c; chi-square and F are skewed, so their lower and upper critical values differ. The values here come from the exact quantile functions, matching statistical tables.

Worked example

For a two-tailed Z test at α = 0.05, each tail holds 0.025, so the critical values are ±1.959964 (the familiar ±1.96) and you reject H₀ if |Z| > 1.959964. A two-tailed t test with 10 degrees of freedom would instead use ±2.228139.

Sources: NIST/SEMATECH e-Handbook — Critical values of the Student’s t distribution · NIST/SEMATECH e-Handbook — Critical values of the chi-square distribution

Frequently asked questions