Critical Value Calculator

A critical value is the cutoff on a test statistic's distribution that separates the rejection region from the acceptance region. If your test statistic is more extreme than the critical value, you reject the null hypothesis at the chosen significance level. It is the reverse of a p-value: the boundary at which p equals alpha.

Use two-tailed when your hypothesis allows a difference in either direction — for example, testing whether a new fertilizer changes yield at all. Use one-tailed when only one direction matters, such as whether the new fertilizer increases yield. At the same alpha, the two-tailed critical value is larger because the alpha is split across both tails.

For a two-tailed test at the 5% significance level, the z critical value is 1.96 — you reject the null if |z| exceeds 1.96. For one-tailed at 5%, it is 1.645. These two numbers appear constantly in quality control, Six Sigma and research papers, and match the 95% and 90% confidence z-values.

Use the t distribution when the population standard deviation is unknown and you estimate it from a small sample — the common case in experimental research. With 10 degrees of freedom the two-tailed t critical at 5% is 2.228, larger than the z value of 1.96 because small samples have fatter tails. As degrees of freedom grow, t approaches z.

Chi-square critical values gate goodness-of-fit and independence tests. With 1 degree of freedom the 5% upper-tail critical value is 3.8415; with 2 it is 5.991 and with 4 it is 9.488. Reject the null if your computed statistic exceeds the critical value for your degrees of freedom and significance level.