Critical Value Calculator

How it is calculated

The tool finds the point where the tail area equals your significance level α (a number strictly between 0 and 1, such as 0.05). With a two-tailed test, α/2 goes in each tail; with a one-tailed test the whole α sits in one tail. For z the cutoff is the normal quantile for that tail area. For Student's t, the tail probability is computed from the exact t distribution with df degrees of freedom and the cutoff is found by bisection, so it matches a printed table to four decimals (for example 12.7062 at df = 1, two-tailed, α = 0.05). Chi-square always uses the right tail and the full α, so the tail choice has no effect; for t and chi-square a missing or non-positive df gives no result. The value is shown with four decimals.

Common mistakes

• Entering 5 instead of 0.05 for a 5% level. Alpha must be a proportion between 0 and 1, otherwise the tool shows no result.
• Choosing one-tailed when the test is two-tailed. At α = 0.05 the z cutoff is 1.6449 one-tailed but 1.9600 two-tailed, which can change the decision.
• Using the sample size as the degrees of freedom. For a one-sample t test df is n − 1, not n.
• Expecting the tail option to change a chi-square result. At α = 0.05 and df = 5 it returns 11.0705 whichever tail is selected.
• Leaving the degrees of freedom empty for t or chi-square. Only z works without df.