Critical value calculator
z, t, chi-square or F — at any α, no table needed.
What is a critical value?
The cutoff for a hypothesis test: if the test statistic falls beyond it, you reject the null hypothesis at significance level α. It is the point where the area in the tail of the distribution equals α (one-tailed) or α/2 on each side (two-tailed).
How do I find the t critical value?
Pick α, the tails and the degrees of freedom (usually n − 1). For a two-tailed test at α = 0.05 with 10 degrees of freedom, the critical value is ±2.228; one-tailed it is 1.812. These match the NIST t table.
What are the common z critical values?
Two-tailed: ±1.645 at α = 0.10, ±1.960 at α = 0.05 and ±2.576 at α = 0.01. One-tailed: 1.282, 1.645 and 2.326. The t critical values approach these as the degrees of freedom grow.
Why does chi-square have two different critical values?
The chi-square and F distributions are not symmetric, so a two-tailed test needs a separate lower and upper cutoff. With 10 degrees of freedom at α = 0.05 they are 3.247 and 20.483. Most chi-square and F tests (goodness of fit, independence, ANOVA) are right-tailed.
How do I read an F critical value?
F has two degrees of freedom: the numerator (df1) and the denominator (df2). For a right-tailed test at α = 0.05 with df1 = 5 and df2 = 10, F = 3.326; with df1 = 10 and df2 = 10, F = 2.978.
Where do the numbers come from?
They are computed from the distributions themselves, to many more decimals than a printed table. The t, chi-square and F values are checked against the tables in the NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.6.7; F uses the regularized incomplete beta function (Numerical Recipes; NIST Digital Library of Mathematical Functions, section 8.17).
How do I decide between one-tailed and two-tailed?
Use a two-tailed test when a difference in either direction counts, which is the default in most analyses. Use a one-tailed test only when the alternative hypothesis names a direction before you collect the data. A one-tailed critical value is smaller in magnitude at the same α, so choosing the tail after seeing the data inflates the false-positive rate.
What is the relation between the critical value and the p-value?
They are two views of the same decision. The test statistic beyond the critical value is the same as a p-value below α. The critical value is fixed by α and the degrees of freedom before you see data; the p-value depends on the statistic you observed.
What goes in and what comes out
You choose the distribution (z, t, chi-square or F), the significance level α, the number of tails and the degrees of freedom. The calculator returns the point (or the two points) that cut off a tail area of α, or α ÷ 2 in each tail for a two-tailed test. For the symmetric z and t distributions the two cutoffs are mirror images; for chi-square and F, which are skewed and start at zero, the lower and upper cutoffs differ.
Worked example: the same α, three distributions
At α = 0.05 two-tailed, the z critical value is ±1.960. For t with 10 degrees of freedom it is ±2.228, larger because the t distribution has heavier tails, and with 120 degrees of freedom it is ±1.980, nearly z. A one-tailed t test at the same α and 10 degrees of freedom uses 1.812. For chi-square with 4 degrees of freedom, right-tailed at 0.05, the cutoff is 9.488. If a computed t statistic is 2.5 with 10 degrees of freedom, it exceeds 2.228, so a two-tailed test rejects the null hypothesis at the 5% level.
Common mistakes
Using the sample size instead of the degrees of freedom is common: for a one-sample t test it is n − 1, and other tests have their own formulas. Another is mixing up the tails: a two-tailed test at α = 0.05 uses 0.025 in each tail, so looking up 0.05 in a one-tailed table gives the wrong value. For chi-square and F tests the usual hypothesis test is right-tailed, so the left cutoff is rarely needed. Do not use the z cutoff in place of t for a small sample with an estimated standard deviation.
Sources: NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.6.7 (tables of critical values for t, chi-square and F); OpenStax Introductory Statistics 2e, chapters 8 to 13 (use of the distributions in tests). Related: p-value calculator, t-test calculator, chi-square calculator and confidence interval calculator.