Written by Dackohn; last reviewed November 2026.
A critical value is the cut-off a test statistic must pass to be significant at a chosen level. For a two-tailed z-test at α = .05 it is ±1.96. If your statistic goes beyond the critical value, you reject the null hypothesis. It leads to the same decision as comparing the p-value with α.
How it works
Choose α, the false-positive rate you accept. The critical value is the point where the tail area of the distribution equals α, so results that extreme would happen only α of the time if the null hypothesis were true. That region is called the rejection region.
Common values
| Test | Situation | Critical value |
|---|---|---|
| z | Two-tailed, α = .05 | ±1.960 |
| z | One-tailed, α = .05 | 1.645 |
| z | Two-tailed, α = .01 | ±2.576 |
| t | Two-tailed, α = .05, df = 20 | ±2.086 |
| t | One-tailed, α = .05, df = 10 | 1.812 |
| χ² | α = .05, df = 4 | 9.488 |
| F | α = .05, df = (3, 36) | 2.866 |
Worked example
A paired t-test gives t = 2.45 with df = 11. For a two-tailed test at α = .05, the critical value is 2.201. Since 2.45 > 2.201, the result is significant. The p-value route reaches the same conclusion: p = .032, below .05.
Critical value or p-value?
They always agree. A critical value is on the scale of the statistic, and a p-value is on the probability scale. p-values tell you more, because they show how far past the line you are, so most reports use them. Critical values are still used in hand calculations, in exams and for building confidence intervals: a 95% CI is the estimate ± critical value × SE.
Which tail?
- Two-tailed: α is split between both tails. This is the usual choice for z and t.
- One-tailed: all of α is in one tail, so the cut-off is lower. Choose this only if the direction was fixed in advance.
- Chi-square and F: usually right-tailed, since only large values count against the null hypothesis.
How to find one
- Tables: the z, t, chi-square and F tables. If your df is missing, use the next smaller one.
- Excel:
=T.INV.2T(0.05,20),=NORM.S.INV(0.975),=CHISQ.INV.RT(0.05,4),=F.INV.RT(0.05,3,36).
Common mistakes
- Using the one-tailed column for a two-tailed test.
- Comparing a negative statistic directly. For a two-tailed test, use its absolute value.
- Using z when df is small, where t critical values are noticeably larger.
Try it yourself
Critical Value Calculator →Exact values for t, z, chi-square and F. P-Value Calculator →Go the other way, from a statistic to a p-value.Frequently asked questions
What is a critical value in statistics?
The value a test statistic must exceed for the result to be significant at a chosen alpha.
What is the critical value for 95% confidence?
For z it is 1.96. For t it depends on the degrees of freedom.
Do critical values and p-values ever disagree?
No, for the same test they lead to the same decision. They express the same comparison on different scales.
Related reading: Statistical tables · How to interpret a p-value · One-tailed vs two-tailed tests
Sources & further reading: NIST/SEMATECH e-Handbook of Statistical Methods · OpenStax Introductory Statistics. How we compute & validate: methodology. Written & reviewed by Dackohn; last reviewed August 2026.