Critical Value Calculator
Find critical values of z, t, chi-square and F for any significance level and degrees of freedom.
About the Critical Value Calculator
A critical value marks the edge of the rejection region of a hypothesis test: if your test statistic is beyond it, the result is significant at level α. For a two-tailed test, α is split between both tails. Values are computed from the exact distributions, not looked up from rounded tables.
The formula
Two-tailed: Φ⁻¹(1 − α/2); one-tailed: Φ⁻¹(1 − α)
Common values
| α = 0.10 | α = 0.05 | α = 0.01 | |
|---|---|---|---|
| z two-tailed | 1.645 | 1.960 | 2.576 |
| z one-tailed | 1.282 | 1.645 | 2.326 |
| t two-tailed, df 10 | 1.812 | 2.228 | 3.169 |
| χ², df 1 | 2.706 | 3.841 | 6.635 |
Worked example
Example: Critical t = ±2.146297
- Inputs: Distribution: t; Significance level α: 0.055; Test: Two-tailed; Degrees of freedom: 11.
- Formula: Two-tailed: Φ⁻¹(1 − α/2); one-tailed: Φ⁻¹(1 − α)
- Result: Critical t = ±2.146297
Questions people ask
Why 1.96?
95% of a normal distribution lies within ±1.96 standard deviations.
Chi-square and F are always right-tailed?
In the usual goodness-of-fit, independence and ANOVA tests, yes.