Critical Value Calculator
Exact critical values for the z, t, χ² and F distributions from your α and degrees of freedom.
Reject H₀ when |t| > 2.0860 at df = 20.
A critical value is the cut-off that separates the rejection region from the rest of a distribution. For a two-tailed z-test at α = 0.05 it is ±1.96: any test statistic beyond that rejects the null hypothesis, because only 5% of the curve lies outside those bounds.
What a critical value is
Every hypothesis test produces a statistic and needs a threshold to judge it against. The critical value is that threshold. You choose a significance level α — the risk of a false positive you are willing to accept — and the critical value is the point with exactly α of the distribution beyond it. Land past it and the result is significant; land short and it is not.
This is the mirror image of the p-value approach. A p-value asks how much of the distribution lies beyond the statistic you actually got; a critical value fixes the area first and asks where the boundary sits. The two always agree, because p < α happens precisely when the statistic passes the critical value. Which one you report is convention, not mathematics.
Choosing the distribution
Use z when the population standard deviation is known or the sample is large. Use t when σ is estimated from the sample, which needs degrees of freedom. Use χ² for tests on counts and on a single variance, and F for comparing variances or running an ANOVA — F needs two separate degrees-of-freedom values, one for each variance in the ratio.
for a two-tailed test the area α is split evenly, leaving α ÷ 2 in each tail
- 1 Pick the distribution the test uses. A two-sample comparison of means with an estimated standard deviation uses t, not z.
- 2 Set the significance level. α = 0.05 is the usual default and corresponds to a 95% confidence level.
- 3 Decide how many tails the test has. A directional hypothesis is one-tailed and keeps all of α at one end; a non-directional one splits it, leaving 0.025 in each tail at α = 0.05.
- 4 Supply the degrees of freedom. Comparing two groups of 11 with pooled variance gives df = 11 + 11 − 2 = 20.
- 5 Read off the cut-off and apply it. At df = 20, two-tailed α = 0.05, the critical value is ±2.086, so a calculated t of −2.31 falls in the rejection region.
Critical values you will meet constantly
z values are exact; t is shown at df = 20, χ² at df = 1, and F at df₁ = 2, df₂ = 27.
| Significance level | z (two-tailed) | t, df = 20 (two-tailed) | χ², df = 1 | F, df = (2, 27) |
|---|---|---|---|---|
| α = 0.10 | ±1.645 | ±1.725 | 2.706 | 2.511 |
| α = 0.05 | ±1.960 | ±2.086 | 3.841 | 3.354 |
| α = 0.01 | ±2.576 | ±2.845 | 6.635 | 5.488 |
Reading the result correctly
Passing the critical value means the data would be unusual if the null hypothesis were true — nothing more. It is not the probability that the null is false, and it says nothing about how big the effect is. A tiny, meaningless difference clears any critical value once the sample is large enough, which is why an effect size belongs next to every significant result.
Two practical cautions. Choose the number of tails before seeing the data: switching to a one-tailed test afterwards because the two-tailed result missed doubles your real false-positive rate. And when you run several tests on the same data, the chance that at least one clears its critical value by luck grows quickly, so adjust for it with a Bonferroni correction or an equivalent procedure.