What the critical value marks
A critical value is the cutoff on a test distribution that separates the rejection region from the rest. If your test statistic lands past it, you reject the null hypothesis. Which distribution you read depends on the test, and the value depends on your significance level (alpha), the tails, and the degrees of freedom.
z critical values (fixed, no degrees of freedom)
The z distribution is the standard normal, so its critical values are constants. Two-tailed values are simply the one-tailed value at alpha over 2.
| α | One-tailed z | Two-tailed z |
|---|---|---|
| 0.10 | 1.282 | ±1.645 |
| 0.05 | 1.645 | ±1.960 |
| 0.01 | 2.326 | ±2.576 |
The famous ±1.96 is the two-tailed z at α = 0.05, the default in most fields.
t, chi-square and F depend on degrees of freedom
These distributions change shape with sample size, so the critical value moves with the degrees of freedom. Chi-square and F are right-tailed only. The values below are illustrative points at α = 0.05.
| Distribution | Degrees of freedom | Critical value (α = 0.05) |
|---|---|---|
| t (two-tailed) | df = 10 | ±2.228 |
| t (two-tailed) | df = 30 | ±2.042 |
| Chi-square (right) | df = 5 | 11.070 |
| F (right) | df1 = 3, df2 = 30 | 2.922 |
Smaller degrees of freedom mean fatter tails and a larger critical value, so a small sample demands stronger evidence to reject the null.
Things that quietly change the answer
- Tails. One-tailed and two-tailed use different cutoffs. Decide which before you collect data, not after seeing the result.
- Alpha. 0.05 is standard, 0.01 for medical and pharma work, 0.10 in some social science. Moving alpha after the fact is p-hacking.
- Degrees of freedom. For t and chi-square that is usually n − 1; F takes two, a numerator and a denominator df.
Common questions
What is the difference between a one-tailed and a two-tailed test?
A one-tailed test looks for an effect in one direction only; a two-tailed test looks for a difference in either direction. In a two-tailed test the significance level is split, so alpha over 2 goes in each tail, which pushes the critical value further out.
When do I use the t distribution instead of z?
Use t when the sample is small (under about 30) or the population standard deviation is unknown, which is most real cases. As the degrees of freedom grow, the t value closes in on the z value.
Why are chi-square and F critical values always positive?
Both statistics come from squared quantities, so they cannot be negative. Their distributions are right-skewed and the test is always right-tailed, meaning there is a single upper critical value.
