Critical Value Calculator
The cut-off your test statistic must beat — z, t, chi-square and F critical values for any significance level and degrees of freedom.
A critical value is the point on a distribution beyond which you reject the null hypothesis.
How the critical value calculator works
A critical value is the point on a distribution beyond which you reject the null hypothesis. For a two-tailed z-test at 5% it is ±1.96; for a t-test it depends on the degrees of freedom, and it converges on the z-value as they grow.
All four are computed here by inverting the distribution numerically, so any α and any degrees of freedom work — not just the values in a printed table.
Formula: the value beyond which the tail area equals α
Worked examples
| Inputs | Critical value | Note |
|---|---|---|
| Two-tailed z at 5% | 1.959963 | 1.96 — the most quoted number in statistics |
| t at 5%, 20 df | 2.085963 | 2.086 — wider than z |
| Chi-square at 5%, 1 df | 3.841459 | 3.841 |
FAQFrequently asked questions
Why is 1.96 everywhere?
It is the two-tailed 5% critical value of the normal distribution — the cut-off behind every "95% confidence interval".
When do I use t rather than z?
When the standard deviation is estimated from the same sample. The t critical value is larger, which widens the interval to pay for that uncertainty.
Why does t depend on degrees of freedom?
Because a small sample estimates the spread poorly. With 5 df the 5% value is 2.57; with 100 it is 1.98; at infinity it is exactly z.
Is the critical value the same as the p-value?
They are two views of the same comparison. Reject when the statistic beats the critical value, or equivalently when the p-value falls below α.
Should I use one-tailed or two?
Two, unless you fixed the direction before collecting data. A one-tailed test at 5% uses the two-tailed 10% cut-off, which is why it is so often abused.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- National curriculum in England — Mathematics — the terms and methods taught in UK schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.