Quick answer
A critical value is the quantile of a distribution at a tail probability. For a two-tailed z test at 5% significance the critical value is ±1.959964.
Choose a distribution, significance level and tails (and degrees of freedom for t, χ² and F). The calculator returns the critical value.
Related tools: P-Value Calculator: P-value from a z, t, chi-square or F statistic. T Statistic Calculator: t statistic from summary statistics. Degrees of Freedom Calculator: Degrees of freedom for one- and two-sample t-tests.
How to use the Critical Value Calculator
- Choose distribution: z (normal), t, chi-square and F.
- Enter significance level (α) (between 0 and 0.5). It starts at 0.05.
- Choose tails: Two-tailed, Right-tailed and Left-tailed.
- Optionally, enter degrees of freedom (df₁) (above 0).
- Optionally, enter df₂ (F only) (above 0).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Distribution | Choose one: z (normal); t; chi-square; F | Starts at z (normal) |
| Significance level (α) | A number | > 0, ≤ 0.5, starts at 0.05 |
| Tails | Choose one: Two-tailed; Right-tailed; Left-tailed | Starts at Two-tailed |
| Degrees of freedom (df₁) (optional) | A number | > 0 |
| df₂ (F only) (optional) | A number | > 0 |
Formula and method
Formula
critical value = quantile of the distribution at the tail probability
IEEE-754 double precision; shown to at most 6 decimal places.
Quantiles by inversion of the exact CDF. Two-tailed uses α/2 in each tail (z and t); chi-square and F report the right-tail value at α (two-tailed gives both).
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Distribution = z (normal); Significance level (α) = 0.05; Tails = Two-tailed.
| Result | Value |
|---|---|
| Critical value | ±1.959964 |
Example 2
Inputs: Distribution = z (normal); Significance level (α) = 0.01; Tails = Two-tailed.
| Result | Value |
|---|---|
| Critical value | ±2.575829 |
Example 3
Inputs: Distribution = t; Significance level (α) = 0.05; Tails = Two-tailed; Degrees of freedom (df₁) = 10.
| Result | Value |
|---|---|
| Critical value | ±2.228139 |
The first line is the headline result (critical value). IEEE-754 double precision; shown to at most 6 decimal places.
Using it
Reject the null hypothesis if the test statistic exceeds the critical value in absolute terms. Common two-tailed z values: 1.645 (10%), 1.96 (5%) and 2.576 (1%). One-tailed 5% is 1.645.
Degrees of freedom
The t, chi-square and F distributions need degrees of freedom; t critical values approach z as df grows (t = 2.064 at df 24, 1.96 at infinity).
How the result changes with significance level
Holding every other input at the first worked example, this table changes only significance level. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Significance level | Critical value |
|---|---|
| 0.025 | ±2.241403 |
| 0.05 | ±1.959964 |
| 0.1 | ±1.644854 |
| 0.2 | ±1.281552 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Degrees of freedom (df₁): left empty shows "Enter the degrees of freedom." beside that field and no result.
- Significance level (α): entering "0.9" shows "Significance level (α) must be at most 0.5." beside that field and no result.
Choosing between the Critical Value Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [P-Value Calculator](/statistics/p-value/) | test statistic, statistic value and test type | p-value |
| [T Statistic Calculator](/statistics/t-statistic/) | sample mean, hypothesised mean and sample standard deviation | t statistic |
| [Degrees of Freedom Calculator](/statistics/degrees-of-freedom/) | test and n₁ | Degrees of freedom |
| [One-Sample T-Test Calculator](/statistics/t-test-calculator/) | sample mean, hypothesised mean and sample standard deviation | t statistic |
| [One-Sample Z-Test Calculator](/statistics/z-test/) | sample mean, hypothesised mean and population σ | z statistic |
Background: hypothesis testing and inference
A hypothesis test compares a test statistic with what the null hypothesis predicts. The decision rule is to reject the null hypothesis when the p-value is at or below the significance level α, which you should choose before seeing the data (0.05 is conventional, not special). For a two-tailed z test the p-value is 2 × (1 − Φ(|z|)); for a one-sample t test the statistic is (x̄ − μ₀) ÷ (s ÷ √n) with n − 1 degrees of freedom.
Failing to reject the null hypothesis does not prove it true, and a statistically significant result can still be too small to matter in practice.
Assumptions and limitations
- df required for t, chi-square and F
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for degrees of freedom (df₁) and significance level (α) to confirm this.
Frequently asked questions
What is alpha?
The significance level.
What is the z critical value for 1%?
For 1% significance the two-tailed z critical value is ±2.576 (one-tailed 2.326).
What is a one-tailed test?
One that looks at a single direction.
What does the Critical Value Calculator calculate?
Critical values for z, t, chi-square and F distributions. It returns Critical value.
What formula does the Critical Value Calculator use?
critical value = quantile of the distribution at the tail probability. Quantiles by inversion of the exact CDF.
What do I need to enter in the critical Value Calculator?
You enter Distribution, Significance level (α) and Tails. Optional: Degrees of freedom (df₁) and df₂ (F only).
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. df required for t, chi-square and F.
What does the “Distribution” option change?
It sets how the critical Value is worked out. The choices are z (normal), t, chi-square and F.
What is an example of a critical Value result?
With distribution z (normal), significance level (α) 0.05 and tails Two-tailed, the critical value is ±1.959964.
Related calculators
- P-Value Calculator — P-value from a z, t, chi-square or F statistic.
- T Statistic Calculator — t statistic from summary statistics.
- Degrees of Freedom Calculator — Degrees of freedom for one- and two-sample t-tests.
- One-Sample T-Test Calculator — One-sample t-test from summary statistics.
- One-Sample Z-Test Calculator — Z-test for a mean with known population standard deviation.
- One-Way ANOVA Calculator — One-way analysis of variance across groups.
- Chi-Square Statistic Calculator — Goodness-of-fit chi-square from observed and expected counts.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods (itl.nist.gov/div898/handbook)
- Standard textbook definitions (e.g. Moore, McCabe & Craig, Introduction to the Practice of Statistics)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.