Critical Value Calculator
Find critical values for z, t, chi-square and F tests at any significance level, one- or two-tailed, with degrees of freedom for ANOVA.
Critical value
1.96
α = 0.05
Rejection region
> 1.96
Beyond both tails
Degrees of freedom
—
Reject the null hypothesis when your test statistic exceeds the critical value. F critical values are used for ANOVA and regression significance tests; for the two-tailed case both the lower and upper bounds of the acceptance region are shown.
Last updated: January 2026
How this calculator is verified
Checked by True Calculator automated test suite on
- Formula verified against a published worked example in the automated test suite
- Edge cases (zero, negative, boundary and unit-mismatch inputs) covered by unit tests
The full verification method is on our how we verify page. Found an error? Tell us and we will re-check it.
When to Use This Calculator
Use the critical value calculator whenever you are about to run a hypothesis test and need the rejection threshold in advance — the standard workflow for z-tests, t-tests, chi-square tests and ANOVA. Rather than leafing through tables that only give a few degrees of freedom, you enter your exact alpha and degrees of freedom and get the precise boundary, including one-tailed and two-tailed variants and both tails for chi-square and F. Students use it to verify coursework and lab reports, while quality and data teams compare process statistics against critical values to decide whether a change is real. The calculator also inverts naturally for confidence intervals, since the two-sided critical value at alpha is exactly the multiplier used to build a (1 − alpha) interval.
How to Use This Calculator
- Step 1: Choose the distribution — normal (z), Student t, chi-square or F.
- Step 2: Enter the significance level alpha as a percentage, such as 5.
- Step 3: For t and chi-square enter the degrees of freedom; for F enter both the numerator and denominator degrees of freedom.
- Step 4: Choose one-tailed or two-tailed and read the critical value(s) and rejection region to compare with your statistic.
Worked Example
A researcher runs a two-tailed t-test with 10 degrees of freedom at the 5% level. Enter t, alpha 5% and df 10: the critical value is 2.2281, so the test statistic must exceed 2.2281 in magnitude to reject the null hypothesis. The same settings on the normal distribution give 1.96, the familiar z cutoff. For an ANOVA with 3 and 20 degrees of freedom at the 1% level, the F critical value is 4.9382. A chi-square test with 1 degree of freedom at 5% gives 3.8415, and with 4 degrees of freedom it rises to 9.4877.
Tips and Common Mistakes
- •Tip 1: Reject the null when your statistic is more extreme than the critical value — beyond it in either direction for two-tailed tests.
- •Tip 2: One-tailed critical values are smaller (z = 1.645 at 5%) because the whole alpha sits in a single tail.
- •Tip 3: Use the two-tailed chi-square or F output when you need both the lower and upper bounds of the acceptance region.
- ✗Mistake 1: Using z critical values for small-sample t-tests — the threshold is too low and inflates false rejections.
- ✗Mistake 2: Confusing significance level with confidence level: 5% significance corresponds to 95% confidence, not 5%.
Frequently Asked Questions
What is a critical value?
A critical value is the cutoff on a test statistic's distribution that separates the rejection region from the acceptance region. If your test statistic is more extreme than the critical value, you reject the null hypothesis at the chosen significance level. It is the reverse of a p-value: the boundary at which p equals alpha.
How do I choose one-tailed or two-tailed?
Use two-tailed when your hypothesis allows a difference in either direction — for example, testing whether a new fertilizer changes yield at all. Use one-tailed when only one direction matters, such as whether the new fertilizer increases yield. At the same alpha, the two-tailed critical value is larger because the alpha is split across both tails.
What is the z critical value for 95% confidence?
For a two-tailed test at the 5% significance level, the z critical value is 1.96 — you reject the null if |z| exceeds 1.96. For one-tailed at 5%, it is 1.645. These two numbers appear constantly in quality control, Six Sigma and research papers, and match the 95% and 90% confidence z-values.
When do I need a t critical value instead of z?
Use the t distribution when the population standard deviation is unknown and you estimate it from a small sample — the common case in experimental research. With 10 degrees of freedom the two-tailed t critical at 5% is 2.228, larger than the z value of 1.96 because small samples have fatter tails. As degrees of freedom grow, t approaches z.
What are chi-square critical values used for?
Chi-square critical values gate goodness-of-fit and independence tests. With 1 degree of freedom the 5% upper-tail critical value is 3.8415; with 2 it is 5.991 and with 4 it is 9.488. Reject the null if your computed statistic exceeds the critical value for your degrees of freedom and significance level.
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