About the Critical Value Calculator
This critical value calculator finds the cut-off points of the four distributions used in most hypothesis tests: the standard normal (z), Student’s t, chi-square (χ²) and F. Pick the distribution, enter the significance level α, choose a two-tailed, left-tailed or right-tailed test and supply the degrees of freedom, and it returns the critical value(s), the rejection region and a table of critical values for other common α levels.
It replaces the printed z, t, χ² and F tables at the back of a statistics textbook, and it works for any α and any degrees of freedom, not just the rows a table happens to include. Students use it for homework and exams, researchers use it to check a test statistic against a cut-off, and analysts use the z and t values to build confidence intervals.
Values are computed by numerically inverting the exact cumulative distribution functions (via the regularized incomplete gamma and beta functions), so they match statistical software such as R, Excel and SciPy to at least four decimal places. For two-tailed χ² and F tests, α is split equally between the two tails.
With the default inputs, the critical value is 2.2281. Change any value above to recalculate instantly.
How to use the critical value calculator
- 1Choose the distribution your test uses: z, t, chi-square or F.
- 2Enter the significance level α (0.05 is the most common).
- 3Pick two-tailed, right-tailed or left-tailed to match your alternative hypothesis.
- 4Enter the degrees of freedom (two values for F).
- 5Compare your test statistic with the critical value and rejection region.
Formula and method
A critical value is the point on a test distribution beyond which the probability in the tail equals the significance level α. If F⁻¹ is the inverse cumulative distribution function (quantile function), the right-tailed critical value is F⁻¹(1 − α), the left-tailed one is F⁻¹(α), and a two-tailed test puts α/2 in each tail. Because z and t are symmetric around zero, their two-tailed values are ± the same number; χ² and F are skewed, so their lower and upper values differ.
The calculator evaluates the exact CDFs — the normal via the error function, t and F via the regularized incomplete beta function, and χ² via the regularized incomplete gamma function — and inverts them numerically by bisection to about 10 significant digits.
- α
- Significance level (probability of a Type I error)
- df
- Degrees of freedom (e.g. n − 1 for a one-sample t test)
- df₁, df₂
- Numerator and denominator degrees of freedom of the F distribution
- F⁻¹
- Inverse CDF (quantile function) of the chosen distribution
Worked examples
Two-tailed t test, α = 0.05, df = 10
With 10 degrees of freedom, 2.5% of the t distribution lies above 2.2281 and 2.5% below −2.2281. A t statistic beyond ±2.2281 is significant at the 5% level.
Right-tailed z test, α = 0.01
For a right-tailed z test at α = 0.01, the critical value is 2.3263 — the 99th percentile of the standard normal. (For a two-tailed test at α = 0.05 the familiar value is ±1.96.)
Chi-square goodness of fit, α = 0.05, df = 5 (right-tailed)
Chi-square tests are almost always right-tailed. With 5 degrees of freedom, 5% of the distribution lies above 11.0705, so a χ² statistic larger than that rejects the null hypothesis.
Two-tailed chi-square for a variance, α = 0.05, df = 10
The χ² distribution is skewed, so the two cut-offs are not symmetric: 2.5% lies below 3.2470 and 2.5% above 20.4832.
ANOVA F test, α = 0.05, df₁ = 3, df₂ = 20
Comparing 4 group means with 24 observations gives df₁ = 3 and df₂ = 20. The F statistic must exceed 3.0984 to be significant at the 5% level.
Frequently asked questions
What is a critical value in statistics?+
It is the cut-off on the test distribution that separates the rejection region from the rest. If your test statistic falls beyond the critical value, the result is statistically significant at the chosen α level.
What is the t critical value for a 95% confidence interval?+
It depends on the degrees of freedom: 12.706 for df = 1, 2.228 for df = 10, 2.086 for df = 20, 2.042 for df = 30 and it approaches the z value 1.960 as df grows. Use α = 0.05, two-tailed.
When should I use z instead of t?+
Use z when the population standard deviation is known or for large-sample proportion tests. Use t when you estimate the standard deviation from a sample, especially with small samples; t has heavier tails and larger critical values.
Should I use a one-tailed or two-tailed test?+
Use a two-tailed test when your alternative hypothesis is “different from”. Use a one-tailed test only if you specified a direction (greater than or less than) before seeing the data; it puts all of α in one tail.
Why are chi-square and F tests usually right-tailed?+
Both statistics are sums of squares, so they are never negative and large values signal a poor fit or real differences between groups. Two-tailed χ² is mainly used for confidence intervals and tests on a single variance.
How do I find degrees of freedom?+
For a one-sample t test df = n − 1; for a chi-square goodness-of-fit test df = categories − 1; for a contingency table df = (rows − 1)(columns − 1); for one-way ANOVA df₁ = groups − 1 and df₂ = total observations − groups.