Critical Value Calculator
Choose a distribution (Z, t, chi-square, or F), enter your significance level, select a one-tailed or two-tailed test, and the calculator returns the critical value and rejection region instantly. The steps panel shows the exact formula used, and the distribution curve highlights the rejection region so you can see where your test statistic must fall to reject the null hypothesis.
What is a critical value?
A critical value is the boundary point that separates the rejection region from the non-rejection region in a hypothesis test. If your computed test statistic (for example, a Z-score or a t-statistic) falls beyond the critical value, you reject the null hypothesis (H0). The critical value is determined by two things: the significance level (alpha) you choose before collecting data, and the sampling distribution of the test statistic under H0. Lowering alpha makes the threshold more extreme, so you need stronger evidence to reject H0.
How to use this calculator
Select the distribution that matches your hypothesis test: Z for tests where the population standard deviation is known; t-Student for tests based on a sample (one-sample t-test, two-sample t-test, regression coefficients); chi-square for goodness-of-fit and independence tests; F for ANOVA and variance-ratio tests. Then enter the significance level (alpha), choose one-tailed or two-tailed, and supply the degrees of freedom when required. The calculator returns the critical value, the rejection region written as an inequality, and a worked explanation in the steps panel.
One-tailed versus two-tailed tests
A two-tailed test asks "is the parameter different from the null value?" and places alpha/2 in each tail. A right-tailed test asks "is the parameter larger than the null value?" and places all of alpha in the upper tail. A left-tailed test asks "is the parameter smaller?" and places all of alpha in the lower tail. For the same alpha, a one-tailed test has a less extreme critical value (closer to zero), which makes it easier to reject H0, but only in the direction you predicted. If you had no directional prediction before collecting data, a two-tailed test is the appropriate choice.
Degrees of freedom and which distribution to use
Degrees of freedom (df) control how heavy the tails of a distribution are, and therefore how large a test statistic must be to reach significance. For a one-sample t-test, df = n - 1. For an independent two-sample t-test, df = n1 + n2 - 2. For a chi-square goodness-of-fit test, df = number of categories - 1. For an F-test in one-way ANOVA, the numerator df = k - 1 (where k is the number of groups) and the denominator df = N - k (where N is the total sample size). As df grows, the t-distribution approaches the standard normal, so for df above about 120 the difference between t and Z critical values is negligible.
Common critical values reference
| Distribution | df | alpha = 0.10 | alpha = 0.05 | alpha = 0.01 |
|---|---|---|---|---|
| Z | n/a | 1.2816 | 1.6449 | 2.3263 |
| t | 5 | 1.4759 | 2.0150 | 3.3649 |
| t | 10 | 1.3722 | 1.8125 | 2.7638 |
| t | 20 | 1.3253 | 1.7247 | 2.5280 |
| t | 30 | 1.3104 | 1.6973 | 2.4573 |
| t | 60 | 1.2958 | 1.6706 | 2.3901 |
| chi-sq | 5 | 9.2364 | 11.0705 | 15.0863 |
| chi-sq | 10 | 15.9872 | 18.3070 | 23.2093 |
| chi-sq | 20 | 28.4120 | 31.4104 | 37.5662 |
| F(3,20) | n/a | 2.3801 | 3.0984 | 4.9382 |
| F(5,30) | n/a | 2.0492 | 2.5336 | 3.6990 |
One-tailed critical values for the most common significance levels. For two-tailed tests, use alpha/2 to look up the positive boundary.
Frequently asked questions
What is the critical value for a 95% confidence interval?
A 95% confidence interval corresponds to a two-tailed test with alpha = 0.05. For the standard normal, the critical value is 1.96 (often rounded). For a t-distribution, it depends on degrees of freedom: with df = 10, the two-tailed t critical value at alpha = 0.05 is about 2.228; with df = 30, it is about 2.042.
What is the difference between a critical value and a p-value?
A critical value is a fixed threshold determined by your chosen alpha and distribution. A p-value is a probability computed from your actual data. If your test statistic exceeds the critical value, the p-value is less than alpha, and both lead to the same conclusion: reject H0. The two approaches are equivalent and give identical decisions.
When should I use the Z distribution instead of t?
Use Z when you know the population standard deviation (sigma), which is rare in practice, or as a practical approximation when the sample size is large (generally n > 100). Use t whenever you estimate sigma from the sample, regardless of sample size. For large samples the two critical values converge, so the distinction matters most for small samples.
Why does the chi-square distribution only produce positive critical values?
The chi-square distribution is defined only for non-negative values because it is the sum of squared standard-normal random variables. Its critical values are always positive. For a two-tailed chi-square test (used in, for example, testing a population variance), the rejection region has a lower critical value and an upper critical value, both positive.
What do the two degrees of freedom parameters mean in the F-distribution?
The F-statistic is the ratio of two chi-square variables each divided by their own degrees of freedom. The numerator df (df1) comes from the between-group variance, and the denominator df (df2) comes from the within-group variance. In one-way ANOVA with k groups and N total observations, df1 = k - 1 and df2 = N - k. Both parameters shape the critical value, so you need both to look up or calculate the threshold.