Critical Value Calculator | Z-test, t-test, Chi-Square

A critical value is the cutoff point on a probability distribution that marks the edge of the "rejection region" in a hypothesis test. If a test statistic falls beyond the critical value, the result is judged statistically significant, and the null hypothesis is rejected. This calculator finds critical values for the three most common tests: the Z-test, the t-test, and the Chi-squared test.

What is a critical value?

In statistics, researchers often start with a null hypothesis, a plain assumption that nothing unusual is happening (for example, that a new drug has no effect). A test statistic is calculated from sample data, and it is compared against a critical value to decide whether the data provide enough evidence against the null hypothesis.

The critical value depends on three things: the choice of test, the significance level (written as α, the Greek letter alpha), and whether the test is one-tailed or two-tailed. The significance level is the chance of wrongly rejecting a true null hypothesis, often set at 0.05 (5%) or 0.01 (1%).

A one-tailed test checks for an effect in a single direction, such as "greater than." It has one critical value. A two-tailed test checks for an effect in either direction and has two critical values, one on each side of the distribution.

How to calculate a critical value

1. Pick the test: Z-test (large samples or known population variance), t-test (small samples with unknown variance), or Chi-squared test (categorical data or goodness-of-fit).
2. Pick the tail type: one-tailed or two-tailed.
3. Choose the significance level α, a number between 0 and 1. Common choices are 0.10, 0.05, and 0.01.
4. For a t-test or Chi-squared test, enter the degrees of freedom (df). For a t-test, df equals the sample size minus 1. For a Chi-squared goodness-of-fit test, df equals the number of categories minus 1.
5. Look up (or calculate) the value that cuts off the chosen tail area under the distribution curve. This is the critical value.

Critical value formulas

Each formula uses the inverse of a cumulative distribution function: the value of x for which the area under the curve up to x equals a target probability p.

Z-test formula

$Z_c = \Phi^{-1}(1 - \alpha) \quad \text{(one-tailed)}$ $Z_c = \Phi^{-1}\left(1 - \frac{\alpha}{2}\right) \quad \text{(two-tailed)}$

Φ⁻¹ is the inverse of the standard normal cumulative distribution function.

t-test formula

$t_c = t^{-1}(1 - \alpha,\ df) \quad \text{(one-tailed)}$ $t_c = t^{-1}\left(1 - \frac{\alpha}{2},\ df\right) \quad \text{(two-tailed)}$

t⁻¹(p, df) is the inverse of the t-distribution's cumulative distribution function with df degrees of freedom.

Chi-squared test formula

$\chi^2_c = \chi^2_{1-\alpha,\ df} \quad \text{(one-tailed)}$ $\chi^2_{\text{lower}} = \chi^2_{\alpha/2,\ df}, \qquad \chi^2_{\text{upper}} = \chi^2_{1-\alpha/2,\ df} \quad \text{(two-tailed)}$

The Chi-squared distribution only takes positive values, so both critical values are positive, and the lower one is smaller than the upper one.

Example: Z-test critical value

A quality-control team wants a one-tailed Z-test critical value at α = 0.05.

$Z_c = \Phi^{-1}(1 - 0.05) = \Phi^{-1}(0.95) \approx 1.6449$

Any test statistic above 1.6449 falls in the rejection region.

Example: t-test critical value

A researcher runs an experiment with 20 participants, so df = 19. They want a two-tailed critical value at α = 0.01.

$t_c = t^{-1}(1 - 0.005,\ 19) = t^{-1}(0.995,\ 19) \approx 2.8609$

The rejection regions are t < -2.8609 and t > 2.8609.

Example: Chi-squared critical values

An analyst compares observed and expected counts across 5 categories, so df = 4. They want two-tailed critical values at α = 0.05.

$\chi^2_{\text{lower}} = \chi^2_{0.025,\ 4} \approx 0.4844$ $\chi^2_{\text{upper}} = \chi^2_{0.975,\ 4} \approx 11.1433$

A Chi-squared statistic below 0.4844 or above 11.1433 would fall in the rejection region.

Common critical values

As degrees of freedom grow, t-distribution critical values shrink toward the corresponding Z-test values, because the t-distribution approaches the normal distribution.

Frequently asked questions

What is a critical value used for? It sets the boundary for deciding whether a test statistic is extreme enough to reject the null hypothesis. If the statistic crosses the critical value, the result is statistically significant at the chosen α level.

What is the difference between a one-tailed and a two-tailed critical value? A one-tailed test looks for an effect in one direction and has a single critical value. A two-tailed test looks for an effect in either direction and has two critical values, positioned symmetrically around the center for the Z-test and t-test. For the same α, a two-tailed critical value is larger in absolute size than a one-tailed one, because the same total probability is split across two tails instead of one.

When should a Z-test be used instead of a t-test? A Z-test is used when the population variance is known or the sample is large, typically 30 or more observations. A t-test is used for smaller samples with unknown population variance. As sample size grows, t-test critical values converge toward Z-test critical values.

Can a critical value be negative? Yes. Z-test and t-test distributions are symmetric around zero, so a two-tailed test has one negative and one positive critical value. Chi-squared critical values are always positive, because the Chi-squared distribution is only defined for values of zero or more.

How do degrees of freedom affect a critical value? Degrees of freedom measure how much independent information is in the data, usually the sample size minus the number of parameters already estimated. Lower degrees of freedom produce wider, heavier-tailed distributions and larger critical values. Higher degrees of freedom bring the t-distribution and Chi-squared distribution closer to the normal distribution.

Which critical value corresponds to a 95% confidence level? A 95% confidence level uses α = 0.05. For a two-tailed Z-test, the critical values are ±1.9600. For a one-tailed Z-test, the critical value is 1.6449.

History

Karl Pearson introduced the Chi-squared test in 1900. William Sealy Gosset, publishing as "Student," developed the t-distribution in 1908 to handle small samples at the Guinness brewery. Ronald Fisher formalized significance testing and critical values in the 1920s in his book Statistical Methods for Research Workers. Before computers, researchers looked up critical values in printed tables; calculators and software now compute them directly from the distribution formulas.

References

1. Pearson, K. (1900). "On the criterion that a given system of deviations... is such that it can be reasonably supposed to have arisen from random sampling." Philosophical Magazine, Series 5, 50(302), 157–175.
2. Student [Gosset, W. S.] (1908). "The probable error of a mean." Biometrika, 6(1), 1–25.
3. Fisher, R. A. (1925). Statistical Methods for Research Workers. Edinburgh: Oliver & Boyd.
4. NIST/SEMATECH e-Handbook of Statistical Methods, "Critical Values."