Critical Value Calculator

What a critical value tells you

A critical value is the threshold test statistic where the rejection region begins. If your observed statistic is more extreme than the critical value, you reject the null hypothesis. That makes the critical value a decision boundary, not a probability on its own.

For a two-tailed z test at α = 0.05, the critical values are ±1.96. That means the most extreme 5% of the standard normal distribution is split evenly across both tails. The same idea applies to t and chi-square distributions, but the cutoff changes because the shape of the reference distribution changes.

How to choose z, t, chi-square, or F

Use a z critical value when the model is standard normal or when the normal approximation is appropriate. Use a t critical value when the population standard deviation is unknown and the sample size is small enough that degrees of freedom matter. Use a chi-square critical value for variance tests, goodness-of-fit work, and contingency-table style procedures. Use an F critical value for ANOVA, regression model F tests, and variance-ratio workflows that depend on numerator and denominator degrees of freedom.

The calculator now keeps z, t, chi-square, and F cutoffs in one workflow because the broad critical value calculator search intent usually includes all four reference distributions. If you need the raw test statistic from data, use the relevant z-score, t-test, chi-square, ANOVA, or variance workflow first, then compare that statistic with the cutoff here.

Tail choice and significance level

Tail direction comes from the alternative hypothesis, not from the calculator menu alone. A right-tailed test puts the full α in the upper tail, a left-tailed test puts the full α in the lower tail, and a two-tailed test splits α between both tails.

That means the same significance level can produce different cutoffs. A right-tailed z test at α = 0.05 uses 1.645, while a two-tailed z test at α = 0.05 uses ±1.96. Chi-square critical values are treated as right-tailed in this calculator because the distribution is non-negative and larger values are the ones that push into the rejection region. F tests are usually right-tailed too, especially for ANOVA and variance-ratio testing, but the calculator can also show left-tail or two-tail cutoffs when a specific textbook or software workflow asks for them.

Worked examples

For a two-tailed z test at α = 0.05, the calculator returns ±1.96. If your test statistic is 2.10, it falls in the rejection region because it is beyond the upper cutoff. If the test statistic is 1.70, it does not clear the threshold.

For a t test at α = 0.05 with 10 degrees of freedom, the critical values are about ±2.228. That cutoff is more extreme than the z value because small-sample t tests account for extra uncertainty in the estimated standard deviation.

For a chi-square test at α = 0.05 with 10 degrees of freedom, the upper critical value is about 18.307. That value is useful for goodness-of-fit tests, variance checks, and contingency-table questions where the test statistic is always non-negative.

For a right-tailed F test at α = 0.05 with numerator df = 5 and denominator df = 10, the upper critical value is about 3.326. That is the cutoff you would compare with an ANOVA-style F statistic when deciding whether the observed ratio is unusually large under the null hypothesis.

Use the observed test statistic, not just the cutoff

A critical value becomes more useful when you immediately compare it with the observed test statistic from your analysis. If the statistic crosses the cutoff in the correct tail, it lies in the rejection region and the null hypothesis is rejected at the chosen alpha level. If it does not cross the cutoff, you fail to reject the null hypothesis at that threshold.

That comparison step is where many quick critical value tables stop too early. A stronger workflow keeps the cutoff and the observed statistic together so you can see the decision rule directly instead of mentally checking whether 2.31 is beyond 1.96, whether −1.40 is far enough into the left tail, or whether a chi-square statistic is still below the upper-tail boundary.

Critical value reference points at 90%, 95%, and 99% confidence

Many searches for a critical value calculator are really asking for a common threshold such as a 90%, 95%, or 99% confidence cutoff. For a two-tailed z test, those standard reference points are about ±1.645, ±1.96, and ±2.576. A smaller alpha level pushes the rejection region farther out, which makes it harder to reject the null hypothesis but lowers the risk of a Type I error.

The same pattern holds for t, chi-square, and F distributions, but the exact critical value depends on degrees of freedom. That is why the page now keeps confidence-level shortcuts and a reference-style alpha comparison table on the calculator itself instead of forcing you to rerun the same setup three times just to compare α = 0.10, 0.05, and 0.01.

Critical value vs p-value

A critical value tells you where the rejection region starts. A p-value tells you how extreme your observed statistic is under the null hypothesis. Both answer the same hypothesis-testing question, but they do it from different directions.

If you already know the critical value, you can compare the test statistic directly against the cutoff. If you already have the p-value, you can compare it to α. In both cases, the decision rule is the same: reject when the result is more extreme than the chosen threshold.

Why degrees of freedom matter

Degrees of freedom change the shape of the t, chi-square, and F distributions. As the degrees of freedom increase, t critical values move closer to z critical values. For chi-square, the distribution shifts right and becomes less skewed as df grows, which changes the upper critical value. For F, both numerator df and denominator df matter because the test statistic is a ratio of two variance estimates.

That is why the same test statistic can be significant in one study and not significant in another. A chi-square statistic of 10 is unusual with 2 degrees of freedom but much less unusual with 10 or 12 degrees of freedom. An F statistic also needs both degrees of freedom before you can decide whether it clears the cutoff, so the calculator evaluates the statistic together with the relevant df inputs.

What this page covers and what it does not

This calculator is designed for the critical values most often used in introductory hypothesis testing: z, t, chi-square, and F. It is a good fit when you already know the distribution, alpha, tail type, and degrees of freedom, and you want the cutoff plus the rejection region.

It does not compute the test statistic from raw data or replace a complete ANOVA, regression, chi-square, or t-test workflow. If you need that step, use a dedicated z-score, t-test, chi-square, or ANOVA calculator first and then compare the result to the critical value here.