Find the critical value for a hypothesis test from the Z (standard normal), t-Student, χ² (chi-square), or F distribution. Choose your distribution, the test type, and the significance level, and the calculator returns the critical value and the rejection region.
The critical value calculator finds the critical value for a statistical hypothesis test in seconds. It supports the four distributions you meet most often in inferential statistics — the Z (standard normal), t-Student, χ² (chi-square), and F (Fisher-Snedecor) distributions — and every common test type: left-tailed, right-tailed, and two-tailed. Choose your distribution, enter the significance level and any degrees of freedom, and the calculator returns the critical value (or values) along with the rejection region. It is built for students, researchers, and anyone running a hypothesis test who needs the cut-off point that separates "reject" from "do not reject."
This page explains what a critical value is, how it relates to the significance level and the rejection region, the exact definition for each distribution and tail type, worked examples you can reproduce, and answers to the questions people ask most. Whether you are checking homework or analyzing real data, this tool gives you accurate, instant results.
A critical value is the point on a test statistic's distribution that marks the boundary of the rejection region — the range of values so extreme that, if your test statistic falls there, you reject the null hypothesis. In practice you compute a test statistic from your data (a Z-score, t-score, chi-square statistic, or F-ratio) and compare it against the critical value. If the statistic is more extreme than the critical value, the result is statistically significant and you reject the null hypothesis; if not, you fail to reject it. The critical value therefore turns an abstract probability threshold — the significance level — into a concrete number you can compare your data against.
The significance level, written α (alpha), is the probability of rejecting the null hypothesis when it is actually true — a "false positive." It is the total area in the tails of the distribution that you set aside as the rejection region, and by convention it is most often 0.05 (5%), though 0.01 and 0.10 are also common. The critical value is simply the value that cuts off that tail area. In a one-tailed test all of α sits in a single tail, so the critical value is the quantile that leaves an area of α beyond it. In a two-tailed test the area α is split evenly between the two tails, α/2 in each, producing two symmetric critical values. Lowering α pushes the critical value further out, making it harder to reach significance and reducing the chance of a false positive.
The critical value is a quantile (an inverse cumulative distribution function value) of the chosen distribution. Writing Q for the quantile function, the definitions the calculator uses are:
| Distribution | Left-tailed | Right-tailed | Two-tailed |
|---|---|---|---|
| Z | Q(α) | Q(1−α) | ±Q(1−α/2) |
| t | Q(α) | Q(1−α) | ±Q(1−α/2) |
| χ² | Q(α) | Q(1−α) | Q(α/2), Q(1−α/2) |
| F | Q(α) | Q(1−α) | Q(α/2), Q(1−α/2) |
The Z and t distributions are symmetric around zero, so their left and right critical values are mirror images and the two-tailed test gives a ± pair. The χ² and F distributions are non-negative and skewed, so their two-tailed tests give two different positive bounds — a lower and an upper critical value. The calculator evaluates these quantiles with accurate numerical algorithms, matching statistical tables to several decimal places.
Suppose you are running a right-tailed t-test with 15 degrees of freedom at a significance level of α = 0.05. The right-tailed critical value is the 0.95 quantile of the t distribution with 15 degrees of freedom:
t critical value = Q(1 − 0.05) = Q(0.95) ≈ 1.7531
So the critical value is about 1.7531, and the rejection region is (1.7531, ∞). If your calculated t statistic exceeds 1.7531, you reject the null hypothesis. These are the calculator's default inputs, so you can press Calculate to confirm the result and then change the distribution, tail, or significance level for your own test.
Often you start from a sample size rather than degrees of freedom. For a one-sample t-test, the degrees of freedom equal the sample size minus one. With a sample of 5 observations, the degrees of freedom are 5 − 1 = 4. For a one-tailed test at α = 0.05, the t critical value with 4 degrees of freedom is about 2.132. Notice that with fewer degrees of freedom the critical value is larger than the 1.7531 from the previous example — smaller samples require more extreme results to reach significance, because they carry more uncertainty.
Choosing the right distribution depends on your test. The Z distribution is used when the population standard deviation is known or the sample is large, for example in proportion tests and large-sample mean tests. The t distribution is used for means when the population standard deviation is unknown and estimated from the sample — the usual case in practice — and it accounts for the extra uncertainty by having heavier tails, converging to the Z distribution as the sample grows. The χ² distribution is used for tests of variance, goodness-of-fit tests, and tests of independence in contingency tables. The F distribution compares two variances and underlies the analysis of variance (ANOVA) and regression significance tests. Matching the distribution to the test is essential, because each has a different shape and therefore different critical values.
The number of tails follows from your alternative hypothesis. A right-tailed test asks whether a parameter is greater than a value, placing the whole rejection region in the upper tail. A left-tailed test asks whether it is less than a value, placing the region in the lower tail. A two-tailed test asks only whether the parameter differs from a value in either direction, so it splits the rejection region between both tails. Because a two-tailed test divides α in half between the tails, its critical values sit further from the center than a one-tailed critical value at the same α — which is why a two-tailed test is more conservative and needs stronger evidence to reach significance. Always decide whether your test is one- or two-tailed before looking at the data, based on the question you are asking.
There are two equivalent ways to decide a hypothesis test, and they always agree. The critical value method, which this calculator supports, compares your test statistic against the critical value: if the statistic falls in the rejection region, the result is significant. The p-value method instead computes the probability of getting a result at least as extreme as yours and compares it to α: if the p-value is less than α, the result is significant. The two approaches are mirror images — your test statistic exceeds the critical value exactly when its p-value is below α. The critical value method is especially handy when working from statistical tables or when you want a fixed threshold to compare against, while the p-value tells you how strong the evidence is on a continuous scale.
A few Z critical values come up so often they are worth memorizing: for a two-tailed test at α = 0.05 the critical values are ±1.96, at α = 0.01 they are ±2.576, and at α = 0.10 they are ±1.645. For one-tailed Z tests at α = 0.05 the critical value is 1.645, and at α = 0.01 it is 2.326. The t distribution's critical values are always a little larger than the corresponding Z values for the same α, shrinking toward them as the degrees of freedom increase. The calculator produces all of these — and the far larger set of χ² and F values — without you needing a printed table.
The rejection region (also called the critical region) is the set of test-statistic values for which you reject the null hypothesis, and the critical value marks its boundary. The shape of the region follows directly from the test type and distribution. For a right-tailed test it is everything above the critical value, written (critical value, ∞); for a left-tailed test it is everything below, written (−∞, critical value); and for a two-tailed test it is the two outer pieces beyond the pair of critical values. For the non-negative χ² and F distributions the lower boundary is 0 rather than −∞. The calculator prints the exact rejection region for your test so you can see at a glance where your statistic must fall to be significant. If your computed test statistic lands inside this region, the evidence is strong enough to reject the null hypothesis at your chosen significance level; if it lands outside, you do not have enough evidence to reject it.
The significance level and the critical value move in opposite directions. Choosing a smaller α — demanding stronger evidence before you reject the null — pushes the critical value further into the tail, shrinking the rejection region and making significance harder to achieve. A larger α pulls the critical value inward and makes rejection easier, at the cost of a higher chance of a false positive. This is why the choice of α is a genuine trade-off between two kinds of error: a stricter threshold guards against wrongly rejecting a true null hypothesis (a Type I error) but raises the risk of missing a real effect (a Type II error). The conventional value of 0.05 is a widely accepted compromise, but fields that demand high certainty — such as particle physics or clinical trials — often use far smaller significance levels and correspondingly larger critical values. Experimenting with different α values in the calculator makes this relationship concrete.
For the t, χ², and F distributions, the degrees of freedom control the exact shape of the curve and therefore the critical value. The t distribution with few degrees of freedom has heavy tails, so its critical values are noticeably larger than the corresponding Z values; as the degrees of freedom grow, the t distribution tightens and its critical values converge on the standard normal ones, which is why for large samples the t and Z tests give almost identical thresholds. The χ² distribution shifts rightward and grows more symmetric as its degrees of freedom increase, and the F distribution depends on two separate degrees-of-freedom values — one for the numerator and one for the denominator — reflecting the two variances being compared. Because a small change in degrees of freedom can meaningfully move the critical value, entering the correct value is essential; the calculator handles the underlying distribution mathematics so you only need to supply the right numbers.
It is the cut-off point that marks the edge of the rejection region. If your test statistic is more extreme than the critical value, you reject the null hypothesis at the chosen significance level.
Take the appropriate quantile of your distribution: for a right-tailed test, the (1 − α) quantile; for left-tailed, the α quantile; for two-tailed, split α between the tails. This calculator does it for the Z, t, χ², and F distributions.
For a two-tailed Z test at α = 0.05 the critical values are ±1.96; for a one-tailed Z test the critical value is 1.645. The t, χ², and F values depend on the degrees of freedom.
Degrees of freedom describe how much independent information is available to estimate a parameter. For a one-sample t-test they equal the sample size minus one; the F distribution uses two, one for the numerator and one for the denominator.
Smaller samples carry more uncertainty, so the t distribution has heavier tails and the critical value moves further out. You need more extreme evidence to reach significance with a small sample.
They always give the same decision. Your statistic exceeds the critical value exactly when its p-value is below α, so the two methods are equivalent ways of testing the same hypothesis.
This Critical Value Calculator is provided for general educational purposes. It computes critical values using accurate numerical approximations of the distribution quantile functions, which may differ from printed tables in the last decimal place. Always confirm critical results for high-stakes analysis with authoritative statistical software.