Critical Value Calculator

How to use the calculator

1. Enter the significance level (α), the error rate you accept, for example 0.05. It must be strictly between 0 and 1.
2. Pick the distribution your test statistic follows: T for a t-test, Z for a z-test, chi-square for variance and goodness-of-fit tests, F for variance ratios and ANOVA.
3. Enter the degrees of freedom. A chi-square or T test needs one value; an F test needs numerator and denominator degrees of freedom. Z needs none.
4. For T, Z and F choose a one-tailed or two-tailed test. Chi-square critical values are always one-tailed, on the right.
5. Press Calculate Critical Value.

The result lists the critical value (or values) and the critical region, the set of statistic values that lead you to reject the null hypothesis. For a one-tailed T or Z test you see both the right-sided and the left-sided cut-off, and you use the one that matches the direction of your alternative hypothesis. For a two-tailed test the α is split in half between the two tails, so each tail holds α ÷ 2.

Critical values explained

A critical value is the point on the scale of a test statistic beyond which the data are surprising enough to reject the null hypothesis at your chosen significance level. If the null hypothesis is true, a statistic lands beyond the critical value with probability α, so α is the false-positive rate you are prepared to tolerate.

The decision rule is simple: compute your test statistic and compare it with the critical value. If it falls inside the critical region, reject the null hypothesis. This is equivalent to checking whether the p-value is below α; you can obtain that p-value from the p-value calculator, or convert between scores and probabilities with the Z-score to P-value and P-value to Z-score calculators.

Critical values are also the multipliers behind confidence intervals: a 95% two-sided interval for a mean uses the two-tailed 5% critical value. See the Z-score calculator for turning a raw observation into a standard score.

Z critical values

The Z distribution is the standard normal distribution and has no degrees of freedom. It applies when the population standard deviation is known or the sample is large enough for the normal approximation to hold.

T critical values

Student's t distribution has heavier tails than the normal distribution, so its critical values are larger, especially with few degrees of freedom. As the degrees of freedom grow it converges to the Z distribution. For a one-sample t-test with n observations, df = n − 1; for a two-sample test with equal variances, df = n₁ + n₂ − 2.

Example: a two-tailed t-test with 15 degrees of freedom at α = 0.05 has critical values ±2.1314. An observed t of 2.31 lies beyond 2.1314, so you reject the null hypothesis; an observed 1.90 would not be enough.

Chi-square critical values

The chi-square distribution takes only positive values and is skewed to the right, so tests such as goodness of fit and independence reject only when the statistic is large. The critical region is therefore a single right-hand tail, [critical value, +∞).

Example: in a chi-square test with 3 degrees of freedom at α = 0.05 the cut-off is 7.815. A statistic of 9.4 is inside the rejection region; 6.2 is not.

F critical values

The F distribution describes the ratio of two variance estimates and is used in ANOVA and in tests comparing two variances. It has two parameters: the numerator degrees of freedom (df₁) and the denominator degrees of freedom (df₂). The order matters, because F(3, 20) is not the same as F(20, 3).

A one-tailed F test, as in ANOVA, looks only at the upper tail. A two-tailed F test, as in a test of equal variances, has a lower and an upper cut-off; the calculator reports both. For df₁ = 4 and df₂ = 12 at α = 0.05 the two-tailed cut-offs are 0.1143 and 4.1212, while the one-tailed cut-off is 3.2592.

Common mistakes

• Mixing up α and α ÷ 2. A two-tailed 5% test uses the 2.5% point in each tail, which is why its Z critical value is 1.96 rather than 1.645.
• Wrong degrees of freedom. Using n instead of n − 1 for a one-sample t-test gives a slightly too small critical value.
• Looking in the wrong tail. Choose the tail that matches the direction stated in the alternative hypothesis before you see the data.
• Expecting a left tail for chi-square. The usual tests reject only for large statistics, so only the upper cut-off is used.
• Treating the critical value as an effect size. It is a threshold for a test statistic, not a measure of how big or important a difference is.