Critical Value Calculator

StatisticsLast updated: August 22, 2026

A critical value is the boundary of the rejection region in a hypothesis test: if the test statistic crosses it, the null hypothesis is rejected. The value itself comes from the inverse cumulative distribution function (the quantile function) of the chosen distribution; the significance level α sets the probability left in the tail, and the direction of the hypothesis decides whether that probability sits in a single tail or is split between two.

This calculator produces critical values for the four core test distributions without any need for a printed table: z for tests on means and proportions, t for the case where the population standard deviation is unknown, χ² for goodness-of-fit and independence tests, and F for variance ratios and ANOVA. Alongside the critical value you get the rejection rule, the non-rejection region, a verification of the tail area, a table of common α levels, and a density plot with the rejection region shaded.

Choose the sampling distribution your test is based on.
Sets the false-positive risk. In the social sciences 5% is the usual choice.
A directional hypothesis should be chosen only if the direction of the difference can be justified in advance.
n − 1 for a one-sample test, n₁ + n₂ − 2 for two independent samples; with the Welch correction it can be fractional.

Critical Value Formulas

Two-tailed (symmetric):  ±z₁₋α/₂   ·   ±t₁₋α/₂(df)
Right tail:              z₁₋α  ·  t₁₋α(df)  ·  χ²₁₋α(df)  ·  F₁₋α(df₁, df₂)
Left tail:               zα   ·  tα(df)   ·  χ²α(df)   ·  Fα(df₁, df₂)
Two-tailed (χ², F):      lower = quantile(α/2)  ·  upper = quantile(1 − α/2)
Definition:              P(X > critical) = α   (for the right tail)
F reciprocity:           Fα(df₁, df₂) = 1 / F₁₋α(df₂, df₁)
Rejection rule:          |statistic| > critical  →  H₀ is rejected

Critical values are computed from the inverse cumulative distribution function (the quantile function) and agree with the rounded entries of printed tables to three decimal places. Here α is the probability of a Type I error: rejecting an H₀ that is in fact true.

How to Calculate

  1. Choose the distribution your test is based on: z or t for means and proportions, χ² for goodness of fit and independence, F for variance ratios and ANOVA.
  2. Set the significance level (α); 0.05 is the common choice in the social sciences.
  3. Choose the hypothesis direction: a two-tailed test splits α between the two tails, a one-tailed test leaves all of it in a single tail.
  4. Enter the degrees of freedom the distribution requires (a single value for t and χ², separate numerator and denominator values for F).
  5. Read the critical value and the rejection rule, then compare your test statistic with that boundary to reach a decision.
  6. Use the shaded density plot to see where the rejection region lies, and the table to compare values at other α levels.

Worked Examples

Critical value for a two-tailed t test

For df = 24 and α = 0.05 the two-tailed critical t value is ±2.0639, leaving an area of 0.025 in each tail. H₀ is rejected when |t| > 2.0639. The z equivalent for the same probability is 1.9600, so the t value is 0.1039 larger because of small-sample uncertainty.

Critical t value: ±2.0639 · Rejection rule: |t| > 2.0639 · Non-rejection (acceptance) region: [−2.0639; 2.0639]

Chi-square goodness of fit (right tail)

For df = 5 and α = 0.05 the right-tail critical value is χ² = 11.0705. If the computed chi-square statistic exceeds this boundary, the gap between observed and expected frequencies counts as statistically significant. The non-rejection region is [0 , 11.0705].

Critical χ² value: 11.0705 · Rejection rule: χ² > 11.0705 · Non-rejection (acceptance) region: (0; 11.0705]

F critical value for ANOVA

In an ANOVA with four groups and 24 observations in total, df₁ = 3 and df₂ = 20; at α = 0.05 the critical F value is 3.0984. If the F statistic exceeds this value, at least one group mean differs from the others. The non-rejection region is [0 , 3.0984].

Critical F value: 3.0984 · Rejection rule: F > 3.0984 · Non-rejection (acceptance) region: (0; 3.0984]

Frequently Asked Questions

How are the critical value and the p-value related?
They are two ways of stating the same decision. The critical value is the threshold that keeps the tail probability at α; the p-value is the actual probability the observed statistic leaves in the tail. If the test statistic exceeds the critical value, then p < α necessarily. The critical value approach is a legacy of the table era, while the p-value is preferred in modern reporting because it also shows how extreme the result is.
Why is the z value 1.96 while the t value is larger?
The t distribution has heavier tails than the normal distribution because the population standard deviation is unknown and estimated from the sample, which adds uncertainty. For the same α, the boundary has to be pushed farther out. The gap closes as the degrees of freedom grow: the two-tailed critical t is 2.042 at df = 30 and 1.980 at df = 120, while z stays at 1.960.
Why does a chi-square test use only the right tail?
In goodness-of-fit and independence tests the chi-square statistic is built from squared differences between observed and expected frequencies, so it grows as the discrepancy grows. Deviations in any direction push the statistic to the right only, which puts the rejection region in the right tail. A two-tailed chi-square makes sense only when building a confidence interval for a variance or running a variance test.
Why does the order of the degrees of freedom matter for the F distribution?
The F statistic is a ratio of two variances, and the numerator and denominator degrees of freedom play different roles. F(3, 20) and F(20, 3) are entirely different distributions: at α = 0.05 the first has a critical value of 3.0984 and the second 8.6602. In ANOVA the numerator carries the between-groups degrees of freedom (k − 1) and the denominator the within-groups value (N − k); mixing them up leads to the wrong decision.
How do I find a critical value in Excel, SPSS, and R?
In Excel you use NORM.S.INV(1−α/2), T.INV.2T(α, df), CHISQ.INV.RT(α, df), and F.INV.RT(α, df1, df2). In R, qnorm(1−α/2), qt(1−α/2, df), qchisq(1−α, df), and qf(1−α, df1, df2) give the same results. SPSS output does not print the critical value separately; it reports the p-value (Sig.) directly and the decision is made from that.
What changes if I pick α = 0.01 instead of 0.05?
The critical value grows, the rejection region narrows, and rejecting H₀ becomes harder. The Type I error risk drops from 5% to 1%, while the Type II error risk (missing a real difference) rises and the power of the test falls. With a stricter α you need a larger sample size to keep the same power.
Should I change the critical value for multiple comparisons?
Yes. If m independent tests are run on the same data, the probability of seeing at least one false positive rises to 1 − (1 − α)^m. The Bonferroni correction uses α/m for each test; with 5 comparisons, for example, you use 0.01 instead of α = 0.05 and the two-tailed z critical value becomes 2.576 instead of 1.960. Less conservative methods such as Holm and Benjamini-Hochberg are also available.