Find the critical value (or the pair of critical values) that marks the rejection region of a hypothesis test. Choose the z, t, chi-square or F distribution, the significance level α, the tails and the degrees of freedom; no printed tables needed.
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Distribution
Test type
Between 0 and 1, e.g. 0.05
t: n − 1 for one sample · χ²: (rows − 1)(cols − 1) · F: numerator df₁
F only, e.g. N − k in one-way ANOVA
Enter your values to see the result.
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How to use the Critical Value Calculator
Pick the distribution your test statistic follows: z, t, χ² or F.
Choose the test type: two-tailed, right-tailed or left-tailed.
Select α (0.05 is the usual choice) or enter a custom value.
Enter the degrees of freedom (two values for F).
Read the critical value(s) and the rejection region, then compare your test statistic with them.
How critical values are found
A critical value is a quantile of the test statistic's distribution under the null hypothesis. For significance level α:
Right-tailed: the value c with P(X ≥ c) = α, i.e. the (1 − α) quantile.
Left-tailed: the value c with P(X ≤ c) = α, i.e. the α quantile.
Two-tailed: the α/2 and (1 − α/2) quantiles. For the symmetric z and t distributions these are ±c.
Which distribution to use: z for large-sample tests of means and proportions with a known standard deviation; t for means when the standard deviation is estimated from the sample; χ² for goodness-of-fit, independence and variance tests; F for ANOVA, regression and the ratio of two variances.
Common critical values
Distribution
α = 0.10
α = 0.05
α = 0.01
z, one-tailed
1.282
1.645
2.326
z, two-tailed
±1.645
±1.960
±2.576
t (df = 5), two-tailed
±2.015
±2.571
±4.032
t (df = 10), two-tailed
±1.812
±2.228
±3.169
t (df = 30), two-tailed
±1.697
±2.042
±2.750
χ² (df = 1), right-tailed
2.706
3.841
6.635
χ² (df = 4), right-tailed
7.779
9.488
13.277
F (2, 20), right-tailed
2.589
3.493
5.849
Frequently asked questions
What is a critical value?
A critical value is the cut-off on the distribution of a test statistic that separates the rejection region from the non-rejection region. If your test statistic is more extreme than the critical value, the result is statistically significant at level α and you reject the null hypothesis.
What is the z critical value for 95% confidence?
For a two-tailed test at α = 0.05 the z critical values are ±1.96. For a one-tailed test at α = 0.05 it is 1.645 (or −1.645 for a left-tailed test). At 99% confidence the two-tailed values are ±2.576.
How do I find the t critical value?
Use the degrees of freedom of your test (n − 1 for a one-sample or paired t-test, n₁ + n₂ − 2 for a pooled two-sample test) and α. For example, df = 15 and α = 0.05 two-tailed gives ±2.131. As df grows the t value approaches the z value of 1.96.
Why are the chi-square and F critical values not symmetric?
Chi-square and F distributions only take positive values and are skewed to the right, so the lower and upper cut-offs are not mirror images. A two-tailed test needs both the α/2 and the 1 − α/2 quantiles, which this calculator computes separately. Most chi-square and ANOVA F tests are right-tailed.
Which degrees of freedom do I use for an F test?
df₁ is the numerator degrees of freedom and df₂ the denominator. In a one-way ANOVA with k groups and N observations in total, df₁ = k − 1 and df₂ = N − k. For comparing two variances, df₁ = n₁ − 1 and df₂ = n₂ − 1.
Is comparing to the critical value the same as using a p-value?
Yes, the two approaches always agree: the test statistic exceeds the critical value exactly when the p-value is smaller than α. The p-value tells you how extreme the result is; the critical value tells you where the boundary lies.
How accurate are these values?
The calculator inverts the exact cumulative distribution functions (built from the incomplete beta and gamma functions) numerically, so results match statistical tables and software to at least four decimal places.
Are all the tools on FreeToolCabin tested for quality and accuracy?
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