Critical Value Calculator — z, t, χ², F
Enter a significance level (α), choose your distribution and tail type, and get the exact critical value — the threshold beyond which you reject the null hypothesis.
Distribution
Tail type
Upper critical value — rejection region is both tails
How does this calculator work?
A critical value is the test-statistic threshold beyond which you reject H₀ at significance level α. For z: use the inverse normal CDF (1.96 for two-tailed α = 0.05). For t, χ², or F: it depends on degrees of freedom. Enter α, distribution, and tail type to get the exact cutoff and rejection region.
Formula
How this is calculated
A critical value is the boundary of the rejection region: if your test statistic exceeds it (in the relevant direction), you reject the null hypothesis at the chosen significance level α. For a right-tailed test the critical value c satisfies P(X > c) = α; for a left-tailed test P(X < c) = α; for a two-tailed test P(|X| > c) = α, which splits α/2 into each tail.
For the standard normal (z) distribution this calculator uses Peter Acklam's rational approximation of the inverse normal CDF, accurate to better than 1.15 × 10⁻⁹. For t, chi-square, and F distributions it uses a bisection search on their respective upper-tail probabilities (evaluated via the regularised incomplete beta and gamma functions) to locate the exact critical point to 11 decimal places of precision.
Typical critical values for α = 0.05: z = 1.645 (one-tailed) or ±1.96 (two-tailed); t varies by degrees of freedom and approaches the z values as df → ∞. Chi-square and F critical values are always positive; for a two-tailed chi-square test, both a lower and an upper critical value are returned because the distribution is asymmetric.
Frequently asked questions
A critical value is a fixed threshold set before the test based on your chosen α: if the test statistic exceeds it, you reject H₀. A p-value is the probability of observing a statistic at least as extreme as the one computed, given H₀ is true. Both lead to the same decision but the approaches are complementary — critical values are often easier to look up in tables.
The t distribution has heavier tails than the normal distribution, and those tails thin as sample size (and degrees of freedom = n − 1) increases. With few df you need a larger t to cut off the same tail area α; as df → ∞ the t distribution approaches the standard normal and t_c → z_c.
Use a two-tailed test when your alternative hypothesis is simply "different from the null" (H₁: μ ≠ μ₀). Use one-tailed when you predict a specific direction before seeing the data (H₁: μ > μ₀ or H₁: μ < μ₀). Two-tailed tests are more conservative and are the default in most scientific contexts.
Also known as
TG we-Calculate Editorial Team. (2026). Critical Value Calculator — z, t, χ², F [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/critical-value-calculator
TG we-Calculate Editorial Team. "Critical Value Calculator — z, t, χ², F." TG we-Calculate. 2026. https://we-calculate.com/calculator/critical-value-calculator.
TG we-Calculate Editorial Team, "Critical Value Calculator — z, t, χ², F," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/critical-value-calculator
@misc{wecalculate_critical_value_calculator, title = {Critical Value Calculator — z, t, χ², F}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/critical-value-calculator}}, year = {2026}, note = {TG we-Calculate} }
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