Critical Value Calculator

What is a critical value?

A critical value is the cut-off point that separates the values of a test statistic that lead you to reject the null hypothesis from those that do not. After you pick a significance level and a tail direction, the critical value marks the edge of the rejection region. If your computed statistic falls beyond that edge, the result is statistically significant at the chosen level.

This calculator returns the critical value for the four distributions you meet most often in hypothesis testing: the standard normal (Z), Student’s t, the chi-square, and the F distribution. Choose the distribution, the test type (two-tailed, right-tailed, or left-tailed), the significance level, and the degrees of freedom where the distribution requires them.

How does the calculator work?

Every critical value is a quantile of the distribution’s cumulative distribution function. If $F$ is the cumulative distribution function of the chosen distribution, the quantile (inverse) function $F^{-1}$ turns a probability back into the value that sits at that probability. The calculator evaluates $F^{-1}$ at the probability dictated by your significance level $\alpha$ and tail choice.

For a symmetric distribution like Z or t, the three test types map to these probabilities:

The chi-square and F distributions are not symmetric, so a two-tailed test produces two different bounds, a lower and an upper one:

Computing the quantiles

The standard normal quantile $\Phi^{-1}$ has no closed form, so the calculator uses a rational approximation (the Acklam method) refined by a Halley step, giving the inverse normal to full double precision. The t, chi-square, and F quantiles are found by inverting their cumulative distribution functions numerically, which are built from the regularized incomplete beta and gamma functions.

Worked examples

1. Z, two-tailed, $\alpha = 0.05$. Split the significance level across both tails and evaluate the normal quantile at $1 - \tfrac{0.05}{2} = 0.975$: $\Phi^{-1}(0.975) = 1.959964 \approx \pm 1.96$ The rejection region is everything below $-1.96$ or above $1.96$.

2. Z, right-tailed, $\alpha = 0.05$. A single upper tail: $\Phi^{-1}(0.95) = 1.644854 \approx 1.64$

3. t, right-tailed, $d = 15$, $\alpha = 0.05$. Evaluate the t quantile at $0.95$ with 15 degrees of freedom: $t^{-1}(0.95;\, 15) \approx 1.7531$ The rejection region is $(1.7531, \infty)$.

4. t, two-tailed, $d = 10$, $\alpha = 0.05$. Evaluate at $0.975$: $t^{-1}(0.975;\, 10) \approx \pm 2.228$

5. Chi-square, two-tailed, $d = 10$, $\alpha = 0.05$. The lower and upper bounds come from $0.025$ and $0.975$: $\chi^{2-1}(0.025;\, 10) \approx 3.247 \qquad \chi^{2-1}(0.975;\, 10) \approx 20.483$

6. F, right-tailed, $d = 5$, $d_2 = 10$, $\alpha = 0.05$. With 5 numerator and 10 denominator degrees of freedom: $F^{-1}(0.95;\, 5,\, 10) \approx 3.326$

Practical notes

• The significance level $\alpha$ must lie strictly between $0$ and $1$. Common choices are $0.10$, $0.05$, and $0.01$.
• Use the Z distribution when the population standard deviation is known or the sample is large; switch to the t distribution for small samples with an estimated standard deviation.
• The chi-square distribution is used for variance and goodness-of-fit tests, and the F distribution for comparing two variances or for analysis of variance.
• Degrees of freedom shape the t, chi-square, and F distributions. As the t degrees of freedom grow, its critical values approach the matching Z values.

FAQ

What is the difference between a one-tailed and a two-tailed critical value?

A one-tailed test places the whole rejection region in a single tail, so it uses $F^{-1}(1 - \alpha)$ (right) or $F^{-1}(\alpha)$ (left). A two-tailed test splits $\alpha$ across both tails, pushing each critical value further from the center.

Why does the chi-square critical value need degrees of freedom?

The chi-square distribution changes shape with its degrees of freedom, so a single significance level corresponds to different cut-off points for different degrees of freedom. The same is true for the t and F distributions.

How does the critical value relate to the p-value?

They are two sides of the same decision. You reject the null hypothesis when the test statistic exceeds the critical value, which is exactly when the p-value is smaller than $\alpha$.

Can a critical value be negative?

Yes. A left-tailed Z or t critical value is negative because it sits in the lower tail. Chi-square and F values are always non-negative, since those distributions are defined only for non-negative numbers.