Critical Value Calculator

Critical value calculator

Free online critical value calculator for the four distributions used in nearly every hypothesis test: the normal distribution (z critical value), Student's t distribution (t critical value), the chi-square distribution and the F distribution. Pick the distribution, enter its parameters and the significance level α, and the calculator returns the critical value(s), the rejection region drawn on an interactive chart, a step-by-step calculation, and the equivalent R and Excel code.

What is the critical value?

The critical value is the boundary between the acceptance region (green) and the rejection region (red). Assuming H₀ is true, the probability of getting the critical value or a more extreme value — the red area — is the significance level α.

What counts as “more extreme” depends on the alternative hypothesis:

• Two-tailed — two critical values; more extreme means smaller than the lower one or larger than the upper one.
• Left-tailed — one critical value; more extreme means smaller than it.
• Right-tailed — one critical value; more extreme means larger than it.

How to find the critical value

A critical value is a quantile: the score whose cumulative probability equals the critical probability. You can read it off a table, or compute it directly with the score calculator. For a one-tailed test the critical probability is α (left tail) or 1 − α (right tail); for a two-tailed test it is α/2 and 1 − α/2.

Left-tailed test

• The critical probability is α — for example 0.05.
• Calculate the score at that probability (z-score, t-score, F-score and so on).
• The critical value is negative for distributions symmetric around zero, such as the normal and t distributions, but positive for the chi-square and F distributions, which are supported only on positive values.

Right-tailed test

• The critical probability is 1 − α — for example 1 − 0.05 = 0.95.
• Calculate the score at 0.95. The critical value is usually positive.

Two-tailed test

• Split α evenly between the tails: each holds α/2.
• The lower critical value is the score at α/2; the upper one is the score at 1 − α/2.

Z critical value calculator

The z critical value calculator gives the critical value for a z statistic (normal distribution). Choose Normal and enter the mean and standard deviation; for the standard normal distribution use μ = 0 and σ = 1. The familiar two-tailed value at α = 0.05 is ±1.959964. You can also look it up in the Z table.

t critical value calculator

The t critical value calculator gives the critical value for a t statistic. Choose Student's t and enter the degrees of freedom (usually n − 1). The t distribution has heavier tails than the normal, so its critical values are larger — and they converge to the z values as the degrees of freedom grow. See also the t table.

F critical value calculator

The F critical value calculator gives the critical value for an F statistic. Choose F and enter the numerator degrees of freedom (n_numerator − 1) and the denominator degrees of freedom (n_denominator − 1). The F distribution is right-skewed and defined only for positive values, so its two-tailed critical values are not symmetric.

Chi-square critical value calculator

The chi-square critical value calculator gives the critical value for a chi-square statistic. Choose Chi-Square and enter the degrees of freedom. Like the F distribution it is right-skewed and positive only, so the two critical values of a two-tailed test are not mirror images of each other.

Critical value or p-value?

They are two routes to the same decision. The critical-value approach fixes α first and compares the statistic against the boundary; the p-value approach computes the probability of a result at least as extreme as the one observed and compares it against α. Both reject H₀ in exactly the same cases.