Critical Value Calculator

A critical value is the boundary of the rejection region of a hypothesis test. It is chosen so that, when the null hypothesis is true, the test statistic lands beyond it with probability exactly α, the significance level. A statistic past the cutoff is rare enough, by your own standard, to reject the null hypothesis.

Formally it is a quantile of the distribution of the test statistic under the null hypothesis. Match the distribution to your test: z for means with a known standard deviation, large samples and proportions; t for means with an estimated standard deviation; chi-square for goodness of fit, independence and variance tests; F for ANOVA and for comparing variances. The t, chi-square and F distributions also need their degrees of freedom.

F⁻¹ is the inverse cumulative distribution function (the quantile function) of the test statistic. A two-tailed test shares α between the two tails, α/2 each, which pushes each limit further out: at α = 0.05 the z cutoff is 1.96 two-tailed but only 1.645 one-tailed. The z and t distributions are symmetric, so their two limits are −c and c. Chi-square and F are skewed, so the lower and upper limits have to be found separately. Each limit is found by inverting its own tail probability, so a very small α such as 1e-10 stays accurate.

The two-tailed value at α = 0.05, 1.96, is also the multiplier of a 95% confidence interval, and 2.5758 is the one for 99%. For the full grid of areas, use the z-table, or run it backwards with the inverse normal calculator.

The t cutoff shrinks toward the z value as the sample grows, because a larger sample estimates the standard deviation better. With df = 5 the two-tailed 5% cutoff is 2.5706, with df = 20 it is 2.0860 and with df = 1000 it is 1.9623, essentially the normal 1.96. The full grid is in the t-table, and how to read a t-table explains the layout.

Goodness-of-fit and independence tests are right-tailed, so only the upper cutoff matters. A two-sided test of a variance needs both limits. The whole distribution shifts right as df grows, which is why the cutoff rises. More values are in the chi-square table. For an F test the cutoff depends on both degrees of freedom: at α = 0.05 the right-tailed F(3, 12) cutoff is 3.4903 and F(3, 20) is 3.0984, see the F-table for more.

The two approaches always give the same decision. The critical value fixes a threshold on the scale of the statistic before the test: reject if the statistic is more extreme than the cutoff. The p-value works on the probability scale after the test: reject if the tail probability of the observed statistic is at most α. Critical values are handy when the same threshold is reused, as in control charts, acceptance sampling and printed tables. The p-value tells you how strong the evidence is, not just whether it crossed the line, so many reports give both. Compute the observed p-value with the p-value calculator.

A researcher compares n = 21 reaction times with a published benchmark using a two-tailed one-sample t-test at α = 0.05. The degrees of freedom are df = n − 1 = 20. Splitting α gives 0.025 in each tail, so the cutoff is the t value with 0.025 above it: t = 2.086 (Load example fills this in). The decision rule is: reject H₀ if |t| > 2.086. A test statistic of t = 2.30 exceeds it, so the null hypothesis is rejected at the 5% level. With only df = 5 the cutoff would be 2.5706 and t = 2.30 would not be significant: small samples need stronger evidence.

Replace alpha with your significance level, for example 0.05. The TI-84 has inverse functions for z and t (see the TI-84 statistics guide); the inverse t calculator does the same for t.