Article — Critical Value Calculator
Critical Value Calculator
For a two-tailed z-test at alpha = 0.05, the critical value is plus or minus 1.960. Reject the null hypothesis when the test statistic falls outside that interval. Different distributions and tail choices change the threshold; the calculator above handles z, Student t, chi-square, and F at any alpha.
The concept itself is straightforward, but the choice of distribution, the number of tails, and the degrees of freedom each shift the answer. This page walks through each piece and explains where the numbers come from.
What a critical value is
A critical value is the boundary on the test-statistic axis between the rejection region and the non-rejection region. Pick a significance level alpha (commonly 0.05), and the critical value is the point on the chosen distribution beyond which only alpha of the probability mass lies. If your observed statistic lands beyond that point, reject the null hypothesis. If not, fail to reject.
Critical values are paired with a distribution. The standard normal gives the z critical value, Student t (W. S. Gosset, 1908) gives the t critical value, the chi-square distribution gives the chi-square critical value, and the F distribution (Fisher, 1924; Snedecor, 1934) gives the F critical value.
How the critical value calculator works
Behind every entry is a quantile of a probability distribution. For the normal we use the inverse CDF directly; the calculator implements P. J. Acklam's rational approximation, which is accurate to about one part in a billion. For the t, chi-square, and F distributions we invert the cumulative distribution by bisection on the regularized incomplete beta or gamma function (the algorithms are the ones in Numerical Recipes).
The R language returns the same numbers with qnorm, qt, qchisq, and qf. SciPy users will recognise scipy.stats.norm.ppf and friends. Our outputs agree with both to four to six decimal places, which is well beyond textbook precision.
William Sealy Gosset published the t distribution in 1908 under the pen name "Student" because his employer, Guinness Brewery, prohibited staff from publishing trade secrets. The brewery's small sample sizes were the motivating problem.
Critical value: z vs. t
Choose z when the sample is large (often n above 30) and the population standard deviation is known. Choose t when the sample is small or you have to estimate the standard deviation from the sample, which is almost always. The t distribution has heavier tails to absorb the extra uncertainty.
As degrees of freedom grow, the t critical value converges to z. At df = 120 the difference is only 0.02. For most modern data sets, both work, but using t is the safer default when sigma is estimated.
Chi-square critical value
The chi-square distribution is the sum of squared standard normal variables. Because squares cannot be negative, the distribution lives on the positive axis and is always right-tailed. The chi-square critical value is the point beyond which the upper alpha of probability lies.
It is the workhorse of goodness-of-fit tests (Karl Pearson, 1900), independence in contingency tables, and tests on variance. Degrees of freedom depend on the test: (rows - 1) x (columns - 1) for a contingency table, k - 1 minus estimated parameters for goodness-of-fit, n - 1 for a variance test.
F critical value and ANOVA
The F distribution is the ratio of two scaled chi-square variables. Like chi-square, F is positive and right-tailed. The F critical value depends on two degrees of freedom: df1 for the numerator (between-group variance) and df2 for the denominator (within-group variance).
The classic use is one-way ANOVA, where you ask whether the means of three or more groups differ. With four groups of 25 subjects each, df1 = 3, df2 = 96, and the F critical value at alpha = 0.05 is roughly 2.70. If the observed F ratio exceeds that, the group means differ at the chosen significance level.
Decide tails and alpha before seeing the data. Switching from two-tailed to one-tailed (or from 0.05 to 0.10) after looking at the numbers is the classic p-hacking move, and it is exactly what pre-registration and replication efforts are designed to prevent.
Critical value vs. p-value
The two are siblings. A critical value is a fixed threshold on the statistic axis; a p-value is a number computed from the data that says how far into the tail your statistic landed. Reject the null if the test statistic exceeds the critical value, which is equivalent to rejecting if the p-value is below alpha.
Critical-value testing is the historical approach used in statistical tables (the classic Fisher F-tables, the chi-square table, the t-table). P-value testing is more flexible because software can return a continuous probability without you having to look up a threshold. Modern stats classes teach both because each illuminates a different intuition.
- Two-tailed z, alpha 0.05 = ±1.9600
- One-tailed z, alpha 0.05 = ±1.6449
- Two-tailed t, df=20, alpha 0.05 = ±2.0860
- Chi-square, df=5, alpha 0.05 = 11.0705
- F, df1=3 df2=30, alpha 0.05 = 2.9223
- Two-tailed z, alpha 0.01 = ±2.5758
Common critical value mistakes
The most frequent error is using z when t is appropriate. With sigma estimated and a sample size of 15, the z critical value of 1.96 is too small and rejecting at that boundary inflates Type I error. The t critical value of 2.145 (df = 14) is the right threshold.
Cell sizes below 5 in a contingency table break the chi-square approximation. ANOVA assumes equal variances across groups. Failing those assumptions makes the published critical value inaccurate even if you typed everything correctly. Use Fisher's exact test or Welch's F for those cases.
known sigma, big n zest. sigma, small n t (df = n-1)variance / fit chi-squarevariance ratio / ANOVA F (df1, df2)