P-Value & Critical Value Calculator

About the P-Value & Critical Value Calculator

Type a test statistic — z, t, χ², F or Pearson’s r — with its degrees of freedom, choose the tail that matches your alternative hypothesis, and get the p-value: the probability, if the null hypothesis were true, of a statistic at least as extreme as yours. The page also gives both tail areas, the decision at your significance level α and a chart of the distribution with the p-value area and the rejection region shaded.

Switch to Critical value to find the cut-off for any α — two-tailed, right-tailed or left-tailed — or to Tables for complete z, t, χ², F and r tables, for every α and any number of degrees of freedom you would otherwise look up in a printed book. Probabilities are worked out from the regularized incomplete beta and gamma functions, so they stay precise far into the tails, where printed tables stop.

Common uses

• Getting the exact p-value of a t, z, chi-square or F statistic from a paper, a homework question or software output that gives only the statistic.
• Finding critical values for any α and degrees of freedom, including the ones printed tables leave out (ν = 37, α = 0.002).
• Testing whether a correlation coefficient differs from zero, from r and the number of pairs.
• Checking a two-tailed against a one-tailed result, or seeing the rejection region of a test on a chart.

How the p-value is computed

• z: Φ(z) = ½·[1 + erf(z/√2)]; the right tail is 1 − Φ(z), worked out directly so that P(Z ≥ 10) = 7.62 × 10⁻²⁴ keeps its digits.
• t with ν degrees of freedom: P(|T| ≥ |t|) = I_x(ν/2, ½) with x = ν/(ν + t²), where I is the regularized incomplete beta function (NIST DLMF §8.17). One tail is half of it.
• χ² with k degrees of freedom: P(χ² ≤ x) = P(k/2, x/2), the regularized lower incomplete gamma function (NIST DLMF §8.2); the right tail is Q(k/2, x/2).
• F with d₁ and d₂ degrees of freedom: P(F ≤ x) = I_y(d₁/2, d₂/2) with y = d₁x/(d₁x + d₂).
• Pearson’s r from n pairs: t = r·√(n − 2)/√(1 − r²), which follows a t distribution with n − 2 degrees of freedom when the true correlation is zero.

The distributions are those of the NIST/SEMATECH e-Handbook (t, chi-square, F). Each tail is computed from its own formula, never as 1 minus a number close to 1, and a p-value too small for a computer’s numbers (below about 10⁻³⁰⁸) is shown from its logarithm instead of as 0.

One-tailed or two-tailed?

Use the tail your alternative hypothesis points to, chosen before you see the data. For “the mean differs from 50” (≠) use two-tailed: extreme values in either direction count against the null hypothesis, so the p-value is twice the area beyond |t|. For “the mean is greater than 50” (>) use right-tailed, and for “less than” (<) left-tailed.

Chi-square tests of independence and goodness of fit, and F tests in ANOVA and regression, are right-tailed: only large values of the statistic count against the null hypothesis. A two-tailed χ² or F p-value — twice the smaller tail — belongs to tests of one variance or of two variances against “not equal”.

Critical values and printed tables

A critical value cuts off an area α in the rejection region: for a two-tailed t-test at α = 0.05 with 10 degrees of freedom it is t = ±2.228 (2.5 % in each tail), and you reject the null hypothesis when |t| ≥ 2.228. A p-value and a critical value always agree: p ≤ α exactly when the statistic falls in the rejection region.

The Tables view rebuilds the classic tables in full: t (ν = 1 to 30 and up to ∞, one-tailed α from 0.25 to 0.0005 with the two-tailed equivalents), χ² (k = 1 to 100, areas from 0.995 to 0.001), F for six values of α (d₁ and d₂ from 1 to ∞) and critical values of r for n = 3 to 1,000. They agree with the printed tables of the NIST/SEMATECH e-Handbook; in the few cells where the last digit differs — F(0.01; 2, 1) is exactly 4999.5, printed there as 4999.52 — the printed figure is the one that is off.

What a p-value does not tell you

The statement on p-values of the American Statistical Association lists the usual misreadings. A p-value says how incompatible the data are with the null hypothesis; it is not the probability that the null hypothesis is true, and it does not measure the size or the importance of an effect — a tiny, unimportant difference gives a tiny p-value in a large enough sample. Report the effect size and a confidence interval with it, and decide on more than whether p falls below 0.05.

Limitations

• Degrees of freedom can be any number from 0.1 to 10,000,000, with decimals (Welch’s t-test gives them). Beyond that t, χ² and F are practically their normal limits.
• Results carry 13 to 15 significant digits for the usual degrees of freedom and about 10 above 100,000; the page shows 4 to 8 of them.
• The calculator turns a statistic into a probability. It cannot check the assumptions of the test that produced the statistic — normal data for a t-test, large enough expected counts for a χ² test, independent observations.
• The r test is the one for Pearson’s correlation; for Spearman’s rank correlation it is an approximation that is fine from about 10 pairs.

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