P-Value Calculator

A p-value is the probability, computed on the assumption that the null hypothesis is true, of getting a test statistic at least as extreme as the one you observed. Small values mean your data would be unusual if the null hypothesis held, which is why a p-value below the significance level α leads you to reject it. The calculator takes the statistic your test produced (z, t, chi-square or F), the degrees of freedom and the direction of the test, and returns the exact tail area together with the critical value for comparison.

The second mode works in reverse. Choose a distribution, a tail and α, and it returns the critical value or values that bound the rejection region. This replaces the printed z, t, chi-square and F tables in the back of a statistics textbook, and it works for any α and any degrees of freedom, including fractional ones such as the Welch t-test produces.

For the symmetric z and t distributions, a two-tailed p-value doubles the area beyond the absolute value of the statistic:

two-tailed p = 2 × P(T ≥ |t|) right-tailed p = P(T ≥ t) left-tailed p = P(T ≤ t)

Here T follows the standard normal distribution for a z-test or Student's t distribution with ν degrees of freedom for a t-test. Chi-square and F statistics cannot be negative and their distributions are skewed, so most tests built on them (goodness of fit, independence, ANOVA, regression F) use the right tail only. When you do need both tails, as in a test of a single variance, the calculator doubles the smaller of the two tail areas:

two-tailed p = 2 × min(P(X ≤ x), P(X ≥ x))

The areas come from the regularized incomplete beta function (t and F) and the regularized incomplete gamma function (chi-square), following the algorithms in Numerical Recipes, and from the complementary error function for z. Right-tail areas are computed directly rather than as one minus the cumulative probability, so a p-value of 10⁻¹⁰ keeps its leading digits instead of rounding to zero. Critical values invert the same functions: for a two-tailed test at α the bounds are the α/2 and 1 − α/2 quantiles.

A z-test, two-tailed and one-tailed

A z statistic of 2.1 leaves 0.0179 of the standard normal curve to its right. A two-tailed test counts both tails, so p = 2 × 0.0179 = 0.0357, below 0.05. If the hypothesis had been directional and set before seeing the data, the right-tailed p-value would be 0.0179.

A t-test with 30 degrees of freedom

A two-sample comparison gives t = 2.5 with 30 degrees of freedom. The two-tailed p-value is 0.0181, and the critical value at α = 0.05 is ±2.042. The statistic is beyond it, so the difference is significant at the 5% level but not at 1%.

A chi-square goodness-of-fit test

A survey with four answer categories gives χ² = 7.81 on 3 degrees of freedom. The right-tail p-value is 0.0501, just over 0.05, because 7.81 sits almost exactly on the 5% critical value of 7.8147. Results this close to the line are a reminder that α is a convention, not a cliff edge.

An ANOVA F statistic

The NIST/SEMATECH e-Handbook one-way ANOVA example (three levels, five observations each) reports F = 9.59 with 2 and 12 degrees of freedom. The calculator gives p = 0.0032, against a critical F of 3.885 at α = 0.05.

A test of one variance, two-tailed

With 25 measurements and a claimed variance, the statistic (n − 1)s²/σ₀² comes out at 9.2 on 24 degrees of freedom. That value is in the left tail, so the two-tailed p-value is twice the left-tail area: 0.0057. The two critical values at α = 0.05 are 12.401 and 39.364, and 9.2 falls below the lower one, so the variance is smaller than claimed.

Every value in this table is computed by the same engine the calculator uses. As the degrees of freedom grow, t approaches the standard normal values in the last row.

The American Statistical Association's 2016 statement on p-values lists the most common misreadings, and they are worth checking before you report a result:

• A p-value is not the probability that the null hypothesis is true, and 1 − p is not the probability that your hypothesis is right.
• It does not measure the size or importance of an effect. A tiny difference measured on a huge sample can give p below 0.001; report an effect size and a confidence interval alongside it.
• The threshold α is a decision rule chosen in advance. A p-value of 0.049 and one of 0.051 carry almost the same evidence.
• Choosing a one-tailed test after seeing which way the data point halves the p-value and inflates the false positive rate. Decide the direction before collecting data.

The evidence wording in the result (weak, moderate, strong, very strong) is a common convention tied to the 0.10, 0.05, 0.01 and 0.001 cut-offs, not a formal standard.

Results are accurate to roughly 12 significant digits for ordinary inputs, more than any printed table. p-values below 0.0001 are shown in scientific notation. Degrees of freedom may be fractional and up to ten million; for very large df the t distribution is numerically indistinguishable from z. The calculator assumes your statistic really does follow the stated distribution under the null hypothesis, which depends on the assumptions of the test that produced it (independent observations, approximate normality, expected counts of at least about five for chi-square), so check those first.