P-Value & Critical Value Calculator

What this tool is used for

• Getting a p-value for a test statistic you have already computed.
• Checking a reported p-value against the statistic and degrees of freedom.
• Comparing one-tailed and two-tailed results for the same statistic.
• Working through a hypothesis-testing exercise with the tails chosen explicitly.
• Confirming a statistics package's output against the distribution directly.

Frequently Asked Questions

Which distributions can it evaluate?

Four: the standard normal (z), Student's t with one degrees-of-freedom parameter, chi-square with one, and F with two — numerator and denominator, which is why the F-test exposes a second df field the others do not. Picking the wrong family gives a plausible number that answers a different question.

How are the tail probabilities actually computed?

The normal CDF uses the Abramowitz & Stegun 26.2.17 rational approximation, accurate to better than 7.5×10⁻⁸. Chi-square goes through the regularised incomplete gamma as a series expansion; t and F both go through the regularised incomplete beta by continued fraction. No lookup table, so any df works.

Why is the two-tailed p-value not simply double the one-tailed one?

It is 2 × min(CDF, 1 − CDF), which is doubling the SMALLER tail. Doubling the raw CDF for a positive statistic would give a number greater than 1. Taking the minimum first makes the result symmetric — a statistic of +2 and one of −2 return the same two-tailed p, which is the whole point of a two-tailed test.

How is the critical value found if there is no inverse formula?

By bisection on the CDF. Every distribution here has a monotonically increasing CDF, so the solver repeatedly halves an interval until the CDF at the midpoint matches the target α. That is why a critical value is available for any df you type rather than only for the tabulated 0.01/0.05/0.10 columns.

Does a p-value below 0.05 mean my hypothesis is true?

No. It means data this extreme would be unusual IF the null hypothesis were true — a statement about the data given the hypothesis, not the hypothesis given the data. It says nothing about effect size either: with a large enough sample, a difference too small to matter will still clear any α you choose.

Common errors and gotchas

• Reading the p-value as the probability the hypothesis is true, which is a different statement entirely.
• Choosing one-tailed after seeing the data, which invalidates the test.
• Getting the degrees of freedom wrong, which changes the answer materially.
• Treating 0.05 as a bright line rather than a convention.
• Running many tests and reporting the significant ones without correcting for multiplicity.