How to Calculate and Read a P-Value
A p-value is a tail probability calculated under a null model. The distribution and tail direction must match the statistic and alternative hypothesis before the number can be interpreted responsibly.
Method
- Identify whether the statistic follows a standard normal, Student t or chi-square distribution under the null model.
- For t or chi-square, enter the correct degrees of freedom.
- For z or t, choose left, right or two-tailed from the alternative hypothesis defined before seeing the result.
- For common Pearson chi-square tests, use the upper-tail probability.
- Report the p-value with the test statistic, degrees of freedom when applicable and the assumptions of the test.
Worked example
A z statistic of 1.96 gives a two-tailed p-value near 0.05.
A chi-square statistic of 3.841 with 1 degree of freedom has an upper-tail p-value near 0.05.
Notes
- A p-value is not the probability that the null hypothesis is true.
- A threshold does not measure effect size, practical importance or study quality.
- Choose the test and tail before inspecting the observed result.
A p-value is conditional on a statistical model
A p-value answers a narrow question: assuming the null hypothesis and the test's other assumptions are correct, how unusual is a test statistic at least as extreme as the one observed? It does not tell you the probability that the null hypothesis is true, the probability that the result happened by chance, or the probability that a replication will succeed.
The same numerical statistic can lead to a different p-value when the reference distribution, degrees of freedom or tail choice changes. Always identify whether the input is a z, t, chi-square or another statistic and whether the test is one-sided or two-sided before interpreting the result.
Tail direction must be chosen before seeing the result
A two-sided test looks for departures in either direction and is appropriate when both a positive and negative effect would contradict the null. A one-sided test places the rejection region in one direction and should come from a directional hypothesis chosen in advance, not from whichever side produces a smaller p-value after the data are seen.
Changing from two-sided to one-sided after inspecting the result changes the question being asked. That can make a result look more significant without adding evidence. Record the planned alternative hypothesis and use the tail that matches it.
Statistical significance is not effect size
Large samples can produce small p-values for differences that are too small to matter in practice, while small samples can produce large p-values even when the observed effect is meaningful but imprecisely estimated. A complete interpretation should include the effect estimate, uncertainty and subject-matter context.
Thresholds such as 0.05 are conventions, not a boundary between truth and falsehood. Treat the p-value as one piece of evidence, and consider confidence intervals, data quality, multiple testing, prior evidence and the consequences of the decision being made.
Editorial review and limitations
Continue with the related tools
Use the linked tools for your own inputs and scenarios.
- T-Test Calculator — Run one-sample, Welch two-sample or paired two-tailed t tests with t statistic, degrees of freedom and p-value.
- Chi-Square Test Calculator — Run chi-square independence or goodness-of-fit calculations with expected counts, degrees of freedom and upper-tail p-value.
- Normal Distribution Calculator — Calculate normal-distribution left-tail, right-tail and between-values probabilities, density, z-scores and inverse percentiles.