From a chi-square statistic to its p-value
Chi-square statistics come from goodness-of-fit tests, tests of independence in contingency tables, and tests about a variance. They are sums of squared standardised gaps, so they are never negative, and only large values argue against the null. The p-value is the right-tail area of the chi-square distribution with your test's degrees of freedom.
χ² = 3.841 with df = 1: right-tailed p = 0.0500. That number is the 5% critical value for df 1, also recognisable as 1.96², because a chi-square on 1 df is literally a squared z. At df = 10 the 5% point is χ² = 18.307.
Finding your df
- Goodness of fit: categories − 1 − (parameters estimated)
- Independence in a table: (rows − 1)(columns − 1)
- Test of one variance: n − 1
Frequently asked questions
Why is there no two-tailed chi-square p-value?
Small chi-square values mean the data fit the null unusually well, which is not evidence against it in these tests. Disagreement only ever inflates the statistic, so the right tail is the test.
Can chi-square be used with small expected counts?
The approximation degrades when expected counts drop below about 5 per cell. Combine categories, or use an exact test for very small tables.