How to Read a Chi-Square Table (df, Critical Values, p-Values)

Why chi-square tables are right-tailed

The chi-square distribution is skewed to the right, and its test statistic is never negative. Goodness-of-fit and independence tests are right-tailed: a large statistic is evidence against the null hypothesis. So the table gives the value that cuts off a chosen area α in the right tail.

Each row is a number of degrees of freedom (df), and each column is a right-tail area α such as 0.05. Some tables, like the NIST one, label columns by the probability less than the critical value, so 0.95 there corresponds to α = 0.05.

Degrees of freedom

How you count df depends on the test.

• Goodness of fit: df = k − 1, where k is the number of categories. Six categories give df = 5.
• Independence (two-way table): df = (r − 1)(c − 1), where r is the number of rows and c the number of columns. A 2 × 2 table gives df = 1. A 3 × 6 table gives df = 10.
• Both tests need expected counts of at least five in each cell for the chi-square approximation to work.

Worked lookups

With df = 1 and α = 0.05, the critical value is 3.841. With df = 10 and α = 0.05, it is 18.307. The critical value grows as df grows because the distribution shifts to the right.

Comparing your statistic with the critical value

Compute the test statistic, then compare it with the critical value for your df and α. If the statistic is larger than the critical value, reject the null hypothesis.

Example 1: a 2 × 2 independence test gives a statistic of 4.2 at df = 1. Since 4.2 > 3.841, reject at α = 0.05. Example 2: a 3 × 6 table gives a statistic of 15.0 at df = 10. Since 15.0 < 18.307, do not reject.

What a p-value means

The p-value is the probability of getting a test statistic at least as extreme as yours if the null hypothesis were true. It is the right-tail area beyond your statistic. If the p-value is smaller than α, reject the null hypothesis; that is the same decision as finding the statistic beyond the critical value.

A p-value is not the probability that the null hypothesis is true. A printed table gives only a few α columns, so it bounds the p-value rather than giving it exactly; software gives the exact right-tail area.