Chi Square Table

The chi square table gives right-tail critical values of the chi-square (χ²) distribution by degrees of freedom — the reference for goodness-of-fit tests, tests of independence, and tests about variances.

How to read the chi square table

Unlike the z and t tables, the chi-square table is used from one side only: its columns are right-tail probabilities directly. Find your degrees of freedom row, run across to the column for your significance level, and the cell is the critical value your test statistic must exceed. Worked example: a goodness-of-fit test with 10 degrees of freedom at the 0.05 level — row 10, column 0.05 — gives χ² = 18.307. A computed statistic of 21.4 exceeds it, so the result is significant at 0.05. A second example, small table: df = 5 at 0.05 reads 11.070, the critical value behind countless textbook problems. Degrees of freedom depend on the test: (rows − 1)(columns − 1) for a contingency table, categories − 1 − (estimated parameters) for goodness of fit, n − 1 for a variance test.

Where chi-square is used

• Goodness of fit: do observed category counts match expected proportions? (Dice fairness, Mendel's peas, digit distributions.)
• Independence: are two categorical variables associated in a contingency table? The workhorse of survey analysis.
• Variance tests: is a population variance equal to a claimed value? The chi-square distribution is the sampling distribution of (n − 1)s²/σ².
• Confidence intervals for σ²: read two columns (for example 0.975 and 0.025 at df = 10: 3.247 and 20.483) to bracket the variance — note the interval is not symmetric, because this distribution is not.

The chi-square distribution is a sum of squared standard normals, which is why it lives only on the positive side and skews right, straightening toward symmetry as df grows — by df = 100 it is nearly a normal curve centred at 100. That family relationship to z is real, not decorative: square the z statistic from a two-sided proportion test and you have the chi-square statistic of the corresponding 2 × 2 table, identical p-value included.

A goodness-of-fit test worked against the table

A die is rolled 60 times, so a fair die expects 10 of each face. The observed counts come out 8, 12, 9, 11, 10, 10. The statistic accumulates squared standardised misses: χ² = ((8−10)² + (12−10)² + (9−10)² + (11−10)² + (10−10)² + (10−10)²) / 10 = (4 + 4 + 1 + 1 + 0 + 0) / 10 = 1.00. With 6 categories, df = 5, and the table's 0.05 column reads 11.070. The statistic is nowhere near the critical value — nowhere near even the 0.90 column's 1.610 region — so there is no evidence against fairness; rolls this uneven (or more so) happen in the large majority of fair sessions. The example is worth keeping because it shows the test's direction: χ² grows with disagreement, and a value near the degrees of freedom (here 5) or below is the signature of a model fitting about as well as chance allows.

Values computed from the chi-square distribution's cumulative function · Verified November 2026

Frequently asked questions