Chi-Square Distribution Table

How to Use the Chi-Square Distribution Table

1. 1. Work out the degrees of freedom. (rows − 1) × (columns − 1) for a test of independence; categories − 1 for goodness of fit, minus one more for each parameter estimated from the data; n − 1 for a variance.
2. 2. Pick the column for α. The table gives the area to the right of each value, so for the usual right-tailed test at α = .05 use the 0.05 column.
3. 3. Compare your statistic. If your χ² is at least the critical value, reject the null hypothesis.
4. 4. Use the lookup for anything else. Any df — 35, 120 or a fraction — any α, and left-tail or two-tailed values for tests and confidence intervals about a variance.

How to Read the Chi-Square Table

Each entry is the value of χ² that has the column's area α to its right. Find your degrees of freedom in the left column and your significance level along the top: with df = 10 at α = .05 the critical value is 18.307, so a χ² statistic of 18.307 or more is significant. The columns near 1 (0.995 to 0.90) hold the small values in the left tail; the columns near 0 hold the right-tail critical values that almost every chi-square test uses.

Chi-square tests of fit and independence are right-tailed: the statistic grows with any departure from the null hypothesis, so only large values count against it. A goodness-of-fit result of χ²(3) = 7.20 falls short of 7.815, so it is not significant at .05 — the P-Value Calculator puts the exact p at .066.

Left Tails and Confidence Intervals for a Variance

The left-hand columns matter when the question is about a variance. For a sample of n = 20 with s = 4.2, a 95% confidence interval for σ² uses both tails of χ² on 19 df: σ² lies between 19 × 4.2² / 32.852 = 10.20 and 19 × 4.2² / 8.907 = 37.63. Choose "Both tails" in the lookup to get the pair of critical values at once.

Common mistakes

• Reading the header as the left-tail area. Every column is headed by the area to the right; the 0.95 column holds the value with 5% of the distribution below it.
• Getting df wrong for a contingency table. It is (rows − 1) × (columns − 1), not the number of cells minus one.
• Halving α for a goodness-of-fit test. The test is one-sided by construction, so α = .05 means the 0.05 column.

Frequently Asked Questions

How do you read a chi-square table?

Find your degrees of freedom in the left column and your significance level α in the top row; the entry where they meet is the critical value. Reject the null hypothesis if your χ² statistic is at least that large. For df = 10 and α = .05 the critical value is 18.307.

What is the critical chi-square value at 0.05 with 1 degree of freedom?

3.841, which is 1.96² — the square of the two-tailed z critical value, because a chi-square variable with 1 df is the square of a standard normal one. With 2 df the critical value is 5.991, and with 3 df it is 7.815.

How do I find the degrees of freedom for a chi-square test?

For a test of independence in an r × c table, df = (r − 1)(c − 1). For goodness of fit with k categories, df = k − 1, minus one for each parameter estimated from the data. For a test or confidence interval about one variance, df = n − 1.

Why does the table have columns like 0.995 and 0.975?

They give values in the left tail of the distribution, which two-sided tests and confidence intervals for a variance need. The header is still the area to the right, so the 0.975 column holds the value with 2.5% of the area below it.

Is a chi-square test one-tailed or two-tailed?

Tests of independence and goodness of fit are right-tailed, because only large values of χ² are evidence against the null hypothesis. Tests about a single variance can be left-, right- or two-tailed, and that is when the left-hand columns and the "Both tails" lookup come in.

What if my degrees of freedom are not in the table?

Use the lookup, which accepts any df: at α = .05 it gives 49.8018 for df = 35, for example. For very large df, χ² is approximately normal with mean df and variance 2 × df.