Chi-Square Table — Definition, Formula & Examples

A chi-square table is a reference chart that lists critical values of the chi-square distribution for different degrees of freedom and significance levels. You compare your calculated chi-square test statistic against these critical values to decide whether to reject the null hypothesis.

The chi-square table provides quantiles $\chi^2_{\alpha, \, df}$ such that $P(\chi^2 > \chi^2_{\alpha, \, df}) = \alpha$ , where $\alpha$ is the upper-tail probability and $df$ is the degrees of freedom of the chi-square distribution. If the computed test statistic exceeds the tabled critical value, the null hypothesis is rejected at significance level $\alpha$ .

Key Formula

\text{Reject } H_0 \text{ if } \chi^2_{\text{calc}} \geq \chi^2_{\alpha,\, df}

Where:

$\chi^2_{\text{calc}}$ = The chi-square test statistic computed from your data
$\alpha$ = Significance level (e.g., 0.05)
$df$ = Degrees of freedom for the test

How It Works

To use the table, first determine your degrees of freedom from your test setup (for a goodness-of-fit test,

df = k - 1

where

k

is the number of categories). Next, choose your significance level

\alpha

, commonly 0.05 or 0.01. Find the row matching your

df

and the column matching your

\alpha

. The number at that intersection is the critical value. If your calculated

\chi^2

statistic is greater than or equal to this critical value, you reject the null hypothesis.

Worked Example

Problem: You perform a chi-square goodness-of-fit test with 3 categories and compute a test statistic of 7.45. At the 0.05 significance level, should you reject the null hypothesis?

Find degrees of freedom: With 3 categories, degrees of freedom equals the number of categories minus 1.

df = 3 - 1 = 2

Look up critical value: In the chi-square table, find the row for df = 2 and the column for α = 0.05. The critical value is 5.991.

\chi^2_{0.05,\, 2} = 5.991

Compare and decide: Your test statistic exceeds the critical value, so you reject the null hypothesis.

7.45 \geq 5.991 \implies \text{Reject } H_0

Answer: Since 7.45 > 5.991, reject the null hypothesis at the 0.05 significance level. The data provides sufficient evidence that the observed distribution differs from the expected distribution.

Why It Matters

Chi-square tables are essential in introductory and intermediate statistics courses whenever you run goodness-of-fit tests, tests of independence, or tests of homogeneity. In fields like genetics, market research, and quality control, practitioners use these critical values to determine whether observed frequencies deviate significantly from expectations.

Common Mistakes

Mistake: Using the wrong degrees of freedom — for example, using the number of categories instead of categories minus 1 in a goodness-of-fit test.

Correction: For a goodness-of-fit test, df = k − 1. For a test of independence on an r × c table, df = (r − 1)(c − 1). Always verify the formula for your specific test type before looking up the table.