How to Read the Chi-Square Table
The chi-square table provides critical values for the chi-square distribution. Each value represents the chi-square statistic that cuts off the specified probability in the right tail of the distribution.
Steps for a Chi-Square Test
- Compute your chi-square statistic: χ² = Σ[(O − E)² / E]
- Find your degrees of freedom. For goodness-of-fit: df = k − 1. For independence: df = (rows − 1)(columns − 1)
- Choose your significance level α (e.g., 0.05)
- Find the critical value at the row and column intersection
- If χ² statistic > critical value → reject the null hypothesis
Example: Goodness-of-Fit Test
Testing whether a 6-sided die is fair: k = 6 categories → df = 5
Significance level: α = 0.05 → Critical value from table: 11.070
If your calculated χ² = 12.5 > 11.070, you reject the null hypothesis (the die is NOT fair at α = 0.05).
When to Use Chi-Square Tests
Goodness-of-Fit
Test if observed frequencies match expected frequencies
Test of Independence
Test if two categorical variables are related
Test of Homogeneity
Test if proportions are equal across groups
FAQ
What is a chi-square distribution?
The chi-square distribution is a continuous probability distribution that arises when you sum the squares of k independent standard normal random variables. It has one parameter — degrees of freedom (df). It is always non-negative and right-skewed, becoming more symmetric as df increases.
How are degrees of freedom calculated?
For a goodness-of-fit test: df = k − 1 (k = number of categories). For a test of independence in a contingency table: df = (r − 1)(c − 1) where r = number of rows and c = number of columns. For a chi-square variance test: df = n − 1.
What is the assumption about expected frequencies?
The chi-square test requires that each expected frequency (E) be at least 5. If some expected frequencies are less than 5, you may need to combine categories, use Fisher's exact test, or use a different statistical approach.