Chi-Squared Test Formula
Test whether observed data matches expected frequencies.
The standard test for categorical data independence.
The Formula
The chi-squared test compares observed frequencies to expected frequencies. A large χ² value means the observed data is significantly different from what was expected.
Variables
| Symbol | Meaning |
|---|---|
| χ² | Chi-squared test statistic |
| O | Observed frequency (actual count) |
| E | Expected frequency (what you would expect by chance) |
| Σ | Sum across all categories |
Example 1
A die is rolled 60 times. Expected: 10 per face. Observed: 8, 12, 10, 11, 7, 12.
χ² = (8-10)²/10 + (12-10)²/10 + (10-10)²/10 + (11-10)²/10 + (7-10)²/10 + (12-10)²/10
χ² = 0.4 + 0.4 + 0 + 0.1 + 0.9 + 0.4
χ² = 2.2 (with 5 degrees of freedom, p > 0.05 — the die appears fair)
Example 2
Survey: 200 people chose colors. Expected 50 each. Observed: Red=70, Blue=55, Green=40, Yellow=35
χ² = (70-50)²/50 + (55-50)²/50 + (40-50)²/50 + (35-50)²/50
χ² = 8.0 + 0.5 + 2.0 + 4.5
χ² = 15.0 (with 3 df, p < 0.01 — strong evidence of preference)
When to Use It
Use the chi-squared test when:
- Testing if a die, coin, or random process is fair
- Determining if two categorical variables are independent
- Comparing survey responses across different groups
- Checking if observed genetic ratios match expected Mendelian ratios