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Understanding Chi-Square Tests
What is a Chi-Square Test?
A chi-square (χ²) test is a statistical hypothesis test that compares observed frequencies with expected frequencies:
χ² = Σ [(O - E)² / E]
where:
- O = observed frequency
- E = expected frequency
- Σ = sum over all categories
Types of Chi-Square Tests
Goodness of Fit Test
Tests if sample data fits a hypothesized distribution
- Compares observed frequencies with expected frequencies
- Tests categorical variables
- One-way classification
Test of Independence
Tests relationship between categorical variables
- Uses contingency tables
- Tests association between variables
- Two-way or higher classification
Key Components
Degrees of Freedom (df):
- Goodness of Fit: df = k - 1
- Independence: df = (r-1)(c-1)
- where:
- k = number of categories
- r = number of rows
- c = number of columns
Critical Values:
- Based on df and α level
- Found in chi-square distribution table
- Used to determine significance
Assumptions and Requirements
- Random Sampling:
- Independent observations
- Representative sample
- Sample Size:
- Expected frequencies ≥ 5
- Adequate cell counts
- Categorical Data:
- Mutually exclusive categories
- Exhaustive categories
- Independence:
- No repeated measures
- Independent categories
Interpretation Guidelines
| P-value | Result | Interpretation |
|---|---|---|
| p < α | Significant | Reject null hypothesis |
| p ≥ α | Not Significant | Fail to reject null hypothesis |
Effect Size Measures
Cramer's V
Measures strength of association
V = √(χ² / (n * min(r-1, c-1)))
Phi Coefficient
For 2x2 tables only
φ = √(χ² / n)
Contingency Coefficient
Alternative measure of association
C = √(χ² / (χ² + n))
Real-World Applications
Market Research
Consumer preferences and brand associations
Medical Research
Treatment effectiveness and disease associations
Social Sciences
Demographic studies and survey analysis
Quality Control
Product defect analysis and process improvement