Chi-Square Tests Complete Guide
Test independence, goodness of fit, and analyze categorical data relationships with chi-square analysis.
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Introduction
The chi-square test is a statistical method for analyzing categorical (nominal) data. It determines whether there's a significant relationship between categorical variables or whether observed frequencies differ from expected frequencies.
Types of Chi-Square Tests
Test of Independence
Tests whether two categorical variables are independent.
Example: Is gender related to voting preference?
Goodness of Fit
Tests whether observed frequencies match expected frequencies.
Example: Does a die roll follow a uniform distribution?
Test of Independence
Hypotheses
H₀ (Null):
Variables are independent (no relationship)
H₁ (Alternative):
Variables are dependent (there is a relationship)
Expected Frequencies
Formula
Expected frequencies represent what we'd expect if the variables were truly independent.
Calculation Steps
Chi-Square Statistic
Where O = observed frequency, E = expected frequency
Steps
- Create contingency table with observed frequencies
- Calculate row and column totals
- Compute expected frequencies for each cell
- Calculate (O-E)²/E for each cell
- Sum all values to get χ²
- Find degrees of freedom: df = (rows-1)(columns-1)
- Compare to critical value or find p-value
Interpretation
p-value < 0.05
Reject H₀. Variables are likely dependent. There is a significant relationship.
p-value ≥ 0.05
Fail to reject H₀. Insufficient evidence of relationship.
Summary
Key Takeaways
- 1.Chi-square tests analyze categorical data relationships.
- 2.Test of independence compares observed vs expected frequencies.
- 3.χ² = Σ(O-E)²/E
- 4.Degrees of freedom = (rows-1)(columns-1)