How to Use the Chi-Square Calculator

Choose the appropriate test type based on your research question. Enter your observed frequencies and the calculator will compute the chi-square statistic and p-value.

Types of Chi-Square Tests

Goodness of Fit Test

Tests if observed frequencies match expected frequencies:

H₀: Observed matches expected distribution
H₁: Observed differs from expected distribution
Example: Testing if a die is fair (equal frequencies)
df: k - 1 (number of categories minus 1)

Test of Independence

Tests if two categorical variables are independent:

H₀: Variables are independent (no association)
H₁: Variables are not independent (associated)
Example: Is smoking related to lung disease?
df: (r - 1) × (c - 1)

The Chi-Square Formula

χ² = Σ [(O - E)² / E]

O: Observed frequency
E: Expected frequency
Sum over all categories or cells

Effect Size: Cramer's V

Measures the strength of association between categorical variables:

0.0 - 0.1: Negligible association
0.1 - 0.3: Weak association
0.3 - 0.5: Moderate association
0.5+: Strong association

Assumptions

Random sampling: Data from random sample
Independence: Observations are independent
Expected frequency: Each cell should have E ≥ 5 (rule of thumb)
Categorical data: Variables must be categorical

Interpreting Results

p < α: Reject H₀ (significant result)
p ≥ α: Fail to reject H₀ (not significant)
Large χ²: Greater deviation from expected
Check contributions: Identify which categories differ most

Common Applications

Market research (preference studies)
Medical research (disease associations)
Genetics (Mendelian ratios)
Quality control (defect analysis)
Survey analysis (response patterns)
Social sciences (demographic associations)

When Not to Use Chi-Square

Continuous data (use t-test or ANOVA)
Small expected frequencies (< 5)
Non-independent observations
When exact probabilities are needed (use Fisher's exact test)