Chi-Square Test Calculator

Calculate chi-square statistics for goodness of fit and test of independence with p-values, critical values, and step-by-step analysis.

Critical Values Reference

Chi-square critical values for common significance levels:

Understanding the Chi-Square Test

What is the Chi-Square Test?

The chi-square test is a statistical method used to determine if there is a significant association between categorical variables or if observed frequencies differ from expected frequencies. The test calculates a statistic that measures the discrepancy between what we observe and what we expect under the null hypothesis.

Chi-Square Formula

The chi-square statistic is calculated as:

Where O is the observed frequency and E is the expected frequency for each category or cell.

Types of Chi-Square Tests

1. Goodness of Fit Test

Tests whether observed sample data fits a hypothesized distribution. Used when you have one categorical variable.

• Null hypothesis: The data follows the expected distribution
• Degrees of freedom: k - 1 (where k is the number of categories)
• Example: Testing if a die is fair by comparing observed rolls to expected uniform distribution

2. Test of Independence

Tests whether two categorical variables are independent or related. Used when you have two categorical variables in a contingency table.

• Null hypothesis: The two variables are independent
• Degrees of freedom: (rows - 1) × (columns - 1)
• Example: Testing if gender and voting preference are independent
• Expected frequency for each cell: (row total × column total) / grand total

Interpreting Results

P-Value Interpretation

• p < 0.05: Strong evidence against null hypothesis (typically significant)
• p < 0.01: Very strong evidence against null hypothesis
• p < 0.001: Extremely strong evidence against null hypothesis
• p >= 0.05: Insufficient evidence to reject null hypothesis

Cramer's V Effect Size

For test of independence, Cramer's V measures the strength of association:

• 0.00 - 0.10: Negligible association
• 0.10 - 0.30: Weak association
• 0.30 - 0.50: Moderate association
• 0.50+: Strong association

Assumptions and Requirements

1. Independence: Observations must be independent of each other
2. Sample Size: Expected frequency in each cell should be at least 5
3. Random Sampling: Data should be collected through random sampling
4. Categorical Data: Variables must be categorical (not continuous)

Common Applications

• Market research: Testing if product preference differs by demographic group
• Medical studies: Testing if treatment outcome is independent of patient characteristics
• Quality control: Testing if defect rates are consistent across production batches
• Genetics: Testing if observed offspring ratios match Mendelian expectations
• Social sciences: Testing relationships between categorical variables like education level and political affiliation

Example Calculation

Scenario: Testing if a die is fair using 60 rolls.

Observed: 1→8, 2→12, 3→9, 4→11, 5→10, 6→10

Expected: Each face should appear 10 times (60/6)

Calculation:

• χ² = (8-10)²/10 + (12-10)²/10 + (9-10)²/10 + (11-10)²/10 + (10-10)²/10 + (10-10)²/10
• χ² = 0.4 + 0.4 + 0.1 + 0.1 + 0 + 0 = 1.0
• df = 6 - 1 = 5
• Critical value at α=0.05: 11.070
• Since 1.0 < 11.070, fail to reject H₀ - the die appears fair