Chi-Square Homogeneity Calculator Online

The Chi-Square Homogeneity Test examines if there is a significant difference in the distribution of a categorical variable across different samples. This calculator aids researchers and statisticians by automating the computations, making the analysis more accessible and less prone to error.

Formula of Chi-Square Homogeneity Calculator

The formula for the Chi-Square Test for Homogeneity is:

Where:

• chi-square is the test statistic,
• O_ij represents observed frequencies,
• E_ij denotes expected frequencies calculated based on marginal totals.

This formula helps in determining the presence of discrepancies between observed and expected frequencies, indicating potential differences across groups.

Steps to Calculate Chi-Square Test for Homogeneity

To use the Chi-Square Homogeneity Calculator effectively, follow these steps:

1. Collect Data: Gather observed frequencies for each category in each sample.
2. Calculate Expected Frequencies: Use the formula:
   • E_ij = (Row_Total_i * Column_Total_j) / Grand_Total
   • Here, Row_Total_i and Column_Total_j are the totals of respective rows and columns, while Grand_Total is the sum of all observed frequencies.
3. Compute Chi-Square Statistic: Apply the earlier mentioned formula to find the chi-square statistic.
4. Determine Degrees of Freedom: Calculate as:
   • df = (r - 1) * (c - 1)
   • With r as the number of rows and c as the number of columns.
5. Compare with Critical Value: Check if the statistic exceeds the critical value from chi-square distribution tables at your chosen significance level.
6. Conclusion: If the statistic is higher than the critical value, reject the null hypothesis, suggesting a difference in distributions.

Example of Chi-Square Homogeneity Calculator

Survey to determine volunteer activity preferences in two cities, City A and City B.
Activities: Environmental, Educational, Health Support.

Observed Frequencies:
City A: Environmental (100), Educational (150), Health Support (50)
City B: Environmental (120), Educational (120), Health Support (60)

Steps:

1. Calculate Totals:
   Environmental: 220 (100 from City A + 120 from City B)
   Educational: 270 (150 from City A + 120 from City B)
   Health Support: 110 (50 from City A + 60 from City B)
   Total for City A: 300
   Total for City B: 300
   Grand Total: 600
2. Calculate Expected Frequencies for City A:
   Environmental: (220 * 300) / 600 = 110
   Educational: (270 * 300) / 600 = 135
   Health Support: (110 * 300) / 600 = 55
3. Calculate Chi-Square Statistic:
   Formula: Chi-Square = sum((Observed - Expected)^2 / Expected)
4. Degrees of Freedom:
   Formula: (Rows - 1) * (Columns - 1) = (2 - 1) * (3 - 1) = 2
5. Critical Value:
   For 2 degrees of freedom at a 0.05 significance level, the critical value is 5.99.
6. Decision:
   If the Chi-Square statistic is greater than 5.99, conclude there is a significant difference in preferences.

Useful Table for Quick Reference

To assist users in understanding the critical values needed for decision-making in the Chi-Square Test, here is a quick reference table showing critical chi-square values at common significance levels for different degrees of freedom:

This table helps users quickly determine the critical values necessary to interpret their chi-square statistic based on the degrees of freedom and desired level of significance.

Most Common FAQs