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-squareis the test statistic,O_ijrepresents observed frequencies,E_ijdenotes 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:
- Collect Data: Gather observed frequencies for each category in each sample.
- Calculate Expected Frequencies: Use the formula:
- E_ij = (Row_Total_i * Column_Total_j) / Grand_Total
- Here,
Row_Total_iandColumn_Total_jare the totals of respective rows and columns, whileGrand_Totalis the sum of all observed frequencies.
- Compute Chi-Square Statistic: Apply the earlier mentioned formula to find the chi-square statistic.
- Determine Degrees of Freedom: Calculate as:
- df = (r - 1) * (c - 1)
- With
ras the number of rows andcas the number of columns.
- Compare with Critical Value: Check if the statistic exceeds the critical value from chi-square distribution tables at your chosen significance level.
- 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:
- 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 - Calculate Expected Frequencies for City A:
Environmental: (220 * 300) / 600 = 110
Educational: (270 * 300) / 600 = 135
Health Support: (110 * 300) / 600 = 55 - Calculate Chi-Square Statistic:
Formula: Chi-Square = sum((Observed - Expected)^2 / Expected) - Degrees of Freedom:
Formula: (Rows - 1) * (Columns - 1) = (2 - 1) * (3 - 1) = 2 - Critical Value:
For 2 degrees of freedom at a 0.05 significance level, the critical value is 5.99. - 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:
| Degrees of Freedom | Critical Value (0.05 level) | Critical Value (0.01 level) |
|---|---|---|
| 1 | 3.84 | 6.63 |
| 2 | 5.99 | 9.21 |
| 3 | 7.82 | 11.34 |
| 4 | 9.49 | 13.28 |
| 5 | 11.07 | 15.09 |
| 6 | 12.59 | 16.81 |
| 7 | 14.07 | 18.48 |
| 8 | 15.51 | 20.09 |
| 9 | 16.92 | 21.67 |
| 10 | 18.31 | 23.21 |
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
It's the threshold probability at which you reject the null hypothesis, commonly set at 0.05 or 5%.
Yes, it is particularly effective for large samples, providing more reliable results as sample size increases.
If it's lower, you do not reject the null hypothesis, suggesting no significant difference in distributions across the samples.