Calculator Input
Create a contingency table, label categories, and calculate association strength between two categorical variables.
Example Data Table
This sample compares purchase channel by region. It is included in the example loader above.
| Purchase Channel | North | Central | South |
|---|---|---|---|
| Store | 42 | 35 | 28 |
| Website | 18 | 27 | 33 |
| App | 14 | 21 | 39 |
Formula Used
Chi-square statistic: χ² = Σ (O − E)² / E, where O is the observed count and E is the expected count.
Expected frequency: E = (row total × column total) / grand total.
Cramér's V: V = √[ χ² / (N × min(r − 1, c − 1)) ], where N is total observations, r is row count, and c is column count.
Bias-corrected V: This version adjusts φ² and the effective table dimensions to reduce upward bias in smaller samples.
Residuals: Standardized residual = (O − E) / √E. Large absolute residuals highlight cells contributing most to χ².
How to Use This Calculator
- Choose the number of rows and columns for your contingency table.
- Rename each row and column to match your categories.
- Enter observed frequencies into every table cell.
- Set the significance level you want to test.
- Optionally enable bias correction or Yates correction.
- Press the calculate button to view V, χ², p-value, and diagnostics.
- Review expected counts and residuals to understand where the association is strongest.
- Use the CSV or PDF buttons to export the calculation report.
FAQs
1. What does Cramér's V measure?
Cramér's V measures the strength of association between two categorical variables. It ranges from 0 to 1, where 0 means no association and values closer to 1 indicate stronger association.
2. When should I use Cramér's V?
Use it after building a contingency table for two categorical variables, especially when you want an effect size alongside the chi-square test. It works well for surveys, audits, classification results, and demographic breakdowns.
3. Is Cramér's V the same as the chi-square test?
No. Chi-square tests whether an association exists, while Cramér's V measures how strong that association is. They complement each other and are often reported together.
4. Why does the calculator show expected counts?
Expected counts help you check whether chi-square assumptions are reasonable. Very small expected counts can weaken the p-value approximation, so the calculator flags those situations for caution.
5. What is bias-corrected Cramér's V?
Bias correction adjusts the standard estimate upward tendency in smaller samples or larger tables. It often produces a slightly smaller, more conservative effect-size estimate.
6. Why are standardized residuals useful?
Residuals show which cells differ most from independence. Large positive residuals mean observed counts exceed expectation, while large negative residuals mean observed counts fall below expectation.
7. Should I use Yates correction?
Yates correction is usually considered only for 2×2 tables and smaller samples. It makes the chi-square test more conservative, but it is not used for larger tables.
8. Can I export the results for reports?
Yes. The calculator includes CSV export for spreadsheet work and PDF export for clean reporting. Both outputs include the summary metrics and table values.