1. What is Cross-Tabulation?
Cross-tabulation (also called contingency table analysis or crosstab) is a statistical technique used to analyze the relationship between two or more categorical variables. It displays data in a matrix format showing the frequency distribution of variables. For a deeper reference with worked examples and research methodology context, see the Cross-Tabulation documentation.
Example: Survey Data
A survey asks respondents about their voting preference and age group:
| Candidate A | Candidate B | Total |
|---|
| 18-35 | 45 | 55 | 100 |
| 36-55 | 60 | 40 | 100 |
| 56+ | 70 | 30 | 100 |
| Total | 175 | 125 | 300 |
2. When to Use Cross-Tabulation
Cross-tabulation is appropriate when:
- Both variables are categorical (nominal or ordinal)
- You want to test if two variables are independent
- You need to explore relationships in survey data
- You're conducting market research or social science studies
3. Creating a Contingency Table
Step-by-Step Process
- Identify variables: Choose your row variable (usually independent) and column variable (usually dependent)
- Count frequencies: Tally observations in each cell
- Calculate marginals: Sum rows and columns
- Compute percentages: Row %, column %, or total %
Tip: Row percentages are useful when comparing groups. Column percentages show the composition of each category.
4. Chi-Square Test of Independence
The chi-square test determines whether there's a statistically significant association between variables. For the full mathematical derivation, see the Chi-Square Test documentation.
How It Works
- Calculate expected frequencies: E = (row total × column total) / grand total
- Compare observed to expected: χ² = Σ(O - E)² / E
- Find degrees of freedom: df = (rows - 1) × (columns - 1)
- Determine p-value from chi-square distribution
Interpreting Results
- p < 0.05: Significant association exists (reject null hypothesis)
- p ≥ 0.05: No significant association (fail to reject null hypothesis)
Warning: Chi-square requires expected frequencies ≥ 5 in each cell. For small samples, use Fisher's exact test instead.
5. Effect Sizes
P-values tell you if an effect exists, but effect sizes tell you how large it is. For a full overview including bias-corrected measures, see the Effect Sizes documentation.
Common Effect Size Measures
| Measure | Range | Best For |
|---|
| Cramér's V | 0 to 1 | Any table size |
| Phi (φ) | -1 to 1 | 2×2 tables |
| Odds Ratio | 0 to ∞ | 2×2 tables |
| Gamma | -1 to 1 | Ordinal data |
Interpreting Cramér's V
- 0.10: Small effect
- 0.30: Medium effect
- 0.50: Large effect
6. Assumptions and Limitations
For a comprehensive treatment of each assumption and what to do when it's violated, see the Assumptions guide.
Chi-Square Assumptions
- Random sampling
- Independent observations
- Categorical data
- Expected frequency ≥ 5 in each cell (or use Fisher's exact)
Common Issues
- Small expected values: Use Fisher's exact test or combine categories
- Large samples: Everything becomes "significant" - focus on effect sizes
- Multiple testing: Adjust p-values for multiple comparisons
7. Reporting Results
APA Style Example
"A chi-square test of independence was performed to examine the relation between age group and voting preference. The relation between these variables was significant, χ²(2, N = 300) = 15.43, p < .001. Older respondents were more likely to prefer Candidate A (Cramér's V = .23)."
What to Include
- Chi-square value and degrees of freedom
- Sample size
- P-value
- Effect size (Cramér's V or similar)
- Direction of the relationship
Ready to Analyze Your Data?
CrossTabs.com makes cross-tabulation analysis simple and fast:
- Upload CSV or Excel files
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