Chi-Square Test of Independence
Decision-First Design
Test whether two categorical variables are associated using a contingency table. Get instant decision guidance with traffic light indicators, effect sizes, and business-ready interpretations.
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MARKETING SCENARIOS
💼 Real Marketing Chi-Square Tests
Select a preset scenario to explore real-world independence tests with authentic marketing data and business context.
INPUTS & SETTINGS
Select Data Entry / Upload Mode
Observed Counts
Shows a small label under each input.
Upload a contingency table
Provide a CSV/TSV where the first row lists column headers, the first column lists row labels, and every other cell contains observed counts.
Drag & Drop raw data file (.csv, .tsv, .txt)
Contingency table format: first row = column headers, first column = row labels, remaining cells = counts.
No contingency table uploaded yet.
Upload raw data
Upload row-level observations with exactly two columns (category for variable A, category for variable B). We will aggregate them into a contingency table.
Drag & Drop raw data file (.csv, .tsv, .txt)
Two categorical columns with headers (Variable A, Variable B); up to 2,000 rows.
No raw file uploaded yet.
Confidence Level & Advanced Settings
Advanced settings
Options for special cases and small samples.
Use for small-sample 2×2 tables to reduce bias.
YOUR DECISION
Enter data to see your result
We'll calculate whether the variables are independent or associated
VISUAL OUTPUT
Stacked 100% Bar Chart
Visualization Settings
TEST RESULTS
Chi-square (χ²): –
Degrees of freedom: –
p-value (upper tail): –
Effect size (Cramér's V): –
Decision (α): –
Interpretation: –
APA-Style Statistical Reporting
Managerial Interpretation
LEARNING RESOURCES
📚 When to use the Chi-Square Test
Use the Chi-Square test of independence when:
- You have two categorical variables (e.g., Segment × Response, Channel × Outcome)
- You want to test if the variables are independent or associated
- Your data is organized in a contingency table with observed counts
- Each observation belongs to exactly one category for each variable
- Expected cell counts are generally ≥ 5 (for reliable approximation)
Why Chi-Square instead of other tests?
Chi-square is designed for categorical × categorical data. Use t-tests or ANOVA when comparing means of continuous variables across groups. Use regression when predicting a continuous outcome.
⚠️ Common mistakes to avoid
- Using percentages instead of counts: Enter raw observed counts, not percentages. The test calculates expected proportions internally.
- Ignoring small expected counts: When expected counts < 5, the chi-square approximation may be unreliable. Consider combining categories or using Fisher's exact test.
- Confusing association with causation: A significant chi-square shows the variables are related, not that one causes the other.
- Ignoring effect size: A significant p-value with tiny effect size (V < 0.1) may not be practically meaningful—especially with large samples.
- Testing the same subjects multiple times: Each row should represent an independent observation. Don't include the same customer multiple times.
📊 Interpreting Cramér's V (Effect Size)
Cramér's V measures the strength of association between variables:
- V < 0.10: Negligible association
- V = 0.10–0.20: Small association
- V = 0.20–0.40: Medium association
- V ≥ 0.40: Large/strong association
In practice: A Chi-square test might be statistically significant (p < 0.05) but show weak association (V = 0.08). For business decisions, prioritize effect size alongside significance.
🔬 The Chi-Square Statistic Explained
The chi-square statistic measures how much observed counts deviate from expected counts:
χ² = Σ (O − E)² / E
Where:
- O = Observed count in each cell
- E = Expected count if variables were independent
- Σ = Sum over all cells
Large χ² indicates the observed pattern differs substantially from what we'd expect under independence → reject H₀.
DIAGNOSTICS & ASSUMPTIONS
Diagnostics & Assumption Tests
Enter observed counts to check sample size, expected counts, and leverage diagnostics.
Expected Counts
What are expected counts?
Under the null hypothesis (independence), the expected count for each cell is what we would anticipate from the row and column totals alone. These expected values are used in the chi-square statistic by comparing observed and expected counts and summing the squared differences scaled by the expected value: χ² = Σi,j (Oij − Eij)²Eij. When many expected counts are small (e.g., < 5), results should be interpreted with caution.