Pearson Chi-Square · Test of Independence
Contingency Table Analysis
Determine whether two categorical variables are statistically independent. Enter observed frequencies, set your significance level, and compute.
Table configuration
Critical values
Degrees of freedom = (rows − 1) × (columns − 1)
Expected frequencies
Cell contributions to χ² — top cells driving association
Effect size
APA 7th edition reporting statement
Copy-paste template
Calculation details
Chi-square distribution & critical regions
Acceptance region
Rejection region
Interpretation Report
Step-by-step analysis & plain-language conclusions
1
Hypotheses
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2
Test conditions & assumptions check
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3
Computed statistical values
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4
Statistical decision
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5
Effect size & practical significance
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6
Conclusion & reporting
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All interpretations are generated from your data. Review before use in academic submissions.
Chi-Square · Goodness of Fit
Observed vs Expected Distribution
Test whether your observed frequencies match a theoretically expected distribution. Leave expected % blank to assume equal distribution.
Configuration
Expected % — leave blank for equal distribution across all categories. If entering custom values, they must sum to 100%.
Observed vs expected frequencies
Observed
Expected
Full breakdown by category
APA 7th reporting
Copy-paste template
Interpretation Report
Goodness of fit — step-by-step analysis
1
Hypotheses
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2
Test conditions & assumptions
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3
Computed statistical values
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4
Statistical decision
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5
Category-by-category analysis
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6
Conclusion & reporting
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All interpretations are generated from your data. Review before use in academic submissions.
Reference · Statistical Theory
How Chi-Square Works
Complete theoretical foundation, assumptions, effect size interpretation, and primary references.
The chi-square statistic
First described by Karl Pearson (1900), the chi-square (χ²) test is a non-parametric statistical test used to determine whether there is a significant association between categorical variables or whether an observed frequency distribution fits an expected distribution.
χ² = Σ [ (O − E)² / E ]
where:
O = observed frequency (actual count in each cell)
E = expected frequency = (row total × col total) / grand total
df = (rows − 1) × (columns − 1) [for independence]
df = k − 1 [for goodness of fit, k = categories]
Assumptions
- Categorical data — both variables must be measured at the nominal or ordinal scale. Chi-square is not appropriate for continuous variables.
- Independence of observations — each participant or observation contributes to exactly one cell in the table. Repeated measures violate this assumption.
- Expected cell frequencies ≥ 5 — the chi-square approximation becomes unreliable when any expected cell frequency falls below 5. For 2×2 tables with any expected cell < 5, use Fisher's Exact Test instead. For larger tables, consider combining categories.
- No expected cell frequency of zero — a zero expected frequency produces division by zero in the formula and must be resolved by merging categories.
- Adequate sample size — as a general rule, the total N should be at least 5 × (number of cells). Larger tables require larger samples.
Effect size measures
- Phi (φ) — appropriate for 2×2 tables only. φ = √(χ²/N). Ranges from 0 to 1. Interpretation: small ≥ 0.10, medium ≥ 0.30, large ≥ 0.50 (Cohen, 1988).
- Cramér's V — appropriate for all contingency tables. V = √(χ²/(N × min(r−1, c−1))). Ranges from 0 to 1. Same thresholds as phi. Preferred over phi for non-2×2 tables.
- Odds Ratio (OR) — for 2×2 tables only. OR = (a×d)/(b×c). An OR of 1 indicates no association; OR > 1 indicates higher odds in the first group; OR < 1 indicates lower odds.
APA 7th edition reporting format
Template: "A chi-square test of independence was performed to examine the relationship between [Variable A] and [Variable B]. The relationship between the variables was statistically significant, χ²([df], N = [n]) = [χ² value], p = [p value], Cramér's V = [V]."
Report exact p-values (e.g., p = .032) rather than inequality statements (p < .05) unless p < .001, in which case report p < .001. Never write p = .000 — this is a rounding artefact.
Primary references
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Lawrence Erlbaum Associates.
Field, A. (2018). Discovering statistics using IBM SPSS statistics (5th ed.). SAGE Publications.
Fraenkel, J. R., Wallen, N. E., & Hyun, H. H. (2019). How to design and evaluate research in education (10th ed.). McGraw-Hill.
Pearson, K. (1900). On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. Philosophical Magazine, 50(302), 157–175.
Rea, L. M., & Parker, R. A. (2014). Designing and conducting survey research: A comprehensive guide (4th ed.). Jossey-Bass.