Discover whether two categorical variables are associated. Build contingency tables, compute expected frequencies, decompose the χ² statistic cell by cell, and quantify association strength.
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| Success | Failure | Row Total | |
|---|---|---|---|
| Treatment | 60 | ||
| Control | 60 | ||
| Col Total | 75 | 45 | 120 |
Observed
| Success | Failure | |
|---|---|---|
| Treatment | 45 | 15 |
| Control | 30 | 30 |
Expected E = (row × col) / n
| Success | Failure | |
|---|---|---|
| Treatment | 37.5 | 22.5 |
| Control | 37.5 | 22.5 |
| Cell | O | E | (O−E)² | (O−E)²/E |
|---|---|---|---|---|
| Treatment / Success | 45 | 37.50 | 56.25 | 1.5000 |
| Treatment / Failure | 15 | 22.50 | 56.25 | 2.5000 |
| Control / Success | 30 | 37.50 | 56.25 | 1.5000 |
| Control / Failure | 30 | 22.50 | 56.25 | 2.5000 |
| χ² = | 8.0000 | |||
Contribution per Cell
Chi-Square Statistic
8.000
Degrees of Freedom
1(2-1)×(2-1)
p-value
0.005
Effect size quantifies practical significance beyond the p-value. Phi, Odds Ratio, and Risk Ratio are available for 2×2 tables.
Cramér's V
0.258
Moderate association
√(χ² / (n × min(r-1, c-1)))
Phi (φ)
0.258
√(χ² / n), equivalent to Pearson's r for 2×2 tables
Odds Ratio
3.000
(a × d) / (b × c)
Row 1 has higher odds of the first outcome
Risk Ratio (RR)
1.500
(a / row1Total) / (c / row2Total)
Row 1 has higher relative risk
Effect Size Reference (Cramér's V)