Calculate the contingency coefficient C from a contingency table or χ² and N, with adjusted C, Cramér’s V, and p-value for association.
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Contingency Coefficient Formula
The calculator uses Pearson's contingency coefficient and a few related measures of association for categorical data.
C = sqrt(chi^2 / (chi^2 + N))
Cmax = sqrt((k - 1) / k)
C_adjusted = C / Cmax
V = sqrt(chi^2 / (N * (k - 1)))
T = sqrt(chi^2 / (N * sqrt((r - 1)(c - 1))))
- C: Pearson's contingency coefficient
- chi^2: chi-square statistic from the contingency table
- N: total number of observations across all cells
- r: number of rows
- c: number of columns
- k: the smaller of r and c
- Cmax: maximum value C can reach for the given table size
- V: Cramér's V
- T: Tschuprow's T
The "From contingency table" mode takes the raw observed counts, computes expected counts as row total × column total ÷ N for each cell, then sums (observed − expected)² ÷ expected to get chi-square. From there it applies the formulas above.
The "From χ² summary" mode skips the cell math. You give it chi-square, N, and the table dimensions, and it returns C, adjusted C, Cramér's V, and Tschuprow's T directly.
The adjusted C corrects a known weakness of the raw coefficient: C cannot reach 1 even with perfect association. Dividing by Cmax rescales the result onto a 0 to 1 range that is comparable across tables of different sizes.
Reference Tables
Use the first table to interpret the adjusted C value the calculator returns. Use the second to see how Cmax depends on table dimensions.
| Adjusted C | Interpretation |
|---|---|
| 0.00 – 0.09 | Negligible association |
| 0.10 – 0.29 | Small association |
| 0.30 – 0.49 | Moderate association |
| 0.50 – 1.00 | Large association |
| Table size (r × c) | k = min(r, c) | Cmax |
|---|---|---|
| 2 × 2 | 2 | 0.7071 |
| 3 × 3 | 3 | 0.8165 |
| 4 × 4 | 4 | 0.8660 |
| 5 × 5 | 5 | 0.8944 |
| 2 × 5 | 2 | 0.7071 |
Worked Example and FAQ
Example. A 2 × 2 study of 200 people gives chi-square = 12.5. Plug into the formula: C = sqrt(12.5 / (12.5 + 200)) = sqrt(12.5 / 212.5) = sqrt(0.0588) = 0.2425. With k = 2, Cmax = sqrt(1/2) = 0.7071, so adjusted C = 0.2425 / 0.7071 = 0.343. That falls in the moderate range.
Why does C never reach 1? The denominator is chi-square + N, so the ratio is always less than 1. The maximum depends on the smaller table dimension, which is why adjusted C is more useful when comparing tables of different sizes.
When should I use Cramér's V instead? Cramér's V is already scaled to 0 to 1 and is the more common choice in modern reporting. The calculator returns both so you can pick what your audience expects.
What if expected counts are below 5? The chi-square approximation gets unreliable. The calculator flags how many cells fall below that threshold. For small samples consider Fisher's exact test instead.
Does a high C prove causation? No. C measures how strongly two categorical variables are associated. It says nothing about which variable drives the other or whether a third variable is involved.
