Contingency Coefficient Calculator

The Contingency Coefficient is derived directly from the chi-square statistic:

$$C = \sqrt{\frac{\chi^2}{\chi^2 + n}}$$

Where χ² is the Pearson chi-square test statistic and n is the total sample size. This formula ensures that C is always non-negative and increases with χ², approaching but never reaching 1.

The maximum possible value of C depends on the number of categories k (assuming a square k×k table or using the minimum dimension):

$$C_{max} = \sqrt{\frac{k - 1}{k}}$$

For a 2×2 table, Cₘₐₓ = √(1/2) ≈ 0.707. For a 3×3 table, Cₘₐₓ = √(2/3) ≈ 0.816. For a 5×5 table, Cₘₐₓ = √(4/5) ≈ 0.894. This asymptotic behavior means C can never equal 1 regardless of association strength.

The corrected contingency coefficient normalizes C to the full 0–1 range:

$$C_{corrected} = \frac{C}{C_{max}}$$

This corrected version allows meaningful comparison across tables with different numbers of categories and is directly interpretable as a proportion of the maximum possible association.

The raw contingency coefficient C should be interpreted relative to its maximum Cₘₐₓ. The corrected C (C/Cₘₐₓ) provides a standardized measure:

• Corrected C < 0.10: Negligible association
• 0.10–0.30: Weak association
• 0.30–0.50: Moderate association
• ≥ 0.50: Strong association

Note that C is always non-negative and cannot indicate the direction of association. For directional measures with ordinal variables, consider Gamma or Somers' D. The corrected coefficient is particularly useful when comparing results from studies using different numbers of categories.