Effect size measure
Contingency Coefficient Calculator — Pearson's C
Pearson's contingency coefficient (C)measures the strength of association between two categorical variables, derived from chi-square; it ranges from 0 (no association) upward but cannot quite reach 1, and its maximum depends on the table's size.
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Open the workspace →What is the contingency coefficient?
Pearson's contingency coefficient is one of the oldest effect-size measures for categorical data, introduced by Karl Pearson in 1904. It rescales the chi-square statistic so that it does not grow without bound as your sample gets larger, turning a raw χ² into a number between 0 and (almost) 1.
Because it is built from the chi-square statistic, C pairs naturally with a chi-square test: the test tells you whether two variables are associated, and C tells you how strongly.
Formula
Definition
C = √( χ² / (χ² + n) )
- χ²
- = the chi-square statistic for the table
- n
- = the grand total (sample size)
Worked example
Worked example
Suppose 110 people are cross-tabulated into a 2×2 table:
[[30, 10], [10, 60]] → n = 110
The chi-square statistic for this table is χ² = 40.55.
C = √( 40.55 / (40.55 + 110) ) = √( 40.55 / 150.55 ) = √0.2694 = 0.519 (≈ 0.52).
A C of about 0.52 indicates a moderately strong association — but remember its ceiling here is only √(1/2) ≈ 0.707, not 1.
When to use it
Use it when
- You want a quick chi-square-based measure of association.
- You understand its size-dependent ceiling and are reporting a single table.
Not the right tool when
- You are comparing association across tables of different sizes — use Cramér's V instead.
- You need a measure that can actually reach 1 for a perfect association.
How to interpret it
Rule of thumb
Higher C means a stronger association, but because its maximum is below 1 and varies with table size, prefer Cramér's V when comparing tables.
Frequently asked questions
- What is the maximum value of the contingency coefficient?
- The contingency coefficient cannot reach 1; its maximum depends on the table size. For a 2×2 table the ceiling is √(1/2) ≈ 0.707, and larger tables have higher (but still below 1) ceilings, which is why C is hard to compare across tables of different sizes.
- Contingency coefficient vs Cramér's V?
- Both are chi-square-based effect sizes, but Cramér's V is rescaled to range from 0 to 1 regardless of table size, while Pearson's C has a size-dependent maximum below 1. For comparing association strength across tables of different dimensions, use Cramér's V.
- How do you interpret Pearson's C?
- Higher C means a stronger association, with 0 meaning no association. Because the maximum is below 1 and varies with table size, interpret C relative to its ceiling for that table, and prefer Cramér's V when you need comparability.
References & further reading
- Pearson, K. (1904). On the Theory of Contingency and Its Relation to Association and Normal Correlation.
- Contingency table — Wikipedia
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