Calculate the contingency coefficient C from a contingency table or χ² and N, with adjusted C, Cramér’s V, and p-value for association.

Use observed counts from independent observations. Pearson χ² is uncorrected.

2–5 rows and columns. Separate counts with spaces or commas; enter 0 for empty cells.

Related Calculators

Contingency Coefficient Formula

The calculator uses Pearson's contingency coefficient and a few related measures of association for categorical data.

C = sqrt(chi² / (chi² + N))
Cmax = sqrt((k - 1) / k)
Cₐdjusted = C / Cmax
V = sqrt(chi² / (N × (k - 1)))
T = sqrt(chi² / (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 accepts 2 to 5 rows and columns of observed counts, entered one row per line. It computes expected counts as row total × column total ÷ N, then sums (observed − expected)² ÷ expected for Pearson chi-square. No Yates continuity correction is applied.

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.

Raw C is bounded below 1. Dividing by the dimension-based Cmax bound rescales the coefficient, but does not establish universal strength categories or guarantee comparability across studies. Fixed margins may further restrict the attainable maximum.

Reference Tables

The table shows how the upper bound Cmax depends on table dimensions. Interpret association measures using the study context; this calculator does not apply universal adjusted-C strength cutoffs.

Table size (r × c) k = min(r, c) Cmax
2 × 220.7071
3 × 330.8165
4 × 440.8660
5 × 550.8944
2 × 520.7071

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. The asymptotic chi-square p-value is approximately 0.000407 with 1 degree of freedom. Interpretation still requires the sampling and expected-count assumptions.

Why does C never reach 1? The denominator is chi-square + N, so the ratio is always less than 1. Its upper bound depends on the smaller table dimension; adjusted C rescales by that bound, while study context and margins still matter.

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.