It is essential to distinguish the Pearson chi-square statistic from the chi-square distribution. A variable has the chi-square distribution if a) it is the square of a variable that is normally distributed, b) it is the suitably normalized sum of m variables each of which has the same exponential distribution.
The permutation distribution of the Pearson chi-square statistic is often used for two-tailed tests of significance for 2x2 contingency tables and for both one- and two-tailed tests for unordered RxC tables. As the sample size increases the permutation distribution of the Pearson chi-square statistic computed for a balanced contingency table converges to a chi-square distribution. For moderate sample sizes or for sparse or unbalanced tables the chi-square distribution is not an appropriate approximation.
Jaques Cuez