What is the maximum a Pearson's chi square statistic can get?

What is the maximum a Pearson’s chi square statistic can get?

I actually know that the answer is $N(k-1)$ (where $k$ is the minimum between number of rows and number of columns).

However, I can not seem to find a simple proof for why the statistic is bounded by this. Any suggestions (or references?)

flag	asked Jul 31 2011 at 18:11 Tal Galili 21●2

Tal, can you state the problem in more details and add some motivation? – Gil Kalai Jul 31 2011 at 18:35

Sure. The motivation is to understand Cramer's V statistic: en.wikipedia.org/wiki/Cram%C3%A9r's_V – Tal Galili Jul 31 2011 at 19:39