# How to interpret results of a $\chi^2$ test of independence?

I've run a chi-square test of independence on a two-way table with two categories on the rows and four categories on the columns. My $p$-value is below my chosen $\alpha$, so I may conclude that the rows and columns of this two-way table are dependent. I'm having trouble characterizing the dependence among the rows and columns though. How can I make sense of the residuals and what can I say about the probability of being in one of the rows given that I'm in a specific column?

One rule of thumb is to then look at the contribution of each cell to the overall chi-squared statistic ( $\frac{(O-E)^2}{E}$ ), cells with a value greater than 4 are the ones most likely to be significantly different from others.