A limitation of a chi-square test is that it does not reveal whether, or which, specific value of either variable is driving the result. Looking at the descriptives could give you some insight about where differences are greater, but can't be used to draw any conclusions about what drives the p-value/difference. If you have a strong (theoretical) reason to hypothesize that specific values/groups are important then other tests could be used. For instance, one may not really be interested in counties, but just the difference between traditional public schools and charter schools. In that case, one could create some new variables that group all public schools together and all the charter schools together and then calculate the proportion of schools from each group that are high, medium, and low poverty, which would allow one to run 3 t-tests (using charter/tps as the independent variable in all three and then using proportion low poverty, proportion, medium poverty, and proportion high poverty as the dependent variables) to investigate whether charter/tps schools are more/less/equally likely to be low, medium, or high poverty.
How do you determine where the difference is coming from? I mean what is the post hoc for this?
A limitation of a chi-square test is that it does not reveal whether, or which, specific value of either variable is driving the result. Looking at the descriptives could give you some insight about where differences are greater, but can't be used to draw any conclusions about what drives the p-value/difference. If you have a strong (theoretical) reason to hypothesize that specific values/groups are important then other tests could be used. For instance, one may not really be interested in counties, but just the difference between traditional public schools and charter schools. In that case, one could create some new variables that group all public schools together and all the charter schools together and then calculate the proportion of schools from each group that are high, medium, and low poverty, which would allow one to run 3 t-tests (using charter/tps as the independent variable in all three and then using proportion low poverty, proportion, medium poverty, and proportion high poverty as the dependent variables) to investigate whether charter/tps schools are more/less/equally likely to be low, medium, or high poverty.