I agree with DuncaK. I find it too simplistic to take the population density as a measure of how urban or how rural a state is.

I think that a better measure would be to compare the population density of a state with the population density of its conties (the Census Bureau gives a very convenient xls file with all the different populations and densities). See what the variance is for each state. This will give you a measure to see if the distribution of the population is uniform (rural) or is non-uniform (urban). It looks like a fun exercise. It would be nice to complement your analysis with this information. But, unfortunately I am very busy today.

The Census Bureau also gives a nice picture:


So your idea is that this would allow us to compute the average density that the average resident experiencies, rather than the average density of the average square mile of the state? I think that's an interesting and worthwhile exercise. However, we still have plenty of states with hardly any urban centers of consequence, and plenty with hardly anything else, and lots in between, so I would expect to to see the right general kinds of relationships.
County population density works well in the Eastern half of the US where counties are small and uniform in size.

In the west counties are larger (huge in fact).  Some of those large counties house huge cities (Los Angeles) and unihabited mountainous stretches that dilute down the urban density figures.

Some counties house million + cities and yet are over half uninhabited.  You dont get this in the east.