Your use of statistics is meaningless because average density is meaningless. It suffers from the same resolution problem as the coastline paradox [1]. I'm an example of this: If you measure density by the block, I live in a density of 120,000 persons/km2, but if you measure density by the city, I live in a density of 1720 persons/km2, and if you measure density by the MSA, I live in a density of 168 persons/km2. I have gigabit, and the fact that I live in an MSA with 168 persons/km2 is in no way a refutation of the fact that density matters for infrastructure costs.

[1] http://en.wikipedia.org/wiki/Coastline_paradox