Population-Weighted Density, Density-Weighted Population, Granularity,   Paradoxes: a Recapitulation
  Quantifying the population density of an urban area is a fraught issue. Measures of density are often defined differently from place to place or applied inconsistently, and arguments abound over just how much of the land surrounding a city should and should not be classified as `urban'. The prime candidates for a consistent density measure are overall density OD (also known as average density) and population-weighted density PWD (as recently adopted by the US Census Bureau). In this note some less intuitive aspects of PWD are explored, so that the consequences of adopting PWD as a density measure are better understood relative to OD. It will also be seen that one cannot entirely dispense with the need to define urban boundaries, to work preferentially with the smallest parcels of land for which one has data, and to pay careful attention to the delineation of boundaries to ensure high-density and low-density developments are allocated to separate parcels where possible. 
Anthony B. Morton
math.HO