Some time ago I did two posts on population-weighted density, which is the mean of the densities of small areas such as census tracts weighted by their populations. In the first post, About population-weighted density, I described this alternative to traditional, conventional density and discussed some of the issues surrounding the use of this measure. The second post was Population-weighted density and urban sprawl, which argued that both conventional density and population-weighted density were appropriate measures of the extent of urban sprawl, relevant to different consequences associated with sprawl.
Doing these posts got me more interested in population-weighted density and led to my writing a full paper exploring this alternative density measure. In addition to expanding on the topics addressed in those two earlier posts, the paper provides much new information. Two of the highlights are the demonstration of the relationship of population-weighted density to conventional density and the comparison of conventional densities and population-weighted (actually housing-unit-weighted) densities across the 59 large urban areas in my urban patterns research.
In the first post I said that to the extent that people are more heavily concentrated in some tracts with higher densities than in others, those higher density tracts will be given more weight in calculating the population-weighted average. This will cause the population-weighted density to be greater than the simple or conventional density. In the paper, I move beyond this qualitative statement to deriving the mathematical relationship between population-weighted density and conventional density. It is actually quite simple: population-weighted density is equal to conventional density plus the variance in density across the subareas divided by conventional density. To the best of my knowledge, this is the first time this relationship has been demonstrated in this manner.
Looking at conventional and housing-unit-weighted densities for the large urban areas, the distribution of the weighted densities is more highly skewed towards higher values, with New York being an extreme outlier on weighted density. The housing-unit-weighted densities were more strongly related to the size of the urban areas, including size in earlier years, suggesting that the presence of areas of concentrated high densities was established in some urban areas decades ago.
Much more information on can be found in my paper, “On Population-Weighted Density” which can be downloaded here.