Blogger Austin Contrarian has come up with and calculated a very useful metric he calls "weighted density." What this is is the average density of an urbanized area, weighted by the population of each census tract. The relevance of this is that this is, roughly, the population density experienced by the typical resident of the area, as opposed to standard measures of density, which measure the average number of people found in a typical piece of land within an area.
What's the relevance? In brief, it does a better job of measuring the phenomenon of population density as people experience it. The canonical example is that using standard methods of measuring density, the Los Angeles urbanized area is the most densely populated in the country. How could L.A. be more dense than New York City? The island of Manhattan is certainly far more dense than any area of comparable size in L.A., but the suburbs in NYC are much less dense than L.A.'s suburbs.
In L.A., new suburbs on the edge of development have plenty of tracts developed at densities of 8 dwelling units per acre, which is what you see a lot of in L.A.'s older, inner-ring suburbs like Boyle Heights or Echo Park. Now, it would be wrong to say that density in L.A. is completely homogeneous. You see a lot more multi-family housing closer to the center, and there are people in Chatsworth or Malibu living on fairly large lots. But overall, if you were to look at a graph of population density in L.A. vs. distance from the center, it would be a fairly shallow curve that would basically drop off to zero once you hit one of the natural barriers that enclose the urbanized area.
In New York, on the other hand, you'd see a big spike in the center that would fall off rapidly, and then trail out and get a lot less dense than L.A.'s suburbs after a certain distance. Just think of starting in Manhattan and then heading out Long Island to Brooklyn, then Queens, then Nassau and Suffolk Counties. It's like the Long Tail. Out East, if you're traveling from one city to another, you may be go through rural areas, but you're never far from some kind of town or village. Here in the West, once you get out of the city, you can pass by hundreds of miles of almost unpopulated landscape.
This is important to understanding issues related to sprawl and transportation. Mass transit needs certain densities in order to function efficiently, but you can't just look at a region's overall density to get a meaningful analysis. What matters is the distribution of that density, and weighted density is a good overall measure of that.