Standard population density is a useless statistic for metropolitan areas
Standard population density -- total population divided by total land area -- is a useless statistic when applied to metropolitan statistical areas (MSAs).
For starters, it does not tell you anything you really want to know. The population density of a metropolitan area is typical of no particular place within the metropolitan area. Most of any MSA is vacant land because MSAs are made up of counties, not cities; they often include not just rural areas, but entire rural counties. Most of the population is concentrated in a relatively small portion of the land at a relatively high density. But all of that vacant land is added to the denominator, which causes the average density to be much lower than the density of the built-up urban areas -- the density at which most people live -- and much higher than the density of most of the land, which is sparsely settled.
That makes it an uninteresting statistic. It's like averaging the temperature outside on a sweltering summer day with the temperature inside a nicely air-conditioned house. You get a number, but it's not a number that tells you much about either the outdoor or indoor temperature.
Likewise, we might be interested in the density of the built-up, settled area within a metropolitan area, and we might be interested in the percentage of the metropolitan area that has been developed, but combining the two statistics -- which is what standard population density essentially does -- erases the useful information.
A second problem with the statistic is that it does not permit reliable comparisons between cities. Standard population density is too sensitive to how boundaries are drawn. Consider San Antonio and Austin. The San Antonio MSA had a population in 2010 of 2,142,508 spread over a land area of 7,313 square miles for a standard density of 293 ppsm. The Austin MSA, by contrast, had a 2010 population of 1,716,289 spread over 4,220 square miles for a standard density of 406 ppsm.
So the Austin MSA appears to be much denser than San Antonio. But in what sense is that true? If we look merely at the urbanized areas (the contiguous, built-up urban core -- what you would circle as the "city" in a nighttime satellite photo), San Antonio's urbanized area is a bit denser than Austin's, using either weighted or standard density. San Antonio's urbanized area contains a larger share of the metropolitan population (82%) than Austin's (79%), so the San Antonio MSA is also more centralized than the Austin MSA.
I submit that there's no meaningful sense in which the Austin metropolitan area is denser than the San Antonio metropolitan area. It's just that the San Antonio MSA was assigned eight counties while the Austin MSA was assigned five, as a result of which San Antonio has more rural counties. The San Antonio MSA includes sparsely-populated Bandera County (standard density: 26 ppsm), which contains less than 1% of the MSA population. If Bandera County were excluded from San Antonio's MSA, its standard density would jump by more than 10%, from 293 ppsm to 325 ppsm. If rural Atascosa and Medina Counties were also excluded, then the San Antonio MSA's standard density would jump to 511 ppsm, much higher than Austin's.
Atascosa, Bandera and Medina Counties together account for only 5% of the San Antonio MSA population. But they determine whether the San Antonio MSA is 28% less dense than the Austin MSA or 26% more dense. Any statistic that fluctuates so wildly depending on whether such a small and insignificant share of the population is included is not a reliable measure of anything.
Finally, standard population density is practically useless for measuring change within a MSA over time. The reason, again, is that the boundaries are based on county (i.e., political) boundaries, which are static over time. If a metropolitan area is not assigned new counties between one census and the next, then the percentage change in standard density is exactly equal to the percentage change in population (because the denominator -- land area -- does not change). On the other hand, if it is assigned a new county, the MSA inevitably sees a dramatic increase in land area and a plunge in density . . . even if every single county in the MSA adds population (and thus density).
I see even serious urban economists like Ed Glaeser using metropolitan area densities from time to time in regressions. I wish they'd stop. Better to use weighted density or stick to urbanized areas.