What, exactly, designates city largeness?

When one talks about the largest cities, one is usually seeking cities with the largest population (and occasionally, some actually are referencing area). As with everything in statistics, though, the picture isn’t that simple. In looking at the largest cities by population, one is usually attempting to find the largest pools of people, but sometimes parts of cities break away and become independent, and sometimes small adjacent incorporated areas are assimilated. Perhaps one is looking for the size (by population) of the metropolitan area. What, exactly, designates a metropolitan area, though? Where, for example, do you draw the line between New York and Philadelphia (with possibly even cities in between, like Trenton)?

Let’s start simply by listing the twelve largest cities in the United States. They are New York, Los Angeles, Chicago, Houston, Philadelphia, Phoenix, San Antonio, San Diego, Dallas, San Jose, Jacksonville, and Indianapolis, in descending order of population according to the 2010 census. This hardly tells the entire story, as some of these cities are large truly because of their concentration of population and some are large for other reasons. Take a look at the following chart (charts made using Microsoft Excel and Microsoft Paint), consisting of the plotting of the area and population of 137 selected cities in the United States (area is measured in square miles):

From this, one could see that using linear scales is not a good idea, as is usually the case with measuring direct quantities of size; hence, the following:

From this, we can conclude that New York is quite indubitably the largest city, largest by both population and population density (unless you’re one of those that measure city largeness by area). One can argue though, for example, that had Chicago had a comparable size to Los Angeles, the former would have had more population than the latter, which like Houston, San Antonio, and Jacksonville, owe a significant amount of the cause of their population size to the fact that in history they annexed many communities around them. The cities on the upper left of the mass of points are the crowded cities, those that have large amounts of people for the area they cover. San Francisco, for example, has a five-digit people/square mile population density. Some of you may know that many cities surround San Francisco (in fact, one directly to the south, Daly City, is labeled on the chart, also a city of high population density). The reason San Francisco is only 13th in the United States by population is really because the city proper encompasses a very small amount of land at the tip of a peninsula. Miami, Santa Ana, Jersey City, Paterson, and Cambridge also are highly populated cities that are small in land area covered. Oklahoma City, Huntsville, Norman, and Suffolk (VA), on the other hand, cover a large amount of area (in fact, two of them cover more area than New York) but have comparatively low populations (drawing a horizontal line, one can see that Oklahoma city has about the same population as Washington, DC, but it has more than two powers of e times greater area). And if you really want land, land, everywhere, move to Juneau (or more practically, move to an unincorporated area, even though in the outer parts of Juneau one might not be able to tell a difference). Or if you don’t want either extreme, move someone in the middle, like Worcester, Tampa, Syracuse (NY), or the Californian cities of Irvine, Riverside, and Fremont. Keep in mind, though, that even this shouldn’t be taken for face value, as clearly within city boundaries the concentration of people is not homogenous, and downtowns will still have greater concentrations of populations than border areas.

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