What if cities were all the same size?
When I was a younger, I remember seeing lists of cities sorted by population and thinking it wasn't quite right because some cities are so spread out, and some cities are so dense, but that population density didn't quite capture what I was interested in either.
It dawned on me recently that I now have easy access to the data (US Census Tracts) to let me see how much it matters. So, I did. I took a circular "cookie cutter" and figured out what the population of various cities would be if you defined them all to be this standard size. But, I realized I didn't want just a single circle, I wanted nested circles where population right in the center of the city counts most, but I also wanted a big outer circle since a city in the middle of nowhere doesn't feel as big as a city surrounded by dense towns and suburbs around it.
So, my "cookie cutter" is a kernel density estimation approach, and I picked a terraced shape with population within a various radii (1 mile, 2.5 miles, 5 miles, 10 miles, 25 miles) having progressively less weight. I then used census data to find local maxima where this cookie cutter (kernel) had its maximum value. These fall over cities, though not always right in the center of them. A city with a dense suburb to the north would cause it to skew north. For example, the "normalized" center of the Boston, MA city is Central Square in Cambridge, MA. Further, it turns out, the specific cookie cutter I picked matters relatively little, since population densities around cities tend to fall off smoothly, except on cities islands in the ocean like Honolulu or San Juan.
So, how do the top 20 cities in the US by my "all cities are the same size" cookie cutter compare to the actual top 20 cities? Pretty similar, with a handful of cities rising up a lot in the ranks.
Ranks 1, 2 and 3 are the same. New York, LA and Chicago are still the top 3. LA is spread out, but not that spread out. Four cities jumped into the top 10: DC, Boston, San Francisco and Miami. All of these are tiny cities in land area with a lot of dense population both in and around them. Miami in particular is striking. It's got less than half a million people in the city proper, but with a uniform size, it's bigger than Dallas which has a population well over a million. Slipping out of the top 10 are Phoenix, San Antonio, San Diego and San Jose, but all of them stay in the top 20 except for San Antonio which is just extremely spread out compared to all these other cities. The other big climbers in the top 20 by this metric are Baltimore, Minneapolis, Atlanta, and San Juan.
So, in the end the top 20 cities if all cities were a uniform land area are (big movers up marked with a ):
New York, Los Angeles, Chicago, Philadelphia, Washington DC, Boston*, San Francisco*, Houston, Miami, Dallas, Detroit*, San Jose, Phoenix, Baltimore*, San Diego, Minneapolis*, Atlanta*, Denver, Seattle, San Juan*.
When I was a younger, I remember seeing lists of cities sorted by population and thinking it wasn't quite right because some cities are so spread out, and some cities are so dense, but that population density didn't quite capture what I was interested in either.
It dawned on me recently that I now have easy access to the data (US Census Tracts) to let me see how much it matters. So, I did. I took a circular "cookie cutter" and figured out what the population of various cities would be if you defined them all to be this standard size. But, I realized I didn't want just a single circle, I wanted nested circles where population right in the center of the city counts most, but I also wanted a big outer circle since a city in the middle of nowhere doesn't feel as big as a city surrounded by dense towns and suburbs around it.
So, my "cookie cutter" is a kernel density estimation approach, and I picked a terraced shape with population within a various radii (1 mile, 2.5 miles, 5 miles, 10 miles, 25 miles) having progressively less weight. I then used census data to find local maxima where this cookie cutter (kernel) had its maximum value. These fall over cities, though not always right in the center of them. A city with a dense suburb to the north would cause it to skew north. For example, the "normalized" center of the Boston, MA city is Central Square in Cambridge, MA. Further, it turns out, the specific cookie cutter I picked matters relatively little, since population densities around cities tend to fall off smoothly, except on cities islands in the ocean like Honolulu or San Juan.
So, how do the top 20 cities in the US by my "all cities are the same size" cookie cutter compare to the actual top 20 cities? Pretty similar, with a handful of cities rising up a lot in the ranks.
Ranks 1, 2 and 3 are the same. New York, LA and Chicago are still the top 3. LA is spread out, but not that spread out. Four cities jumped into the top 10: DC, Boston, San Francisco and Miami. All of these are tiny cities in land area with a lot of dense population both in and around them. Miami in particular is striking. It's got less than half a million people in the city proper, but with a uniform size, it's bigger than Dallas which has a population well over a million. Slipping out of the top 10 are Phoenix, San Antonio, San Diego and San Jose, but all of them stay in the top 20 except for San Antonio which is just extremely spread out compared to all these other cities. The other big climbers in the top 20 by this metric are Baltimore, Minneapolis, Atlanta, and San Juan.
So, in the end the top 20 cities if all cities were a uniform land area are (big movers up marked with a ):
New York, Los Angeles, Chicago, Philadelphia, Washington DC, Boston*, San Francisco*, Houston, Miami, Dallas, Detroit*, San Jose, Phoenix, Baltimore*, San Diego, Minneapolis*, Atlanta*, Denver, Seattle, San Juan*.
Have you heard of population-weighted density (e.g. http://seattletransitblog.com/2012/10/27/population-weighted-density-how-seattle-stacks-up/)? It's another way to capture what I think you're going for, which is that:
1. Municipal boundaries don't really matter; and
2. Large areas with few residents don't really affect the experience of the typical resident of a city.
The analogy I like to use is a train line that arrives on time 95% of the time, but where 50% of the line's riders complain that the train is always late. How can that be? Simple: The train is generally late during rush hour, and 50% of the line's ridership is during rush hour. Those on-time trains have comparatively few riders, and so it's misleading to treat those trains as being just as important as the ones that everyone rides.
Population-weighted density is just the same thing applied to cities. Imagine if you take New York, and surround it by 100,000 square miles of empty fields. Conventional density measures will suggest that the resulting city is less dense than New York is today, but that's clearly ridiculous -- nothing has changed for anyone who actually lives there. Population-weighted density correctly handles this case.
Yeah, it's also a density measure, but population density for cities is measured over a highly variable municipal area. LA is less dense than Chicago, but if you expanded Chicago to be the size of LA (absorbing surrounding towns) it wouldn't be quite as populous as LA, which is what I was trying to get at.
The population weighted density number is interesting, but suffers from the same problem: the denominator is defined by the boundaries of the MSA, which change a lot.
By that ranking, Honolulu is above Chicago, which is only because the MSA for Honolulu is barely 5% of the size the MSA for Chicago. My preference was to essentially force a fixed denominator. Similarly, if I'm on a cruise ship in the middle of the ocean, the local (and population weighted) population density is double that of the densest census tract in Manhattan, but that doesn't make it a "bigger city" than NY. A lot of what I was trying here was to reconcile the sense of what makes something feel like a "big city" and substantially to me it's "how many people are there within X miles" for small-ish X.
Another alternative would be to fix the numerator and see how large you'd have to get. How big would the "city" have to be to contain, say, 3 million people. Using the population weighted density methodology, you could do a calculation of the weighted density for those nearest 3 million people to avoid the Honolulu/Chicago problem.