Perceived density

Every once in a while -- invariably in a debate over sprawl -- someone will toss out the "fact" that the Los Angeles metropolitan area is denser than the New York metropolitan area. 

It's true.  At least, it's true if one uses the standard definition of density as gross population divided by gross land area.  According to 2000 U.S. census data, the Los Angeles "urbanized area" has a density of 7,068 persons per square mile.  The New York urbanized area has 5,309 persons per square mile.  Ergo, Los Angeles is denser than New York.

But common sense tells us that this coarse statistic is misleading.  Ryan Avent puts it well:

Los Angeles is hemmed in by its geography, so it can’t just keep spreading at ever lower densities out into the wilderness. As such, its density profile is like a plateau–not all that tall at anyone point, but with a respectable average height, because the long tails are excised. New York, by contrast, is like a mountain. It has an enormous peak containing most of the mass, but the flattening sides of the mountain continue on for miles.

In other words, the fact that the last million or so people in the New York metro area occupy an incredibly large area while the last million or so Angelenos are in moderate density suburbs packed against the very edge of the basin, skews the relative density figures, making them pretty uninformative.

A more meaningful metric is "weighted" density or "average perceived density."  Carve the metropolitan area into distinct regions (census tracts, for example), compute the density of each, and then assign each a weight based on its percentage of the total population.  This discounts large, sparsely populated census tracts, and gives extra weight to densely populated tracts.

An extreme but simple example:  Suppose Metropolis consists of a central core of 100,000 residents on 10 square miles, and a suburb of 10,000 on 100 square miles.  Its standard density is 1,000 persons per square mile. 

But this is a meaningless number.  Most of the residents of Metropolis live in a very dense environment.   The roughly 90% who live in the core are packed in at 10,000 per square mile, while just 10% live at the rural density of 100 per square mile.  By giving the core's density a weight of 90%, we get an adjusted density of 9,100 persons per square mile, a much better description of the density perceived by the average resident. 

I show the weighted densities for some U.S. cities below the jump. 

I used 2000 Census Bureau data for "urbanized areas."  In Census-speak, an urbanized area consists of one or more "central places" -- essentially, incorporated cities at the core of a metropolitan area -- and any adjacent territory with a general population density of at least 1,000 per square mile.  The Census Bureau gives the population and density for each central place in the urbanized area.  It also gives the population and density of the area not within one of the central places.  I assigned weights to the central places and the "non-central" place according to their respective percentages of the total urbanized area population.

Surprise.  New York is denser than LA.  NYC's standard density is 5,309 per square mile.  Its weighted density skyrockets to 14,786.  LA's density, on the other hand, merely creeps up from 7,068 to 7,961.

Here are my calculations for some other metropolitan areas:

Boston is less dense than Austin under the standard metric, but 50% more dense using the weighted metric.  Most of the northeastern cities -- Boston, Philadelphia, New York, D.C. -- are significantly denser using this measure.  (Note that Austin, Houston and Dallas have almost exactly the same weighted density.)  Western cities such as Los Angeles and Phoenix see only a slight increase, reflecting the fact that their population is more or less evenly distributed.

The final column in the chart shows the ratio of weighted density to standard density.  It reflects the extent to which the population is concentrated in the core.  A ratio near one means the urbanized area's population is uniformly distributed.  A high ratio reflects a concentrated core.  (Note:  A city's weighted density is always at least as great as its standard density.  That's just a consequence of the math.)

Caveat:  It would be better to use census tracts rather than central places.  An otherwise dense central place could still contain low-density areas that would pull down its average.  Using census tracts would largely solve this problem.  When I find time to run the calculations with census tracts, I will post the results.

Comments

[The Overhead Wire]

AC I saw your comment earlier on the Bellows. This is awesome. Finally a real metric that tells of the true comparison between places and density!

[M1EK]

Cool analysis. I think the census change would make it even more useful too - but you've got a great start here.

[M1EK]

Honolulu would be a very neat test of this too - I bet their weighted density beats even New York's.

[AC]

Honolulu's standard density is 4,660 and its weighted density, using the Census data set I used above,is 4,699.

This is a good illustration of the caveat I gave at the end. The Census Bureau breaks Honolulu down into just two regions -- the "central place" of Honolulu proper and everything else. They have roughly the same density, which is why the weighted density does not rise much.

The bigger metropolitan areas have more central places; with more central places, the weighted density calculation becomes more discriminating. My method is a little too coarse for Honolulu.

I've never been to Honolulu, so can't speculate, but the weighted density would rise, and could rise significantly, if based on census tracts.

It won't beat New York, though. Manhattan's density is 67,000 per square mile. A city can't have that kind of density unless it looks like Manhattan. (Manhattan has about 10% of the NY metro population, which means that Manhattan, all by itself, contributes 6,700 per square mile to the weighted density.)

[M1EK]

Huh. Too coarse data indeed. The dense residential center of Honolulu is typified by high-rises which average around 20 stories; and I would figure they'd have included the whole island in the metro area - which includes a bunch of forest preserve and agricultural areas, since it's all the same county, pretty small, and all the other little bedroom communities basically function as commuter enclaves.

In other words, the dense parts of Honolulu are Manhattan-esque; and to boot, the low-density parts are more like LA (high-density suburban).

Honolulu reportedly had the highest residential density in the country - I don't know what the metric was, but the stat was used in a previous effort to justify rail transit.

[AC]

I pulled up a Census map of Honolulu and it is indeed very dense. There are only 4 census tracts (with a total of about 18,000 people) that are in the Manhattan range of 67,000/square mile. But there are many other tracts at 15,000+/square mile, which is very dense, and a lot of sparsely populated tracts. If I find time to run the calculation with census tracts, I think Honolulu will turn out to be quite dense. (By contrast, I don't expect Austin's or Houston's density to increase much.)

You can get to the Census map here:
http://factfinder.census.gov/servlet/SAFFPopulation?_submenuId=population_0&_sse=on

[AC]

I pulled up the Census map of NYC. The Upper West Side and Upper East Side have lots of tracts at 130,000/square mile. These are twice as dense (or denser) than the densest Honolulu tracts. (There are even a couple of tracts at 200,000+/square mile!).

[M1EK]

Yeah, I had all of Manhattan at super-dense in mjy head but there's actually more 3-5 story stuff in Manhattan than people think; and especially in the other boroughs. I was more thinking that Honolulu's "least dense places" would be even less dense than New York's (if, as I suspect, they include the whole island in the metro area).

Thanks for the link. Will be checking out today as time permits. I see at least that the CDP only includes the southern half of the island - do they have a way to get the census tract map for the metro area? (after entering Honolulu, I only had the option of "county", which didn't have census tracts, or CDP).

[AC]

I'm sure there is some way of linking census tracts to the metropolitan area. I just haven't found the right census file yet.

My plan at the moment is to calculate weighted density using the census tracts for the entire county. One of the beauties of weighted density is that large sparse tracts have essentially zero effect on the final number. This makes the precise definition of the metropolitan area relatively unimportant.

The county has about 150,000 more people than the urbanizes area (it's about 20% larger). I can calculate the gross density of the non-urbanized area and try to back it out of the weighted density calculation for the county as a whole. I ought to be able to establish upper and lower bounds for the urbanized area's weighted density, and it ought to be a pretty narrow range.

All of this assumes I can get past some deadlines at work.

[The Overhead Wire]

Hey AC, send me your email and I can send you a spread sheet with blockgroup population and acreage. I can pull it out of my GIS. theoverheadwire at gmail dot com

[AC]

OK, M1EK, Mr. Honoluluan-wannabe:

I've calculated the weighted density of Honolulu County, Travis County, and Harris County using census tracts.

Honolulu County: 10,652
Travis County: 4,169
Harris County: 4,921

I'll get around to some others later.

True, this includes non-urbanized area. But the nice thing about the weighted density is that it discounts the sparsely populated areas. I can calculate a rough upper bound on the density of the urbanized area by assuming the non-urbanized area has a uniformly distributed population and doing a bunch of algebra. It's only a rough upper bound because some census tracts may straddle urbanized and non-urbanized area, diluting the density of the urbanized area, but I wouldn't expect this to have much effect.

I'm sure that's clear as mud (it's late).

The rough upper bounds on the weighted densities of the urbanized areas within these counties:

Honolulu County: 12,917
Travis County: 4,601
Harris County: 5,032

I was surprised that the weighted density of Harris County is higher than that of Travis County.

[M1EK]

You are a gentleman and a scholar. (How did a guy who enjoys statistics this much end up as a law-talking-guy anyways?)

Amazingly, Honolulu has their own anti-rail Skaggs/Daugherty fellow who insists they're not dense enough for rail. If we were building the New York subway today, I'm sure somebody would claim Manhattan's not dense enough either...

[bittergradstudent]

Somethng still strikes me from weird, but it's anecdotal--I'm from St. Louis and I live in Austin, and it seems bizarre to me that St. Louis would be less spread out than Austin is. The STL metro area is roughly twice the size of Austin's, but when I lived there, I spent half my life driving every which way, and the congestion was of the type where there were lots of people on the road, not the Austin type of congestion where the roads and interchanges are so badly designed that they create lots of artificial congestion.

Then again, I guess Round Rock and Georgetown are a pretty big chunk of the population of the Austin metro area, if they are counted as part of it

[zilfondel]

Census tracts may not be so handy. For instance, several inner-city census tracts in Portland, OR, consist of a large % of water. If 20% of the inner city "land area" consists of water, that is defeating the point of determining what would match what a resident would experience.

[AC]

The Census Bureau breaks down census tract area by land area and water area and calculates density based on land area only.

[low-tech cyclist]

I understand what you're doing. Consider this semi-analogous example: you've got a college with only 2 classes, that's computing its average class size. One class has 30 students, the other has 10 students.

Easy, right? (30+10)/2 = 20. That's certainly the number that'll be in the catalog.

But if we use the weighted-density method, we get 30 x (30/40) + 10 x (10/40) = 25. That's the average class size as experienced by the students - 3/4 of them experience the size-30 class, and only 1/4 of them experience the 10-student class. So from their POV, the average class size really IS 25.

That works really, really well for class sizes, because classrooms are discrete entities. If you had 10 classes of 30 students, and 10 classes of 10 students, you'd get the same weighted average of 25 per class, no matter where the large classes were in relation to each other. And that would reflect truth, because their proximity doesn't matter.

However, it DOES matter where your dense Census tracts are in a city.

Imagine two chessboard-shaped urban areas, composed of 64 square-mile tracts. In each one, 52 of those tracts have 1000 persons apiece, and the other 12 have 10,000 persons each.

They'd both have the same gross density (2,688 persons/sq.mi.), and the same weighted density (7,267/sq.mi.).

But suppose one has its 12 denser tracts in the rectangle with its corners at squares c3, c6, e3, and e6 - a compact 12-square-mile rectangle with 10K density, surrounded by a 52 square mile region of 1K density.

The other has its 12 dense tracts at a1, a4, a8, c3, c6, d8, e1, e3, e6, h1, h5, and h8.

The first is unquestionably more compact in its density than the second, with a 12-sq-mi area at its core with 120,000 people, while the other area has no more than 39,000 people in any similarly compact area. A density measure that misses that has a pretty significant blind spot, IMHO.

I'd argue that a good density measure has to start off with an urban center of mass (which weighted density would be a useful tool for defining and locating), and give greater weight to people who are closer to the center.

[AC]

See the Bertaud paper cited in the comments to the more recent post on this topic. He advocates calculating the center of mass of the city's population. (He treats the city population as a solid, with the city's footprint as the xy-plane and the population density on the z-axis.)

Calculating the center of mass alone won't tell you how compact the city is. Bertaud creates a pretty complicated "dispersion" index using the center of mass that he says is a good measure of sprawl. That's a lot more sophisticated than what I've done.

Weighted density, as I've calculated it, obviously doesn't tell you whether the dense areas are contiguous. (They are not in Austin.) But since census tracts correspond roughly to neighborhoods, they do tell you how densely the average person lives.