Density calculations for U.S. urbanized areas, weighted by census tract

I recently calculated the weighted densities for a number of cities using the Census Bureau's "central place/non-central place" classification.  As I explained in that post and the comments, this is actually a coarse measure of weighted density because central places themselves are usually large cities with pockets of wildly varying density.

I have now calculated the weighted density of 34 large urbanized areas using census tracts.  (See my last post for an explanation of weighted (or "perceived") density.)

I took the 32 largest U.S. urbanized areas and added Austin and Honolulu for good measure.  I pulled the Census data on each census tract partially or completely contained within each of these urbanized areas.  I calculated the standard density (i.e., total population/total land area) for each census tract.  I also calculated each census tract's share of the total population of the urbanized area.  I then assigned each tract's density a "weight" equal to its share of the total population.  I summed the weights to get the weighted density for the urbanized area. 

I'm preparing a permanent page with more information on the methodology and limits of this approach.  (It's a very good method, though, in my opinion.)

But that technical stuff can wait a few days.  Here are the weighted densities, ranked from most dense to least dense:

         

(Chart on Scribd)

This approach obviously does a much better job of reflecting our perceptions of a city's density.  LA's urbanized area is denser than NY's when using the standard density metric; NY's urbanized area is almost three times denser using weighted density.  (LA is still quite dense, though.) 

Weighted densities straighten out a lot of other counterintuitive "facts."  Austin and Tampa are not really denser than Boston (as the standard density figures suggest), and the sprawling suburbs of Riverside County are not actually denser than Chicago. 

Note that Portland's urbanized area is less dense than Houston's.  The urbanized area most like Portland's, viewed strictly by their density profiles, is Riverside-San Bernardino.

New York is an outlier at the upper end, while Atlanta is an outlier at the lower end.

The "density gradient index" is a term I made up to dignify what is a pretty simple calculation:  weighted density divided by standard density.  The index would be an even 1 if a city's population were uniformly distributed across the landscape.  The larger the index, the more uneven the distribution.  I'll go into this in more detail on the permanent page, but you can get a rough idea of a city's urban form just by looking at the weighted density, standard density and density gradient index.

(There really is something called the "density gradient," but it is a curve rather than a number, and there is no "index" for it, as far as I know.)

Related posts:

1. Perceived density (March 16, 2008).
2. More on weighted density for the crackpot blogging stats geeks (March 27, 2008).
3. Another feature of weighted density (May 20, 2008).
4. The association between density and mode of commute (Sept. 21, 2008). 

Note:  I don't want to take credit for the idea of weighted density.  I'm not the first person to think of this.  And it is possible someone has run these calculations before.  I've never seen them, though; the calculations above are mine.  (I suppose I could have done a Google search.  But I really don't care whether this is original work -- this isn't a refereed journal and I don't have to do literature surveys.)

Update/Correction:  The census data express land area in square meters, and I made a careless error when converting square meters to square miles.  It caused me to overstate each city's weighted density by 1.8%.  I've corrected that error.  (Since the error applied to all of the cities, the relative rankings are unaffected.) 

Comments

[Matt L]

Great work. A mapping exercise would give you the best idea of how density compares to other areas, but this is a good method for comparing 30+ urban areas. I think another useful column would be land area. These comparisons would be very meaningful if each urbanized area was the same size (geographically), but they are not. You compare Portland with Houston, yet Houston's census-defined "urbanized area" is almost three times larger than Portland's.

It would also be interesting to apply this method to a different census geography, like MSA or CMSA, that is closer to an area's commute shed. I wonder how much "weight" the low density exurbs that would be captured in an MSA would have in your formula.

[AC]

I mapped densities here: http://austinzoning.typepad.com/austincontrarian/2007/02/comparing_some_.html

I'm glad you raised the point about land area. One of the nice things about weighted density is that land area doesn't matter much. Weighted density lets us make an "apples-to-apples" comparison that does _not_ depend on land area or precisely how the geographic area is defined.

Take Houston and Portland. The average Houston resident lives in a census tract with a density of 4,597/square mile. The average Portlander lives in a census tract with a density of 4,463/square mile. That Houston is much larger than Portland (whether measured in square miles or population) doesn't have anything to do with these averages -- unless we expect larger cities to have larger, denser cores.

I'll take a look at MSAs some time. The MSA population is often much larger than the urbanized area population, at lower average density, so I expect the weighted densities would fall.

[SusanaSanJuan]

I understand that this rooted in numbers and I am only speaking from a superficial perspective, but I don't quite understand how this represents "percieved" density. Do you mean it explains what meets the eye as you move around these areas?

I lived for a long time in the DFW area, and now I live in Portland. Portland seems much denser with people able to walk or bike to work or the store in most areas of Portland proper. DFW on the other hand, required driving for the smallest thing. When I go back and visit without a car, I am literally stranded many times because you can't walk to anything except miles and miles of suburban style housing. Am I simply feeling the effects of poor zoning in DFW that clusters all the commercial zones into one area and segregates it from residential areas? If that's the case, then I would think that zoning is much larger part of the problem than density.

[M1EK]

Susana, the idea is that "perceived density" is "how dense is the area that the typical person lives in"; meaning that if in Manhattan, 9 people live in skyscrapers and one in a huge mansion on 2 acres of woods in Connecticut, the typical density is of course greater than in LA, where 10 people live in McMansions on small lots, even if LA's average density is still higher.

AC, great work. You've done the world a service here. Well, the internet at least. Well, a bunch of crackpot bloggers, at least. If I wasn't so busy at work on a debacle right now, I'd have crackplogged about the McCracken meeting yesterday - but here's the short version: no rail up Guadalupe; still talking about MLK versus Manor; BM has some interesting ideas about governance reform to make this happen. I'm still cautiously optimistic given that this is the last best chance to get something other than commuter rail - something that can actually someday go up Guadalupe and down SoCo in other words.

[AC]

Two or three points:

1. These are for "urbanized areas." These are generally smaller units, geographically, than consolidated metropolitan areas. But they encompass more than the central city. The Portland urbanized area includes Vancouver, Washington. Most of the population of Portland's urbanized area lives outside Portland's city limits.

2. Not all compact, mixed-use, walkable areas are very dense -- e.g., a neighborhood consisting mainly of single family homes, with some modest multi-family and neighborhood retail may be quite walkable and yet have a density of just 5,000 or so per square mile.

3. A census tract can be quite dense and yet be pedestrian unfriendly. Think massive, gated apartment complexes lining busy highways.

This is "perceived" density in the sense that it is the average density experienced by a resident of the urbanized area. The statistic does not measure the "quality" of the density.

[BeyondDC]

This is really fantastic. I love it.

[DCTexan]

Are these 2000 numbers? Would we expect Portland to moved have past Houston by now? DC to have moved past Baltimore? Maybe even approach San Diego? What's holding DC so far down from Boston? 1,000 people less per census tract? Does Boston not have a Loudon County?

[AC]

DCTexan, answers in order.

(1) Yes. (2) It's possible, since they are so close. I'm confident that Portland's weighted density has decreased since 2000, though. (3) Again, it is possible since they are so close. (4) Unlikely, since it's a bigger gap, and ISan Diego's weighted population has probably not decreased. (5) A larger percentage of Boston's population lives in a denser core. (6) These are weighted averages of census tract populations; the straight averages may be more or less. (7) Boston has many "Loudon Counties," which is why its standard density is so low.

[DCTexan]

Thanks for getting back AC. I guess I'm just skeptical of the DC numbers. It seems that the Census misses the transient people who make up such a large percentage of the inner DC population, whether because they're poor or young or both. I suppose that might not make that much of a difference in a large region.

I just don't 'perceive' Boston as 13% denser than DC. Great work though

[Tedd]

Very interesting work. I have a question, though: would places like Portland and Austin that have ample public land (i.e. pabkland) score less favorably on a density ranking such as this? Or should your weighted rankings largely control for this?

[AC]

The weighted rankings largely control for this. Census tracts containing large parks usually have very few people and thus have little effect on the weighted density. An unusually large number of small parks scattered throughout the city would have a larger effect.

[lemuel pitkin]

Why on earth do you present the table as a jpg? it's the worst possible format -- it's designed for phots and other images with smooth gradations of color.

If you must make it an image file, you should use a gif. But why not put it in a format that would allow it to be indexed by search engines, copied from, etc.?

[AC]

Because I'm a luddite who doesn't know any better. Can't you tell from the site decor? ;)

[adam harrell]

It will be interesting to see how this changes with updated census data. Especially with cities like Atlanta that have experienced a sustained trend towards inward growth. I believe most estimates from 2006 show that the heart of the city has added at least 70,000 new residents since 2000, and this number seems to be growing every year.

[arcseed]

DCTexan, really? Having grown up in DC, the central core of Boston really does seem a great deal denser than that of the District. Boston, after all, doesn't have the strict height limit, and is quite a bit taller. And, I suppose, Crystal City and such may be lumped in with more suburby areas for the census tracts?

That said, I do think there's a line-- with Philadelphia definitely on one side, and DC on the other-- dividing what feels "city-like" to me from what doesn't. And I'm not entirely sure which side I put Boston on.

Oh, is there really a good dividing line between San Francisco and San Jose? Seemed to me (from my admittedly limited experience) they just kinda flowed right into each other

[bof]

Just curious:

how does it combine with the density gradient calculated there:
http://alain-bertaud.com/AB_Files/Spatia_%20Distribution_of_Pop_%2050_%20Cities.pdf
(see page 96 of the pdf.)

[AC]

Bertaud uses a negative exponential function to represent population density as one moves out from the city center. He calls the coefficient of the exponent the "density gradient." The bigger the coefficient, the steeper the fall off in density as one moves out from the city center, at least until density plateaus. (I'm not an expert, but I believe this is common terminology; I've also seen the curve itself called the "gradient.")

This is a lot more sophisticated than what I've done. I'm just trying to give a quick-and-dirty snapshot. My index will be a big number when the population is "clumpy" -- e.g., concentrated in a central core surrounded by lots of sparsely populated suburbs. It will be a small number when the population is more evenly distributed. The ratio of weighted to standard density won't necessarily correlate with Bertaud's gradient, though.

One of the commenters on Matt's blog correctly pointed out that since my index uses the standard density in the denominator, it is sensitive to how the geographical boundaries are drawn. I'll probably recalculate using the standard density of the urbanized area in order to compare apples to applies. But it will still be a very crude measure.

[Matt L]

Re: DC vs. Boston (since I've lived in both in recent years)

I don't know how this would affect AC's weighted calculations (or if it would be backed up by hard data), but the inner suburbs of Boston seem much more dense than those of DC. Those suburbs inside Rt. 128 in Boston (Somerville, Medford, Newton, etc.) are mostly 1, 2, or 3 family homes on relatively small lots (1/8 or 1/4 acre). The suburbs inside DC's Beltway tend to be single family homes on larger lots (with notable exceptions along transit corridors like Bethesda and Arlington).

If I had to hazard a (again not backed by research) guess as to why, I'd say that Boston's inner suburbs developed in the early 20th century, while DC's were mostly developed after WWII (coinciding with the growth in federal and defense-related jobs).

[ValkRaider]

I know I am posting this 3/4 of a year after the discussion has stopped...

Quote:
--
"These are for "urbanized areas." These are generally smaller units, geographically, than consolidated metropolitan areas. But they encompass more than the central city. The Portland urbanized area includes Vancouver, Washington. Most of the population of Portland's urbanized area lives outside Portland's city limits."
--

The numbers that are listed in this chart are for what looks to be the entire Oregon side of Portland's metro area. The 2000 census population of Portland proper was only 529,121. The 2000 census has the Metropolitan Statistical Area population about 1.9 million, and the MSA includes Salem, OR and Vancouver, WA. Taking out Vancouver and Salem gets you to about 1.5 million people, and this includes the "urbanized area" of Portland and the suburbs (Troutdale, Gresham, Wood Village, Clackamas, Happy Valley, Damascas, Boring, Milwaukie, Gladstone, Oregon City, Wilsonville, West Linn, Lake Oswego, Tualatin, Sherwood, King City, Tigard, Beaverton, Cornelius, Hillsboro, and Forest Grove, and some unincorporated areas in Washington and Clackamas counties like Aloha, Cedar Mill, Rock Creek, etc... Also includes Sandy and some other outlying areas like Newberg and St. Helens)

So while yes, most of the metropolitan area's population is outside of the city limits of Portland - this density calculation does include all but Vancouver. (I don't know why Salem is counted in the MSA since it is separated from the Portland metro area by 40 miles of farmland...)

[AC]

No, Portland's urbanized area includes Vancouver. If you want to check out the urbanized area boundary for yourself, got to census.gov, type in Portland, Oregon in the box, then click on the link at the bottom of the next page to the 2000 density map for Portland, and then after the map pops up change the boundaries to urbanized area. (The boundaries are in light pink which makes them hard to read.)

The census bureau uses a very precise algorithm for specifying the urbanized area. Basically, it counts contiguous census block groups with a density >1,000 ppsm as part of the urbanized area. The algorithm is much more complicated than this -- sometimes it will sweep up blocks with a density of >500 ppsm, there are complicated rules for jumping over rivers and lakes, etc. The idea, though, is to include the contiguous built-up urban area. Portland's stretches into Clark County, WA.