Another feature of weighted density

Another in an occasional series on weighted density.

Below the jump I discuss a rather odd feature of weighted density:  It can fall even as a city's standard density rises.  This is truly a feature and not a bug, though, because it allows weighted density to yield some interesting information about how a city is growing.

Again, the point of weighted density is to capture the density at which the average person lives.  Rather than divide a city's gross population by its gross area (standard density), we divide the city into small geographic units such as census tracts, calculate the standard density of each, and then assign each a weight equal to its share of the total population.  The sum is the weighted density.  (There is a longer discussion here.) As someone else has put it, standard density measures the average amount of land around each resident; weighted density measures the average number of people around each resident.

Weighted density has a counter-intuitive feature.  Unlike standard density -- which always rises when the city adds more people without expanding its boundaries -- weighted density can actually fall even as the city adds new people. 

This really is strange at first blush.  How can a city's density fall as it adds people within its existing boundaries?  But it makes sense if you bear in mind that weighted density measures the average perceived density.  If a city adds a lot of people to low-density areas, then the percentage of the population living at low density increases and the percentage of people living at high density decreases.  The high-density areas get less weight, in other words; the slight increase in overall density might not make up the difference. 

Let's take a simple case.  Suppose Metropolis  consists of two tracts:  Tract A (the city core) with 20,000 residents living at a density of 10,000 people per square mile (ppsm), and Tract B (the suburb) with 80,000 residents living at a density of 1,000 ppsm.  Tract A's density gets a weight of 20% because it has 20% of the total population; Tract B's gets a weight of 80% because it has 80% of the population.  Metropolis's weighted density is (0.2)10,000 ppsm + (0.8)1,000 ppsm = 2,800 ppsm. 

Suppose Metropolis grows by 10%.  Let's consider three simple variations.

Scenario 1.  Metropolis experiences a uniform 10% growth in population; i.e., both tracts grow by 10%.  This is simple.  The weighted density increases by 10%, or 280 ppsm, to 3,080 ppsm.

Scenario 2.  All of the growth takes place in the suburb.  Tract A remains at a population of 20,000 (and its density at 10,000 ppsm), while Tract B's population rises from 80,000 to 90,000 and its density rises from 1,000 ppsm to 1,125 ppsm.  Tract B is now slightly denser than before.  But its share of the total population has risen from 80% to 82%, and the dense core's share has dropped from 20% to 18%.  Metropolis's weighted density is now 10,000(.18) + 1,125(.82) = 2,722.5.

In Scenario 2, Metropolis sees a slight decrease in its weighted density even though its standard density has risen by 10%.  But there's no paradox here:  More of Metropolis's citizens live at low density than before; the suburb's slightly higher density is still quite low, and not quite enough to offset the dilution of the high-density core's contribution.  Adding a bunch of people at a low perceived density, in other words, tends to lower the average perceived density. 

Scenario 3.  All of the growth is in Tract A, the core.  Tract A's population rises from 20,000 to 30,000 and its density rises from 10,000 ppsm to 15,000 ppsm.  Its share of the total population increases from 20% to 27%.  Meanwhile, Tract B maintains its density but sees its share of the total population fall from 80% to 73%.  Metropolis's weighted density in this scenario is 15,000(.27) + 1,000(.73) = 4,780 ppsm, a 71% increase in weighted density -- even though Metropolis saw only a 10% growth in population.

Metropolis's standard density is exactly the same in all three scenarios.  And that, I think, is a shortcoming of standard density.  It does not tell you anything about how Metropolis is growing.  By comparing the weighted density before and after, it may be possible to determine whether the growth is taking place in the denser areas rather than the far-flung suburbs. 

I have to hedge that last statement because there are a number of variables here.  The larger and denser the core, the more sensitive the weighted density to growth in the suburbs.  Consider this:

Scenario 2':  Metropolis's 100,000 residents are evenly split between the core and suburb.  Its weighted density is (0.5)10,000 ppsm + (0.5)1,000 ppsm = 5,500 ppsm.  Metropolis grows by 10%, all in the suburb.  Its weighted density drops dramatically, to 4,290 ppsm.  By contrast, when the core accounted for only 10% of the city population, the weighted density fell only slightly.  In both cases, though, all of the growth took place in the suburb.

The city's initial distribution of population thus matters a lot.  New York City, for example, almost certainly will see its weighted density decline in the 2010 census, despite the fact that Manhattan's population almost certainly is rising.  A sparsely populated city like Atlanta may see its weighted density hold steady, if only because such a high percentage of the population is already situated in low-density suburbs. 

The Census Bureau unfortunately tallies census tract populations only every ten years (hence the term "census" tract).  It will be interesting to see how the weighted densities change between 2000 and 2010.  My guess is that low-density cities will hold roughly steady, as will cities with relatively uniform standard densities, such as Phoenix.  Cities with dense cores experiencing a lot of growth in the suburbs, such as New York or D.C., will doubtless see declines.  The big surprise will probably be in places like Austin and Portland that think they're getting denser -- I suspect these cities are experiencing more suburban growth than most people realize, and their weighted densities will experience a corresponding decline.

Comments

[M1EK]

Actually, the suburban sprawl in both Austin and Portland is more medium-density than very low density - so I'd continue to expect 'cities' like Houston to lead the way here.

[DSK]

I think you are correct, more or less. Being pretty familiar with growth of both Austin and Atlanta over the last 10 years, I think Austin's weighted density will decline using the 2010 numbers, and Atlanta's weighted density will rise significantly from your 2000 calculation.

[lemuel pitkin]

Why do you say New York's growth is biased toward the suburbs?

During 1990-2000, New York City grew somewhat faster than the New York metropolitan area as a whole. (9.4% vs. 8.4%) And within the city, the New York City planning dept. expects Manhattan to significantly increase its share of the city's population between 2000 and 2010. (http://www.nyc.gov/html/dcp/pdf/census/projections_report.pdf) So I would expect both standard and weighted density to increase for New York.

[AC]

I was making a seat-of-the-pants guess. Generally, it's harder for very dense places to match the percentage increases of the suburbs. If Manhattan really does increase its share of the city population, then, yes, NYC's weighted density ought to rise.

[M1EK]

AC, I guess the point might rest on what we call NYC's suburbs. The outer boroughs didn't grow much if at all, I'd guess, due to NIMBYism. But are we including exurban Connecticut? (Did that grow, either?).

I see ads for high-rises in Manhattan whenever I look through the Times at my in-laws', so I know they finally started building there again; but I haven't seen anything advertised in Queens or Brooklyn.

[AC]

I use urbanized area. The Census Bureau uses a very complicated algorithm for identifying the urbanized area. It takes up 2 single-spaced pages of the Federal Register. I'm being lazy here -- I ought to pull up the Census map -- but I'm sure NYC's urbanized area stretches into Connecticut.

The urbanized area obviously can grow through new development. It can also grow by absorbing pockets of development that at one time were disconnected under the Census Bureau's algorithm.

For example, San Marcos is not in Austin's urbanized area. However, if development eventually fills in the I-35 corridor, it may become part of the urbanized area. Then there will be a big jump in the urbanized area population even though not all of the growth was due to new development.

When I have time, I may calculate selected weighted densities based on 1990 urbanized areas. It won't be a perfect comparison because of changes in census tract boundaries and in the Census Bureau definition of "urbanized area" between 1990 and 2000.

[lemuel pitkin]

For recent years, the pattern is that Manhattan is growing faster than the otehr boroughs, and NYC as a whole is growing faster than the rest of the urbanized area. So the increase in density is conssitent across metrics.

This pattern is, of course, very atypical of the United States and it's not guaranteed to continue into the future -- altho, as noted above, the view of NYC goverment is that it will.

[AC]

I should add that out of necessity I use all census tracts that contain any portion of the urbanized area. This means I'm including some population not in the urbanized area, which will be lower density, exurban population.

This expansion doesn't skew the weighted density calculations much because it typically adds just 3-4% to the urbanized area population. It will have an even smaller effect on the weighted density. I am able to calculate a potential range of the modified weighted densities, and will post them sometime.