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, a71% increasein 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.