One of the counter-intuitive things about weighted density is, well, that it's weighted. To calculate the weighted density of, say, a metropolitan statistical area, you first calculate the density of each census tract and then assign it a weight equal to its share of the total MSA population. A census tract that has 1% of the total population is assigned a "weight" of 0.01, a census tract that has 2% of the total population is assigned a "weight" of 0.02, and so forth. The idea is that a tract with twice the population should have twice as much impact on the final average, because a person chosen at random is twice as likely to live there. (Weighted density is precisely the expected density of the census tract in which a random residents lives.)
This can produce seemingly weird results. One that surprises people is that a city can add people everywhere (and grow denser everywhere) and yet still experience a decline in weighted density.
Austin provides a nice example. The Austin MSA experienced rapid growth during the aughts, growing by 37.3 percent. But its weighted density (as calculated by the Census Bureau) dropped by 3.9%, from 3,259 ppsm to 3,132 ppsm.
Here is a graph of the change in weighted density by distance from City Hall in downtown Austin. (I made the chart using data taken directly from the Census Bureau's Excel spreadsheet.)
Austin experienced a sharp increase in density in the center of town (0-1 miles), a steady increase in density between the 7-mile and 23-mile points, and an outright (noticeable) decline in density in only three tracts. But even these declines were relatively small.
You might be tempted to infer from this chart that Austin's weighted density rose, but you do not, in fact, have enough information to make that inference. You need to know where the growth took place. So consider this chart, which shows the population within the concentric bands at one-mile intervals from City Hall:
This chart shows that while the part of town nearest City Hall experienced healthy growth, almost all of the growth in the MSA population occurred between the 7-mile and 23-mile marks. Now, if we go back to the weighted-density graph, and draw in a line (brown) to mark the 2000 MSA weighted density, it's easy to see what happened:
Almost all of the MSA population growth occurred in the area between the black lines. But this entire area, in 2000, had below-average density, because it lies below the brown line denoting the 2000 MSA weighted density. And although the healthy population growth during the aughts pushed up its density some, it generally did not push it up to the 2000 average. Because this area now comprised a greater percentage of the total population, it received a larger weight and pulled down the average. This merely reflects the fact that a 2010 resident was more likely to live in the low-density 7-23 mile band than a 2000 resident. The MSA's weighted density accordingly declined even though the MSA added population or retained population almost everywhere.