I want to point to this wonderful graph of the density of eight American cities by Fedor Manin at We Alone on Earth. I'm referring specifically to the top graph. (I have a problem with the bottom one.)

Manin computed the density of each census block group in each city's urbanized area. He then calculated the percentage of the population living at at least the density given on the x-axis. The graph gives a snapshot of each city's weighted density: if city A's weighted density is greater than city B's, then city A's curve mostly lies to the right of city B's.

Note the graph is logarithmic.

Manin observes that it is strange that the density graphs for Boston and New York are straight lines. I think it's strange, too. Because the graph is logarithmic, the straight lines mean the densities of Boston and New York obey (approximately) a power law, while the density functions of the typical city are (approximately) linear.

The bottom graph also gives a visual snapshot of weighted density. It depicts the number of people living within 5% of the given density.

It is perfectly reasonable to smooth the curve using a range like this. But the graph is distorted because the range increases with density. E.g., the range for 10^{3}/km^{2} is 950-1,050 persons/km^{2}, while the range for 10^{5}/km^{2 }is 95,00-105,000 persons/km^{2}.^{ }The range for 10^{5}/km^{2} is thus 100 larger than the range for 10^{3}/km^{2}. This distorts the graph, biasing it in favor of high densities.

Manin should have taken the average density over each range; i.e., he should have divided the number of people living within 5% of the given density by the length of the interval.

H/t Market Urbanism and Human Transit.