DC's population-weighted density for the 2010 Census was 17,442 people per square mile. This is the density the average person living in DC experiences.
Population-weighted density, a more sophisticated way to measure density, counts the local density around each person and averages it over all the residents of a jurisdiction. The average resulting from a population-weighted density calculation is less impacted by large unpopulated areas.
Population-weighted density can be calculated by multiplying the population of a census tract by that census tract's population density and then dividing that number by the total population of the jurisdiction. This is done for every census tract within a jurisdiction and then adding all the numbers up at the end (I did the calculation for DC's population-weighted density in an excel spreadsheet using formulas)
The standard density calculation for DC in 2010 calculated out to 9,857 people per square mile when taking DC's total population and then dividing that by DC's total land area. This is the density that DC's total land area experiences, not what the average person experiences.
The U.S. Census Bureau has started using population-weighted density in some of its reports:
http://www.census.gov/prod/cen2010/r...c2010sr-01.pdf
I expect DC's population-weighted density for the 2020 Census to go up considerably compared to 2010, especially since so many people have moved into central DC (especially Columbia Heights, U Street, Shaw, Mount Vernon Triangle, NoMa, and Navy Yard).