Werner's Blog — Opinion, Analysis, Commentary
A better measure of population density captures urbanization

Population density is a hugely important variable in many economic analyses. Yet, empirical work is often very cavalier about measuring it. Simple population density is a rather flawed measure when dealing with large geographic entities such as provinces, states, and countries. The simple measure of population density is population divided by area (more more specifically, land area). However, this measure ignores the fact that most people live in cities. As an economic measure, population density needs to accommodate the fact that most economic agents live much more concentrated in space than simple population density measures suggest. We can do better with a little bit of econometric work. Enter the notion of a population-weighted population density measure. A recent working paper by John Ottensmann from Indiana University/Purdue University makes this point in much greater detail, and I would like to reemphasize the point by showing an important example—the population density of U.S. states.

Population-weighted population density actually introduces two modifications. The first is to weight the individual cells within a geographic area by population size. The second is to use the geometric rather than arithmetic mean. In the case of the United States, I have taken population estimates for 2019 from the U.S. Bureau of Census and combined it with land area data (as provided by the 2010 census). For each U.S. county \(i\) we have population \(P_i\) and land area \(A_i\). Then the population density \(\Phi_i\) is \[\Phi_i=\exp\left[\frac{1}{P}\sum_i P_i\cdot \ln\left(\frac{P_i}{A_i}\right)\right] \quad\mathrm{with}\quad P\equiv\sum_i P_i\] The results of the analysis for each state are shown in the table below. It may be more intuitive to discuss the results in a diagram that shows simple population density on the horizontal axis and weighted population density on the vertical axis. Both axes are scaled logarithmically. Click on the image to see a high-resolution version.

Simple versus Weighted Population Density of U.S. States

click on image for high-resolution PDF version

Every single state in the U.S. has a higher weighted population density than simple density. All points in the scatter plot lie above the 1:1 line. The differences are huge. Consider the U.S. overall. The simple population density puts 36 people on each square kilometer. That would make the U.S. a very rural place, which is of course far from what the U.S. looks like. The population-weighted numbers is almost six times as larger: 206 people per square kilometer. For some states the difference are much larger. Alaska's small population spread over the entire size of the state gives it a puny 0.5 people per square kilometer. But even in Alaska most people live in cities, and the population-weighted density is 5.6—about 11 times larger.

One state stands out the most: New York. The simple population density of 159 people per square kilometer entirely misses the fact that population is concentrated enormously in New York City. The revised population density for New York is 1,323 people per square kilometer, making New York the densest state in all of the U.S. If instead one relies on the simple measure of population density one misses the exceptionalism of New York as the urban metropolis of the United States. And even spacious Texas is much, much denser than simple measures suggest.

StateCode Popu-
lation
[million]
Land Area
[1000 km2]
Population Density [per km2] Difference
[%]
simple weighted
United States   328.240  9147.593  35.883  206.346  475.1% 
Alabama AL 4.903  131.171  37.380  64.194  71.7% 
Alaska AK 0.732  1477.953  0.495  5.625  1036.4% 
Arizona AZ 7.279  294.207  24.740  76.011  207.2% 
Arkansas AR 3.018  134.771  22.392  40.781  82.1% 
California CA 39.512  403.466  97.932  329.123  236.1% 
Colorado CO 5.759  268.431  21.453  121.156  464.7% 
Connecticut CT 3.565  12.542  284.276  371.439  30.7% 
Delaware DE 0.974  5.047  192.951  259.754  34.6% 
Florida FL 21.478  138.887  154.641  286.392  85.2% 
Georgia GA 10.617  148.959  71.277  204.405  186.8% 
Hawaii HI 1.416  16.635  85.116  250.577  194.4% 
Idaho ID 1.787  214.045  8.349  35.138  320.9% 
Illinois IL 12.672  143.793  88.125  410.564  365.9% 
Indiana IN 6.732  92.789  72.554  138.052  90.3% 
Iowa IA 3.155  144.669  21.809  43.676  100.3% 
Kansas KS 2.913  211.754  13.758  64.891  371.7% 
Kentucky KY 4.468  102.269  43.685  91.305  109.0% 
Louisiana LA 4.649  111.898  41.545  97.165  133.9% 
Maine ME 1.344  79.883  16.827  37.415  122.3% 
Maryland MD 6.046  25.142  240.465  453.378  88.5% 
Massachusetts MA 6.893  20.202  341.178  552.354  61.9% 
Michigan MI 9.987  146.435  68.200  201.263  195.1% 
Minnesota MN 5.640  206.232  27.346  128.880  371.3% 
Mississippi MS 2.976  121.531  24.489  37.282  52.2% 
Missouri MO 6.137  178.040  34.472  113.870  230.3% 
Montana MT 1.069  376.962  2.835  6.875  142.5% 
Nebraska NE 1.934  198.974  9.722  70.961  629.9% 
Nevada NV 3.080  284.332  10.833  64.351  494.0% 
New Hampshire NH 1.360  23.187  58.640  95.416  62.7% 
New Jersey NJ 8.882  19.047  466.322  785.426  68.4% 
New Mexico NM 2.097  314.161  6.674  24.910  273.2% 
New York NY 19.454  122.057  159.381  1322.99  730.1% 
North Carolina NC 10.488  125.920  83.292  150.632  80.8% 
North Dakota ND 0.762  178.711  4.264  9.608  125.3% 
Ohio OH 11.689  105.829  110.453  228.178  106.6% 
Oklahoma OK 3.957  177.660  22.273  73.170  228.5% 
Oregon OR 4.218  248.608  16.965  76.903  353.3% 
Pennsylvania PA 12.802  115.883  110.473  275.998  149.8% 
Rhode Island RI 1.059  2.678  395.645  455.294  15.1% 
South Carolina SC 5.149  77.857  66.131  97.514  47.5% 
South Dakota SD 0.885  196.350  4.506  12.533  178.2% 
Tennessee TN 6.829  106.798  63.945  123.679  93.4% 
Texas TX 28.996  676.587  42.856  245.431  472.7% 
Utah UT 3.206  212.818  15.064  135.237  797.7% 
Vermont VT 0.624  23.871  26.140  34.090  30.4% 
Virginia VA 8.536  102.279  83.453  286.827  243.7% 
Washington WA 7.615  172.119  44.242  129.547  192.8% 
West Virginia WV 1.792  62.259  28.785  44.760  55.5% 
Wisconsin WI 5.822  140.268  41.509  112.485  171.0% 
Wyoming WY 0.579  251.470  2.302  3.270  42.1% 

Readers in the United States please take note the use of metric measures. The use of imperial measures is strongly deprecated in the scientific community. However, if you prefer expressing population densities in square miles, divide the square kilometer numbers by 2.59.

It matter which density measure one uses for empirical work. Even though the correlation of the logarithms of the two density measures is 0.91 (or 0.71 without logarithms), the population-weighted measure shifts the rank order of states in important places. If empirical work tries to capture a notion of economic density by way of population density, it is essential to allow for the degree of urbanization of states. Nevertheless, simply using unweighted population density and the urbanization share of states is not sufficient to capture the true density effect because cities vary dramatically in density as well.

If you use population density in any of your research, abandon simple density and embrace population-weighted density. It is not difficult to compute and it is much superior. The U.S. Census Bureau has embraced population-weighted density measures as well (see the report below.)

In the analysis above I used census data from counties. Of course, even better would be to use census tracts (there are more than 73,000) instead of counties (3,143). The measure of population density would be even better with smaller cells.

Further readings and information sources:

Posted on Sunday, April 12, 2020 at 11:15 — #Econometrics | #US
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© 2020  Prof. Werner Antweiler, University of British Columbia.
[Sauder School of Business] [The University of British Columbia]