I’m specifically thinking of population density, but I suppose it could apply to other things. Population density of a country is defined as # of people / area of country. Consider two countries with the same area and population. Country A is evenly distributed while Country B is packed into a mega-city with sparse population in the rest of the country. A citizens feel low-density and B citizens feel high-density. Is there a standard way of representing that?
You are talking about population distribution. E.g., you could compute the Clark–Evans index by computing the distance to the nearest neighbour of each individual; then the mean distance should be around the reciprocal of twice the square root of the density. If R=\text{mean distance}\times 2\sqrt{\text{density}} is approximately 1, then the individuals are randomly distributed; less than one and they are clumped together, greater than 1 and they are dispersed.
If you define specific number of areas of equal size in each country and want a single number to describe the concentration of people you could use the Herfindahl index *also called the Herfindahl-Hirschman index. The HHI is the sum of the squared fraction of the population in each area.
One problem with the index in this usage is the areas are not endogenously defined as they are in most usages.
I’ve often contemplated that a measure like this would be good for violent crime statistics. A violent crime is fundamentally an interaction between two people, an attacker and a victim, and so one would expect that the rate of violent crimes in a region would be proportional to the square of the population density (in other words, high-density areas would have a higher per capita rate than low-density areas). It’s similar to what you see in chemistry, where the reaction rate between two substances is proportional to the product of their density, or, in the rare case where a substance reacts with itself (like two NO2 molecules combining into N2O4), the square of that substance’s density.
I’m not so sure about that. Sure the number of muggers could be assume proportional to the population. But they’re just going to find an appropriate victim. Possible victims is also proportional to population, but if a mugger searches until he finds one, muggings won’t be proportional to possible victims, but just to muggers.
Looks like this is an idea that scientists already use in a number of ways
The traditional and most widely understood method for calculating an aggregate measure of human population density within any geographical region is simply to divide its total population by the total area (i.e. d = ΣP/ΣA). It has long been recognised in the field of geography and by many other scholars that this method has significant shortcomings for certain types of research, particularly in the human sciences and where the subject matter of interest may be related to the typical density levels experienced by the population, such as in epidemiology.
Population Weighted Density (PWD) – proposed by John Craig in 1984 is a family of methods that – as the name suggests – weight the density values by their corresponding population sizes in the aggregation process. We have utilised three distinct methods to generate PWD estimates:
That’s assuming someone who considers “mugging” to be their job. I doubt, however, that that’s very common. Much more violence comes as crimes of opportunity (and density determines how likely an opportunity is), or as a disproportionate reaction to something someone else does, or the like.