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We all know that countries (for example) have population densities, but suppose we want to know what is the actual experienced population density (that is, if, say, in Sweden 95% of the population live in Stockholm, and a couple of hardy Lapps live up North, then the average Swede has the experience of a densely populated country, even though the population density is quite low overall [I don't know if that is actually true of Sweden, by the way].

The obvious way to quantify this is to divide the country into square miles (or whatever). If for the $i$th square miles the population is $p_i,$ then the quantity we seek is reasonably close to

$$\frac{\sum_i p_i^2}{\sum_i p_i},$$ so the ratio of the uncentered second to the uncentered first moment.

My questions are:

  1. Has this been considered?
  2. Does it have a name?
  3. Has it actually been computed/tabulated somewhere (for the geographical setting as above)?
kjetil b halvorsen
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Igor Rivin
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  • I doubt it has a name in this form, because it's arbitrary: its value depends on the number of pieces into which you decompose the population. At one extreme, the entire country fits in one bin and the value is the total population; at the other extreme, every bin contains at most one individual and the value is $1.$ As such, this doesn't even quality as a statistic! – whuber Sep 11 '20 at 18:39
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    @whuber Notice that I specify the density of the subdivision, and in the proposed application there is going to be some natural size (a typical person stays within a mile radius, for example). It is not so much a statistic as a family of statistics. – Igor Rivin Sep 11 '20 at 19:52
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    Perhaps see 'wikipedis on [diversity indexes](https://en.wikipedia.org/wiki/Diversity_index) – BruceET Sep 12 '20 at 05:16
  • You could consider average distance to nearest neighbor (and then ways of approximating that in practice ...) – kjetil b halvorsen Sep 12 '20 at 16:46
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    Here is a link (arXiv) discussing [*lived population density*](https://www.bristolmathsresearch.org/2020/05/11/population-density-as-experienced-by-the-average-person-better-explains-variations-in-the-rate-of-spread-of-covid-19/). – kjetil b halvorsen Sep 12 '20 at 16:51
  • See also https://www.journals.uchicago.edu/doi/abs/10.1086/284994 – Nick Cox Sep 12 '20 at 17:57
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    @kjetilbhalvorsen Thank you! The population weighted density in the preprint uses exactly this definition! – Igor Rivin Sep 13 '20 at 14:04

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