I've got an educational game in the work that challenges players to redraw U.S. state borders in a way that reduce the inequality between big-population and small-population states. As they play, they need to get real-time feedback on the effect of assigning counties from one state to another.

So I need some sort of single parameter of "equalness" that is derived from the distribution of populations in the new hypothetical set of states.

I've thought about just displayed the standard deviation in the state populations. Is there a better way to succinctly express a power distribution? I'd prefer something as simple to understand as possible since I want to appeal to a broad audience. State populations as of 2010 Census below for reference.

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    $\begingroup$ "On average, how many strictly yes-no questions do you need to ask someone to find out their home state?" (The more the better.) $\endgroup$ – Kirill Sep 15 '14 at 4:05

Perhaps the best known measure would be the Gini index.

The R package ineq (See here) implements the Herfindahl and Rosenbluth concentration measures (in function conc).

It also implements a number of inequality indexes (including the Gini) in function ineq -- the Gini coefficient, Ricci-Schutz coefficient (also called Pietra’s measure), Atkinson’s measure, Kolm’s measure, Theil’s entropy measure, Theil’s second measure, the coefficient of variation and the squared coefficient of variation.

This answer mentions the Simpson diversity index, and derives a concentration measure from that. There are numerous other diversity indices (and thereby, other concentration measures). You'll probably note that there's a connection to the Herfindahl index (the Simpson diversity index is the Herfindahl, and the corresponding concentration measure is the normalized Herfindahl. In fact I just edited the other answer to point this out.)

[When dealing with count data, or proportions derived from counts, it's also possible to define measures derived from chi-square goodness-of-fit statistics (they can be normalized to 0-1), for example. For one such measure, see here.]

Many of these are either suitable or can be rescaled to be suitable as measures of the kind of thing you want.

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    $\begingroup$ This is a great start, thank you! I'm still a little worried about the accessibility for a general audience. Would a Pareto distribution be reasonable here? Something like "The top 5 states have 90% of the population" $\endgroup$ – Chris Wilson Sep 15 '14 at 14:59
  • $\begingroup$ But "the top 5 states have 90% of the population" is neither "inequality expressed in one number" nor "a Pareto distribution". It's a descriptive statistic but lacks properties you'd need from an index of concentration. What is it you need to do? $\endgroup$ – Glen_b Sep 15 '14 at 22:35
  • $\begingroup$ One very rough measure of concentration might be something like "proportion of states required to get 80% of the population" (or whatever other % you want to specify) - then the result of evaluating that could be "inequality expressed in one number". But it tends to be a bit "lumpy" ... change the data from one year to the next and nothing changes for years, then suddenly it changes by a relatively large amount. Something like the normalized Herfindahl index or the Gini coefficient is quite smooth; they are able to respond to more subtle trends. It really depends on what you need to achieve. $\endgroup$ – Glen_b Sep 15 '14 at 23:53
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    $\begingroup$ If you define your measure to be "the proportion of the population in the 5 most populous states", that might work okay. $\endgroup$ – Glen_b Sep 16 '14 at 0:00

I've developed a method for quantifying "uniformity" that allows you to do what you are asking. Its helped out a couple other folks too.

See: https://math.stackexchange.com/questions/921084/how-to-calculate-peakiness-or-uniformity-in-histogram/921110#921110

Basically, you are just calculating the path length of the associated CDF by connecting consecutive points by a straight line.

  • $\begingroup$ Some entropy measure ... $\endgroup$ – kjetil b halvorsen Feb 3 '17 at 23:46

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