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If a particular dataset has a lognormal distribution, will it also follow Zipf's law when the items are ranked?

That is, say I have a set of populations of a random sampling of cities (assumed to be a lognormal distribution). If I ordered them by rank, would their values be predicted by some power curve?

If so, given a mean and standard deviation of a lognormal distribution, how can I derive the power curve that Zipf's law describes?

If not, what type of distribution has the quality where when it's items are ranked, they follow Zipf's law? And also what type of curve best approximates a ranked list of items from a lognormal distribution?

Thanks.

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1 Answer 1

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A lognormal distribution will not produce a Zipf power law type result: the bottom half of the distribution will look very different to the top half. Here is an example with 1 million samples from a "standard" lognormal distribution

enter image description here

though if you only look at the highest values (the right-hand tail of the distribution) then you are not that far away

If you want the whole distribution to look like a Zipf law then you want something like a Pareto distribution, with a similar example with the shape parameter 2:

enter image description here

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Fantastic answer, thank you. One follow-up question: What is the curve that describes ranked lognormal values that is drawn above? –  math_idiot Feb 16 '13 at 23:27
    
Follow my link to Wikipedia. I used the CDF: $\Pr(X \le x)=1 -\dfrac{1}{x^2}$ for $x \ge 1$, i.e. with a density $p(x)=\dfrac{2}{x^3}$ –  Henry Feb 22 '13 at 22:18

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