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Even this explanation is incorrect. Evenly sampled data across many orders of magnitude give an even distribution of leading numbers. See my comment for code here: https://news.ycombinator.com/item?id=21541264

There are many distributions which do lead to the distribution of leading digits that follow Benford's law, just not uniform. Nor "uniform over several orders of magnitude".

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> Thus, real-world distributions that span several orders of magnitude rather uniformly

They mean uniformly on a log distribution. Ie, there are a uniform number of elements between 10^x and 10^(x+1), for many different values of x. This is best shown by the graph here:

https://en.wikipedia.org/wiki/Benford%27s_law#Overview

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