## Politics

Does a statistical property named Benford’s Law point toward fraud in the Iranian elections? That’s one possible reading of a new paper (.pdf) by Boudewijn Roukema of Nicolaus Coprenicus University in Toruń, Poland. I think the paper is intriguing, but like Andrew (yes, we’re both writing on the same subject), I also have one or two reservations.

First, let me explain in a bit more detail what Benford’s Law is. Or actually, let me let Wikipedia explain:

Benford’s law, also called the first-digit law, states that in lists of numbers from many (but not all) real-life sources of data, the leading digit is distributed in a specific, non-uniform way. According to this law, the first digit is 1 almost one third of the time, and larger digits occur as the leading digit with lower and lower frequency, to the point where 9 as a first digit occurs less than one time in twenty. This distribution of first digits arises logically whenever a set of values is distributed logarithmically […]

This counter-intuitive result has been found to apply to a wide variety of data sets, including electricity bills, street addresses, stock prices, population numbers, death rates, lengths of rivers, physical and mathematical constants, and processes described by power laws (which are very common in nature).

The specific distribution of first digits (in the number 2,684, two is the first digit) that Benford’s law forecasts is as follows:

Wikipedia calls this property counter-intuitive, but I don’t know that it’s entirely so. For instance, think about the number of daily visitors to the millions of websites that are out there in the world, which classically follows a power-law distribution. There are a lot more websites that have 1,000-some visitors a day than 9,000-some visitors a day, and there a lot more websites that have 100-some visitors a day than 900-some visitors a day. For that matter, there are a lot more websites that have 1 visitor a day than 9 visitors a day. Website traffic very probably obeys Benford’s law or something approaching it.

Or, to give you an example where I actually have some numbers to show you, let’s look at the first digit for all places (cities and down) in California as of the 2000 Census.

This distribution obeys Benford’s Law almost perfectly.

### The Republican Establishment Is Waiting On The SidelinesSep 3, 2015

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