Mathematics and the Iranian Elections

By Steven Miller, July 15, 2009

One of the many things I love about mathematics is that results initially discovered in one realm pop up in surprising places. A terrific example is Benford’s law of digit bias (those taking Math 341 in the fall will get to learn a lot more about this!). Benford’s law says that for many data sets, the probability of a first digit being d (base 10) is about log_{10}( 1 + 1/d), which means there’s about a 30% chance of a leading digit of 1 (the first digit of 1701 is a 1). It was initially discovered by Newcomb in the 1880s, who noticed certain pages of logarithm tables were more worn than others. It was rediscovered and popularized by Benford in the 1930s, who looked at a variety of natural and mathematical data sets.

One of my colleagues, Mark Nigrini, noticed that Benford’s law could be used to detect certain types of corporate tax fraud. The reason is that people are horrible random number generators, and thus certain ‘patterns’ are missing in the data. For example, if people are creating numbers they’ll often cluster the leading digit around 5, or spread them out uniformly from 1 to 9. Benford’s law is frequently used by the IRS and has successfully identified numerous cases of fraud.

Building on its success here, Benford’s law has been applied to numerous other data sets to test for fraud. One very important example is election fraud. While sometimes it is easy to detect fraud (there are instances where there are more votes than registered voters!), often it isn’t initially clear. Walter Mebane (University of Michigan) and others have pioneered applying tests built on Benford’s law to elections.

What does Benford have to say about the recent Iranian elections? The gist of the analyses I’ve read are that it is likely that fraud occurred. If you’re interested, the following links have more details. If you’d like more reading about Benford’s law, just let me know (sjm1 AT williams.edu).