This has all been very interesting but not exactly useful. So, can Benford's Law be useful? The answer is yes. In fact, analyses using Benford's Law is admissible in the United States courts. To get an idea for some uses of this analysis, let's take a look at a moderately well-publicized case where Benford's law was used.

The 2009 Iranian presidential election committee gathered analyses into whether the elections were fraudulent or not. Some of these used Benford's Law. One major article on this was A first-digit anomaly in the 2009 Iranian presidential election by Boudewijn F. Roukema (http://arxiv.org/abs/0906.2789). In this study, the author analyzes the first digit of vote counts in the election results publicized by the Iranian Ministry of the Interior on June 14, 2009. First, he analyzed first-round results for elections in immediately preceding years in other countries. This established a baseline or control to compare with. He also took into account the pre-election polls. This allowed him to establish the immediate political landscape in which the election was conducted.

Roukema then used a bootstrap to obtain a sample of the votes. In applying an analysis of the votes using Benford's Law, he found that there was a significant number of more vote counts beginning with the digit 7 than could be predicated by Benford's Law. In fact, the frequency of 7 was more in line with the frequency of the digit 3.

In another study of the 2009 Iranian elections, Walter R. Mebane, a forensics expert, used Benford's Law to analyze the first and second digits of the vote counts. Based particularly on the second digits, he also found evidence of fraud, especially in the counts of two of the candidates.

This seems like it should be clear-cut. However, several other people looked at this situation with varying degrees of thoroughness and failed to find anything. Several people wrote blog posts about doing cursory inspections of the data using Benford's Law, without finding evidence of any problems.

The Carter Center also questioned whether Benford's Law applied to election data at all, and in The Irrelevance of Benford's Law for Detecting Fraud in Elections, Joseph Deckert, Mikhail Myagkov, and Peter C. Ordenshook looked at election data from Ohio, Massachusetts, and Ukraine as well as at simulations of elections and concluded that Benford's Law does not, in fact, indicate election fraud well. Deviations in the frequencies of first and second digits do not reliably indicate fraud, and actual fraud may push the distributions into more compliance with Benford's Law. Thus, for a number of reasons, Benford's Law may not work well with the election data.