Anomalies in the Iranian election

While the results of the recent Iranian presidential election are currently severely contested, with accusations of fraud and manipulations, and with an level of protest unheard of in Iran, I had not so far seen a statistical analysis of the votes. This is over: Boudewijn Roukema, a cosmologist with the University of Toruń (Poland), has produced an analysis of the figures published by the Iranian Ministry of the Interior, based on Benford’s Law for the repartition of the first digit i in decimal representations of real numbers, which should be

f(i) \propto \log_{10}(1+\frac{1}{i})

for the proportion of votes for a candidate among the four in competition. Roukema exhibits a very unlikely discrepancy on Mehdi Karoubi’s votes, with an extremely high occurence of the digit 7. There is also a discrepancy for Mahmoud Ahmadinejad’s frequencies of 1’s and 2’s that is harder to detect because of the higher frequency of votes for this candidate in the Iranian Ministry of the Interior data. But looking at the most populous districts, Roukema concludes that several million votes could have been added to Ahmadinejad’s votes in those areas, if Benford’s Law holds…

I find this analysis produced merely five days after the election quite astounding, even though the validity of applying Benford’s Law in those circumstances needs more backup…

6 Responses to “Anomalies in the Iranian election”

  1. […] (Some of those recipients are or were researchers at CREST. And Elyès is my colleague in Paris-Dauphine. When he is not minister in Tunisia!) The 2011 winner is Xavier Gabaix, who is professor of economics at NUY. I know nothing of his research and of its impact on Economics, nor do I want to to criticise the 2011 prize in any respect, however in a fairly bland and uninformative interview with Le Monde, Xavier Gabaix focused on the Zipf laws (connected with the Benford law I mentioned a while ago about the Iranian elections): […]

  2. […] to Probability Theory (volume 2) gets Benford’s Law “wrong”. While my interest in Benford’s Law is rather superficial, I find the paper of interest as it shows a confusion […]

  3. […] ArXiv by the cosmologist Boudewijn Roukema, but I first heard about it myself via a pingback from another wordpress blog.  The same blogger has written a subsequent analysis […]

  4. Andrew Gelman has also posted later last night his comments on why using Benford’s Law does not convince him… And there are indeed other analyses on this on the Web, like Nate Silver’s post that date back to the first day the figures were released. But Roukema’s analysis is the first paper on the topic!

  5. One needs to be very, very careful when applying Benford’s law to election results.

    From the Carter Center’s final report about the 2004 referendum in Venezuela, where similar claims were made:


    4. The failure of recall referendum vote totals to
    conform to “Benford’s Law” governing the frequency
    of the first and second digits in those totals.”


    The panel concludes that there is insufficient
    evidence that Benford’s Law applies to election results
    in general. Furthermore, a simple but plausible model
    of the election does not produce results that conform
    to Benford’s Law.”


    The panel believes that there are many reasons to
    doubt the applicability of Benford’s Law to election
    returns. In particular, Benford’s Law is characteristic
    for scale-invariant data, while election machines are
    allocated to maintain a relatively constant number of
    voters per machine.”

    “In short, Benford’s Law does not generally
    apply to electoral data and even in cases where we suspect
    that it might apply, we find that it does not. All in
    all, Benford’s Law seems like a very weak instrument
    for detecting voting fraud. There are many reasons to
    believe that it does not apply to electoral data, and
    empirical tests suggest that deviations from the law are
    not necessarily indicative of fraud.”

  6. A few people have been looking at this stuff, including Nate Silver et al at and Walter Mebane at (see the “Papers for downloading” section). They all point out some discrepancies that suggest vote rigging but perhaps not on the scale that would change the outcome.

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