Benford’s law and the Iranian election

By Michael Banks

What do street house numbers, death rates and election results have in common?

They all follow a law, devised by physicist Frank Benford in 1938, which states that in a list of numbers from real-life data there are more entries that start with the digit “1” than any other number.

According to Benford’s law, numbers that begin with “1” occur almost 30% of the time in a list of numbers that are distributed logarithmically, such as house numbers. The higher the number the less it occurs, to the point where numbers that begin with “9” occur less than 5% of the time.

This law also turns out to be useful for checking fraudulent behaviour, for example, finding out if people have made up number on their tax return forms.

Now, however, cosmologist Boudewijn Roukema, from the Nicolas Copernicus University in Poland, has used this law to test the results from the recent Iranian election.

On 12 June it was announced that Mahmoud Ahmadinejad, the current Iranian president, had won the election beating main rival Mir-Hossein Mousavi. Protests then broke out in Iran disputing the results.

Then on 14 June the Iranian Ministry of the Interior released the results of the 2009 Iranian election for 366 voting areas giving Mahmoud Ahmadinejad over 24 million votes and Mir-Hossein Mousavi around 13 million votes.

Roukema noticed a strange anomaly in the votes for Mehdi Karroubi from the National Trust Party, who came in third place. He found that the number seven occurs as a first digit more often than would be expected by Benford’s law.

He found that this anomaly occurs in three of the six largest voting areas and, moreover, that Mahmoud Ahmadinejad had a greater proportion of votes in these three areas than the others.

Roukema concludes that this could suggest an error in the official count of around one million votes.

However, he says that applying Benford’s law may not be able to find every “anomaly” in the election results – meaning the difference could be more significant.

“The fact that use of the first digit detected a significant anomaly in this particular case only indicates that this anomaly somehow failed to be hidden,” says Roukema. “It certainly doesn’t guarantee it’s the only anomaly.”

Meanwhile, Elham Kashefi and Vincent Danos from Edinburgh University have started collecting signatures for an appeal to call for fresh elections and to oppose violence against protesters.