Statistical proof the Iran Elections were rigged

Since the declaration of Mahmoud Ahmadinejad's landslide victory in Iran's presidential election, accusations of fraud have swelled. Against expectations from pollsters and pundits alike, Ahmadinejad did surprisingly well in urban areas, including Tehran -- where he is thought to be highly unpopular -- and even Tabriz, the capital city of opposition candidate Mir Hussein Mousavi's native East Azarbaijan province.

Others have pointed to the surprisingly poor performance of Mehdi Karroubi, another reform candidate, and particularly in his home province of Lorestan, where conservative candidates fared poorly in 2005, but where Ahmadinejad allegedly captured 71 percent of the vote. Eyebrows have been raised further by the relative consistency in Ahmadinejad's vote share across Iran's provinces, in spite of wide provincial variation in past elections.

Why would fraudulent numbers look any different? The reason is that humans are bad at making up numbers. Cognitive psychologists have found that study participants in lab experiments asked to write sequences of random digits will tend to select some digits more frequently than others.

So what can we make of Iran's election results? We used the results released by the Ministry of the Interior and published on the web site of Press TV, a news channel funded by Iran's government. The ministry provided data for 29 provinces, and we examined the number of votes each of the four main candidates -- Ahmadinejad, Mousavi, Karroubi and Mohsen Rezai -- is reported to have received in each of the provinces -- a total of 116 numbers.

The numbers look suspicious. We find too many 7s and not enough 5s in the last digit. We expect each digit (0, 1, 2, and so on) to appear at the end of 10 percent of the vote counts. But in Iran's provincial results, the digit 7 appears 17 percent of the time, and only 4 percent of the results end in the number 5. Two such departures from the average -- a spike of 17 percent or more in one digit and a drop to 4 percent or less in another -- are extremely unlikely. Fewer than four in a hundred non-fraudulent elections would produce such numbers.