Political scientists Bernd Beber and Alexandra Scacco have found some peculiarities in the Iran election results. While many people around the world are questioning the validity of the results, Beber and Scacco have gone a step further and looked at some statistical anomalies.
Beber and Scacco looked at the final digit of the results from each province for the top four candidates, and found two anomalies. Since there are ten possible digits, we would expect each number 0 - 9 to appear in about 10 percent of all the precinct totals, give or take a few percentage points due to random variability.
We wouldn't expect the number 7 to be the last digit for 17% of all precinct totals, or the number 5 to be the last digit in only 4% of all precincts. But that's exactly what Beber and Scacco found. The chances of seeing two such unusual distributions in a fair election are about 3.7%. On the other hand, when people attempt to create random-looking numbers, they often pick numbers ending with 7.
Another pattern found in human-generated numbers is what are called adjacent digits. People tend to follow a digit with the next higher or next lower digit, e.g. 23 or 54. In the Iran election totals Beber and Scacco looked at the final two digits from each precint, and found that fully 38% of all the pairs of digits were adjacent digits. The chances of seeing this in a fair election are about 4.2%.
So we have two unlikely scenarios, independently appearing in the same set of election results. Beber and Scacco calculate the chances of seeing both of these anomalies at half of a percent. In other words, we would expect to see results like this in only one of every two hundred fair elections.
When we combine these anomalies with the discrepancy between vote results and pre-election polling, and the fact that in many cities the vote totals were greater than the number of eligible voters, it becomes very hard to believe this election is in any way legitimate.