Election fraud by the numbers

Two graduate students at Columbia University, Bernd Beber and Alexandra Scacco, have published a compelling statistical demonstration that the disputed Iranian vote counts are fraudulent. For example, focusing on the final digits in the vote tallies reported province by province for each of the four presidential candidates, they point out that some numerals are significantly over- or under-represented. The numeral 7 appears 17% of the time as the final digit, whereas the numeral 5 appears only 4% of the time (when on average a numeral should appear 10% of the time). Since the final digit in a large vote tally can be treated as a random occurrence, it's possible to calculate the probability that such large divergences in final-digit frequencies would arise in authentic vote tallies. The authors put that probability at just under 4%. A professional statistician once remarked to me that when an occurrence has a calculated probability of no more than 3 to 4%, it's pretty safe to conclude that the event in question was not random.

The authors also describe a second (unrelated) test of probability, which makes it that much less likely that the reported Iranian vote tallies are authentic (random) rather than generated through fraud.

I'll simply note that that authors could have performed a third test, with similar results: Some numerals are significantly over- or under-represented when you look at the second-to-last digit in the vote tallies (i.e. the "tens-column"). The numerals in this column as well ought to be randomly generated in an authentic election. But the numeral 7 once again is greatly over-represented (16%), while the numeral 3 is significantly under-represented (5%). I won't bother to crunch the numbers to demonstrate that this result, too, is rather improbable. To quote Beber and Scacco:

As a point of comparison, we can analyze the state-by-state vote counts for John McCain and Barack Obama in last year's U.S. presidential election. The frequencies of last digits in these election returns never rise above 14 percent or fall below 6 percent, a pattern we would expect to see in seventy out of a hundred fair elections.

Thus the frequencies with which the numerals 7 and 3 appear as the penultimate digits of the Iranian vote tallies further reduce the already slim probability that those results are authentic.

crossposted at unbossed.com