Last week in Math Lint, I discussed how panic set in when the Shanghai Stock Market fell 64.89 points...on a special aniversary of a tragic day: June 4, 1989. Numerology was in effect!
Other instances exist. For example, consider the Iran's presidential election in June of 2009. The primary concern was the published vote counts across Iran's 29 provinces, with a special focus on the last and second-to-last digits of these vote totals...as reported by U.S. poltical scientists Bernd Beber and Alexandra Scacco.
Their first complaint regarded last digits: "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."
Their second complaint regarded lasy digit pairs: "Only 62 percent of the pairs contain non-adjacent digits. This may not sound so different from 70 percent, but the probability that a fair election would produce a difference this large is less than 4.2 percent."
And, overall, their conclusion: "Each of these two tests provides strong evidence that the numbers released by Iran's Ministry of the Interior were manipulated. But taken together, they leave very little room for reasonable doubt. The probability that a fair election would produce both too few non-adjacent digits and the suspicious deviations in last-digit frequencies described earlier is less than .005."
Now, is there a mathematical basis for this number-jumbo? First, research by cognitive psychologists have documented that "humans are bad at making up numbers." When asked to write sequences of random digits, we tend to uses some digits more often than others.
And, specific to two-digit pairs, cognitive psychologists have demonstrated that "humans have trouble generating non-adjacent digits (such as 64 or 17, as opposed to 23) as frequently as one would expect in a sequence of random numbers." Specific to the Irani vote, "if the results had not been manipulated, 70 percent of these pairs should consist of distinct, non-adjacent digits."
Thus, the willingness of suspicious authors to use statistics to claim that they are 99.5% certain that the Irani vote totals were fake (i.e. made up by humans). But, did this use of mathematics have any effect...well, I will let you determine that on your own.
And...use these two notions to monitor the state-by-state vote totals for the pending Presidential Election. Is it also rigged?
Source: B. Beber & A. Scacco's "The Devil Is in the Digits: Evidence That Iran's Election Was Rigged," Washington Post, 6/20/09