Mathematical analysis has been showing some odd outcomes in the Iranian election results. There have been a few posts on this already so I thought I’d try out the analysis for myself.

In a truly random distribution, you would expect to see each of the digits 0 – 9 to appear 10% of the time. When the numbers are manufactured (i.e. by a human) they usually do not meet this equal distribution. Here’s why … humans are awful at picking random numbers. We tend to pick numbers like 7 more often than numbers like 2, 4 & 5. Marketers use this to their advantage. Ever seen a headlines like “7 reasons why…”, “7 ways you can…”, etc.? The number 7 makes us feel that there must be some true reasoning or analysis behind the statement. A headline starting with “10 reasons why…” just isn’t as convincing. It doesn’t feel random enough to most humans.

Taking the results published on the recent Iranian Election (from the Interior Ministry), we can create a random set of data by using the last digit in each province’s vote count. Here’s what the distribution looks like:

You can see that the number 7 occurs disproportionately more frequently than other numbers. The number 5 appears 6% less than where we expect it to be.

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.

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.

The Devil is in the Digits, Washington Post

I made my own comparison with some Australian election data. Here I have used AEC data from the 2007 Federal Election, House of Representatives 2 party preferred vote by Polling Place. I was actually shocked to see how closely the results match the 10% expected distribution. Either Australia is far advanced in creating realistic fraudulent votes or we’re model statistical citizens.

According to Beber and Scacco, the last two digits also highlight some interesting anomolies.

But that’s not all. Psychologists have also found 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. To check for deviations of this type, we examined the pairs of last and second-to-last digits in Iran’s vote counts. On average, if the results had not been manipulated, 70 percent of these pairs should consist of distinct, non-adjacent digits.

Not so in the data from Iran: 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 while our first test — variation in last-digit frequencies — suggests that Rezai’s vote counts are the most irregular, the lack of non-adjacent digits is most striking in the results reported for Ahmadinejad.

The Devil is in the Digits, Washington Post

Benford’s Law

Another type of analysis involves looking at the leading number of the each result. Benford’s Law predicts that these numbers should be distributed in a “specific non-uniform way.” Here are the result of the Iranian election:

Whilst most of the numbers follow the expected pattern, the number 7 leads the results almost twice as much as is predicted. The numbers 6 & 8 are also quite low. Whether this is “statistically significant” I’m not sure – perhaps someone else can help with that bit of maths.

For comparison, here are the Australian results:

This is certainly not proof of a fraudulent election in Iran but does raise some interesting doubts about whether or not the results were doctored by human hands.

I’ve included a copy of my Excel sheets below if you want to do your own maths and graphing.

• Iran Election Results (Excel)
• Australian Election Results (Excel)

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