Rise and Flaw of Internet's Election-Fraud Hunters

Protesters on the streets of Tehran questioning the recent Iranian presidential election results have gotten support from a new breed of election watchers: Internet-enabled anomaly hounds who say the numbers don't add up.

Fraud hunters are no newer to elections than conspiracy theorists are to the Internet. But unlike election monitors seeking voter tampering or intimidation, or local experts who critique faulty ballot design or study pre-election polling data, these statistical analysts don't need to know anything about the dynamics of an individual race. Their toolkit is primarily statistical and can be applied to any numbers, voting or otherwise. The Internet provides quick access to election numbers and speedy dissemination of findings.

Yet their methods are unproven and risky. It's impossible to test their efficacy on prior elections, because we don't know which ones were tainted. Suspected election riggers don't confess their crimes. So these tools aren't sufficient to prove fraud. Also, the more you look for fraud, the more likely you are to stumble on signs of it, even if it isn't there.

"Wouldn't it be nice to have the equivalent of DNA testing for elections?" asks Walter Mebane, a political scientist at the University of Michigan. He says his work and colleagues' can't live up to the certainty of findings on "CSI."

"In television programs, the person always confesses, I guess because they run out of time," he says. "But that certainly doesn't happen in real life."

Prof. Mebane is one of the analysts who dug through vote totals from thousands of ballot boxes in Iran. They suspected that the election wasn't democratic just by looking at the vote counts. Of particular fishiness: There were too many 0s in the second digit of these totals and too few 5s in the last digit.

Just how many 0s and 5s to expect hinges on a law rediscovered by General Electric physicist Frank Benford in 1938, after mathematician Simon Newcomb had noted the same principle in 1881. Both men were clued in to what's now known as Benford's Law by logarithm books, tools used frequently before calculators by engineers and physicists. The early pages of these books, listing the logarithms of numbers with low leading digits (such as 128), were more heavily worn than later pages. They surmised that such numbers occur more often than those with high leading digits (such as 876).

The reason is that many sets of numbers operate like an investment that grows exponentially. Consider a $1 investment that grows at 10% per year. It will remain between $1 and $2 for 7.3 years, but will grow from $9 to $10 in just 1.1 years. Because it passes through that second phase so much faster, a snapshot of all such investments at any one time will be likely to find about 6½ investments with a first digit of 1 for every one with a first digit of 9. [The same principle applies to the length of rivers, and the population of cities.]

It turns out that lower digits should also predominate in the second position, though the effect is less pronounced. Benford's Law dictates about 1.4 numbers with a second digit of 0 (say, 203) for every one number with 9 in that position (293).

While accountants have used this tool to test financial numbers, it isn't known to have caught any prominent book-cookers. That's partly because accountants don't like tipping fraudsters to their techniques. An IRS spokesman, for example, would say only that Benford's Law is one of many techniques that could be used, and that the IRS uses some of these techniques.

[Mark Nigrini, a professor of accounting at the College of New Jersey and popularizer of the law, says he has consulted with companies that have used it successfully. One example: It found too many 4s as leading digits in one employee's expense accounting, which was caused by profligate expensing of daily Starbucks purchases. Of course, the law couldn't account for whether that was an authorized repeat expense. (It wasn't.)]But do election returns behave like investments? Some observers say they don't. The Carter Center, a nonprofit organization that monitors elections for fraud, was among those naysayers after analyzing a disputed Venezuelan recall referendum in 2004.

The Carter Center also pointed out that such a test could ensnare elections deemed clean. One possible reason: Precinct sizes are often similar, which could result in too many 1s or high numbers. In a close two-candidate race where most precincts have around 1,800 votes, for example, many of the candidates' totals by precinct would be between 800 and 999, an apparent violation of Benford.

"Benford's Law seems like a very weak instrument for detecting voting fraud," Henry Brady, a political-science professor at the University of California, Berkeley, wrote at the time. But Prof. Brady today says Benford's Law can help analyze elections. He notes that his pessimistic conclusion applies to analyses using leading digits. But the second digit is a better indicator of election fraud, he says, and other sleuths agree.

Using second digits of vote totals for candidates in Iran, Prof. Mebane found anomalous results, which he's published online with near-daily updates in recent weeks. The discrepancies "strongly suggest there was ballot box stuffing," Prof. Mebane wrote.

Write to Carl Bialik at numbersguy@wsj.com