My final thoughts on those Iran vote analyses:
From Florida in 2000 and Ohio in 2004 to Mexico City in 2006 to Iran in 2009, tightly contested elections are often accompanied by claims of fraud or serious error: that is, that the election outcome does not match the intentions of the people who voted. Sometimes there is direct evidence of fraud: people voting multiple times, tampering with ballot boxes, etc., and often there is evidence of mistakes, including overvotes (such as a person choosing a candidate and also writing in his name), lost ballots, and technical problems with voting machines.
But the the usual sort of evidence for major problems is a discrepancy between the overall election outcome and what was expected from polls or from extrapolation from other elections. A notorious example is Patrick Buchanan's votes on the "butterfly ballot" in Palm Beach in 2000, which were inconsistent with patterns in other Florida counties in that year.
For the Iran election, the natural step is to compare to previous election returns and look for large changes, as was done by political scientist Walter Mebane. Striking patterns found in such a comparison to not prove fraud but can be useful in giving people a sense of where to focus attention if they want to look further.
Scacco and Beber's analysis is based on the idea that, if there is election fraud, the cheaters are probably acting in a hurry and with various constraints on what numbers they can actually manipulate. As a result, the fake numbers might show some patterns that would be highly unlikely to be seen in tallies of real votes. Again, it is hard for such circumstantial evidence to be entirely convincing on its own, but the patterns they find can support particular theories of how the vote totals came to be.
Another way to calibrate our understanding of such statistical tests is to apply them to a large number of actual elections to see where apparent anomalies appear. Are anomalies happening pretty much at random, as might be expected if one were simply trawling through the data looking for patterns, or do they actually coincide with elections known to be suspicious or fraudulent on other grounds?
Even if statistical tests cannot prove fraud, they can help the news media and observers on the ground to focus their inquiries.