Running The Stats Again

A new analysis:

The results of the 2009 Iranian presidential election presented by the Iranian Ministry of the Interior (MOI) are analysed based on Benford's Law and an empirical variant of Benford's Law. The null hypothesis that the vote count distributions satisfy these distributions is rejected at a significance of $p e 0.007$, based on the presence of 41 vote counts for candidate K that start with the digit 7, compared to an expected 21.2--22 occurrences expected for the null hypothesis. A less significant anomaly suggested by Benford's Law could be interpreted as an overestimate of candidate A's total vote count by several million votes.

Possible signs of further anomalies are that the logarithmic vote count distributions of A, R, and K are positively skewed by 4.6, 5.8, and 2.5 standard errors in the skewness respectively, i.e. they are inconsistent with a log-normal distribution with $ p sim 4 imes 10^{-6}, 7 imes 10^{-9},$ and $1.2 imes 10^{-2}$ respectively. M's distribution is not significantly skewed.

From Wiki:

Benford's law, also called the first-digit law, states that in lists of numbers from many (but not all) real-life sources of data, the leading digit is distributed in a specific, non-uniform way. According to this law, the first digit is 1 almost one third of the time, and larger digits occur as the leading digit with lower and lower frequency, to the point where 9 as a first digit occurs less than one time in twenty. This distribution of first digits arises logically whenever a set of values is distributed logarithmically. For reasons described below, real-world measurements are often distributed logarithmically (or equivalently, the logarithm of the measurements is distributed uniformly).