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The Iranian elections of 2009, when Mahmoud Ahmedinejad beat Mir-Hussein Mousavi, among others, were widely believed to have been rigged.

The reader might recall that those elections were followed by riots on the streets of Tehran for several days in protest against the rigging.

One popular method used to determine whether a data set is doctored is to look at the last digits of the values. Normally, the last digit has no influence on the overall â€œwinners" since it is the least significant digit. However, the way the last digits of the numbers are distributed follows some special properties, and if the rigger has not been careful enough (usually, perhaps shockingly, they are not) the last digit of the rigged numbers can give away the rigging.

In an election, for example, there is no reason that the distribution of the last digits of the vote counts of various candidates should not be uniformâ€”given the large number of votes that each candidate gets, the last digit is essentially random, and there is no reason that the probability of a 1 in the units place is more than that of a 2 in the units place. Thus, in a free and fair election, it is likely that the last digit is distributed uniformly.

In figure 1, we look at the distribution of the last digits of the number of votes from the recently concluded Indian general election (data source: AC Nielsen). You can see that the distribution is fairly uniform, lending credence to the belief that Indian elections are largely fair.

Data for the last four Indian elections, too, shows a similar distribution.

Coming back to the Iranian elections of 2009, Bernd Beber and Alexandra Scacco (then of Columbia University) looked at the last digits of the provincial vote counts for each of the four candidates and found that the digit 7 appeared 17% of the time and the digit 5 appeared only 4% of the time, indicating that the numbers were very likely manipulated (you might remember that childhood game where you ask your friend to quickly tell you a random number between 5 and 12, and most of the time the reply is 7).

The question is if we can use the same technique to analyse the recent Afghan elections, which have ended in controversy. In the first round of the election held in April, Abdullah Abdullah got 45% of the vote, short of the 50% that would have made him an outright winner. He went into a run-off in June with second placed Ashraf Ghani, in which Ghani appears to have got more votes (the results of the elections are still in dispute).

We have data for the final vote tally of the preliminary elections held in April. We have data for eight candidates in each of the 34 provinces of Afghanistan. Can we look at the last digits of these and see if something is amiss? Graphing it, like we did for this summerâ€™s Indian elections, gives us this (see figure 2):

Notice how skewed this graph is, relative to the Indian elections. We would expect on an average 27 numbers to end in 0. What we see is that there are 37 numbers that end in 0. Similarly, we have 33 numbers ending in 5 and only 21 numbers ending in 6.

When you have 272 numbers, and each number from 0 to 9 can occur as the last digit with equal probability, if the numbers have been drawn at random, the probability that we have 37 or more zeros is only 2.25%. In other words, even ignoring the spike at 5 and trough at 6, the probability that the last digits of the votes received by the candidates in the Afghan elections follows the distribution shown by the above graph is 2.25%.

Which means that left to chance, there is only a 2.25% probability that the last digit of the vote counts could have followed this distribution. The observant reader will notice that this is incredibly low, lending credence to the hypothesis that the Afghan elections were rigged.

Finally, a note of caution. There are several ways in which an election can be rigged. Speaking broadly, it can be rigged at either the voting or the counting stages. This method of looking at the last digits only gives us an indication of the probability of rigging in the counting stages. Methods such as â€œballot stuffing" (reportedly not uncommon in India) cannot be caught with such methods.