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In a suspicious election involving 5 candidates, the announced votes for each candidate are a multiple of 3. The chance of all five numbers being a multiple of 3 is only 0.41%. Therefore, some people conclude there has been foul play (votes are multiplied by 3 to show the number of voters as higher for political reasons).

I am not sure if we can make this conclusion with certainty. I think the hypothesis we should consider is whether all votes are multiplied by a constant (since if they were multiplied by 5, we would also be suspicious). Therefore, the probability should be calculated for all numbers greater than or equal to 3 (all numbers being a multiple of 2 could be likely).

This is actually a true story that is unfolding right now. The question is whether we can reject the hypothesis of the votes being reported as they were in reality or not. Does anyone know the proper way to answer this question?

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  • $\begingroup$ One thing to consider is that the probability of all five candidates having a multiple of 3 votes is 1/243. There have been far more than 243 elections involving five candidates throughout history, so this scenario should have happened many previous times purely by chance. $\endgroup$
    – JaredS
    Commented Jun 29 at 22:13
  • $\begingroup$ @JaredS In the subject country this is the first election with 5 candidates. Should we condition on the subject country or the entire history of mankind? $\endgroup$
    – Amin
    Commented Jun 30 at 7:13
  • $\begingroup$ It would be more appropriate to evaluate the history of elections in general rather than the subject country in isolation. Statistical anomalies in small samples can be misleading, whereas larger datasets provide a clearer picture of what can be expected by chance. Either way, I doubt any conclusions can be drawn from a statistic like this alone. It would require in-depth investigation, which may or may not happen. $\endgroup$
    – JaredS
    Commented Jul 1 at 13:47
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    $\begingroup$ +1 Any hypothesis formulated post hoc based on a chance observation like this is difficult to evaluate. Arguably, the hypothesis you need to test is $H_0:$ "some kind of arithmetical regularity or pattern in the vote counts or any other aspect of the election exists that would draw the attention of somebody." Because this is so vague, the best you can hope for is to obtain a lower bound for your test. Even worse, the question might need to be broadened to whether such a pattern would be noted in some election, in which case the p-value is essentially 1.0. $\endgroup$
    – whuber ♦
    Commented Jul 1 at 15:16

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Is this a parliamentary election, for a parliament with many seats? If so, a 1/243 likelihood is not very surprising.

Are any details of the counting method available? For example if these are national totals in a presidential election, are the state totals also published? If so, and the addition was accurate, then fixing the national total to a multiple of three would require collaborators in every state, which seems improbable.

It's hard to think of any way to rig an election that would be revealed by this test.

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  • $\begingroup$ These are national totals in a presidential election. State totals are available and they sum to the total number but it's possible that every state's vote count is multiplied by three resulting in a total that is also a multiple of 3. State authorities are assigned by the government itself. $\endgroup$
    – Amin
    Commented Jul 2 at 11:18

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