### Does Benford's Law apply to election results?

Yesterday I blogged about an analysis I did of the Iranian Presidential election results and showed that they seemed to match closely with Benford's Law in the first and second digits.

A few people asked me if it would be expected that election results meet Benford's Law and in reading up on the research in this area I wasn't very convinced that they would always work. I read a number of papers, including this one and thought its justification was weak.

I figured the best way to test this was to go get some real election data from a country that I assume is not electorally corrupt and run exactly the same test. I obtained the per-constituency election results for the 2001 UK General Election and ran the same test against the Labour and Conservative party candidates.

The result is that these results do not match Benford's Law at all in either first or second digits. Here are the relevant graphs.

For the first digit we get chi-squared values well above the critical level for both Labour and Conservative which indicates that there is no correlation and these are not Benford's Law distributed. The same applies to the second digits.

I suspect this is because of the way the size of constituencies is constrained in the UK. They are subject to resizing and try to balance local geography (e.g. borough boundaries) and keep to an average size. In fact, there's not a lot of variation in constituency size. This point is actually made by the paper that I refer to above, although it claims that the second-digit would be significant.

So in this case Benford's Law seems to tell us nothing because of the underlying shape of the data.

So, I'm left with the curious conundrum of the Iranian results which seem to follow Benford's Law rather nicely and the British ones that don't. Could this simply be to do with the way in which the voting areas (British constituencies and Iranian counties) are organized?

At any rate it doesn't seem to me that you can just take election data, apply Benford's Law and come to any useful conclusion. Looks like the Carter Center agrees.

A few people asked me if it would be expected that election results meet Benford's Law and in reading up on the research in this area I wasn't very convinced that they would always work. I read a number of papers, including this one and thought its justification was weak.

I figured the best way to test this was to go get some real election data from a country that I assume is not electorally corrupt and run exactly the same test. I obtained the per-constituency election results for the 2001 UK General Election and ran the same test against the Labour and Conservative party candidates.

The result is that these results do not match Benford's Law at all in either first or second digits. Here are the relevant graphs.

For the first digit we get chi-squared values well above the critical level for both Labour and Conservative which indicates that there is no correlation and these are not Benford's Law distributed. The same applies to the second digits.

I suspect this is because of the way the size of constituencies is constrained in the UK. They are subject to resizing and try to balance local geography (e.g. borough boundaries) and keep to an average size. In fact, there's not a lot of variation in constituency size. This point is actually made by the paper that I refer to above, although it claims that the second-digit would be significant.

So in this case Benford's Law seems to tell us nothing because of the underlying shape of the data.

So, I'm left with the curious conundrum of the Iranian results which seem to follow Benford's Law rather nicely and the British ones that don't. Could this simply be to do with the way in which the voting areas (British constituencies and Iranian counties) are organized?

At any rate it doesn't seem to me that you can just take election data, apply Benford's Law and come to any useful conclusion. Looks like the Carter Center agrees.

Labels: pseudo-randomness

*If you enjoyed this blog post, you might enjoy my travel book for people interested in science and technology: The Geek Atlas. Signed copies of The Geek Atlas are available.*

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