A statistical trick which reveals whether MPs are lying about expenses

Benford's law has many uses. Can it trip up MPs?

Are politicians routinely making up expenses? A simple statistical test suggests not.

Benford's law is a statistical artefact found in numerical data spanning several orders of magnitude. Ben Goldacre explains:

Imagine you have data on, say, the population of every world nation. Now, take only the "leading digit" from each number: the first number in the number, if you like. For the UK population, which was 61,838,154 in 2009, that leading digit would be "six". Andorra's was 85,168, so that's "eight". And so on.

If you take all those leading digits, from all the countries, then overall, you might naively expect to see the same number of ones, fours, nines, and so on. But in fact, for naturally occurring data, you get more ones than twos, more twos than threes, and so on, all the way down to nine. This is Benford's law: the distribution of leading digits follows a logarithmic distribution, so you get a "one" most commonly, appearing as first digit around 30% of the time, and a nine as first digit only 5% of the time.

This pattern should repeat for almost any data which matches the key condition of spanning a large range of sizes. Take the example above, world populations, which goes from 800 in the Vatican City to 1.35 billion in China. But one category of data which rarely obeys the law is that where the numbers are made-up. When people are trying to "randomly" write down numbers, they rarely do it very well, more frequently following the intuition that random data ought to have just as much chance of starting with any given digit.

The value of MP's expenses certainly spans several orders of magnitude. Excluding repaid claims, expenses in the latest tranche, released last week, span from a value of 10p (reconciliation for a travelcard between Euston and Coventry) to £9900 (for staffing costs in Woking constituency office).

So does the data follow Benford's law? It largely does:

 

The largest variation is a 3 percentage point difference between the expected number of leading 2s and the actual number, with most other digits being present in slightly larger quantities than expected.

Scanning through the data, it's easy to see why this is. There are a large number of claims which are made repeatedly. For instance, 18 different MPs claimed £139.26 for the same twin pack of HP toner cartridges; while nearly every claim for petrol costs came in between £10 and £19.99, boosting the 1s' count again. Conversely, there simply weren't that many must-have services which began with a 2 (although a lot of things MPs need do, apparently, cost £20 on the dot, from venue hire to cleaning bills and car parking).

None of which means there may not still be fraud in the expenses. It simply means that the actual values being claimed for have been drawn from real life. MPs are not, on the whole, making up numbers on the spot as the fill in expense forms; whether what they are claiming for ought to be paid out of the public pocket, statistics are less likely to help with.

(As an aside, it's actually surprising that the figures match Benford's law quite so well; while MP's may not be choosing the numbers they submit, the people who set the prices clearly are. That's probably the reason for the slight uptick in the 9s, for instance; a lot of things which may cost £10 instead are charged as £9.99. It seems that there are either enough counter-examples that it gets balanced out, or lots of claims for things like mileage, which have no set price)

Two data CDs, much like the ones which sparked the original expenses scandal. Photograph: Getty Images

Alex Hern is a technology reporter for the Guardian. He was formerly staff writer at the New Statesman. You should follow Alex on Twitter.

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Jeremy Corbyn fans are getting extremely angry at the wrong Michael Foster

He didn't try to block the Labour leader off a ballot. He's just against hunting with dogs. 

Michael Foster was a Labour MP for Worcester from 1997 to 2010, where he was best known for trying to ban hunting with dogs. After losing his seat to Tory Robin Walker, he settled back into private life.

He quietly worked for a charity, and then a trade association. That is, until his doppelganger tried to get Jeremy Corbyn struck off the ballot paper. 

The Labour donor Michael Foster challenged Labour's National Executive Committee's decision to let Corbyn automatically run for leadership in court. He lost his bid, and Corbyn supporters celebrated.

And some of the most jubilant decided to tell Foster where to go. 

Foster told The Staggers he had received aggressive tweets: "I have had my photograph in the online edition of The Sun with the story. I had to ring them up and suggest they take it down. It is quite a common name."

Indeed, Michael Foster is such a common name that there were two Labour MPs with that name between 1997 and 2010. The other was Michael Jabez Foster, MP for Hastings and Rye. 

One senior Labour MP rang the Worcester Michael Foster up this week, believing he was the donor. 

Foster explained: "When I said I wasn't him, then he began to talk about the time he spent in Hastings with me which was the other Michael Foster."

Having two Michael Fosters in Parliament at the same time (the donor Michael Foster was never an MP) could sometimes prove useful. 

Foster said: "When I took the bill forward to ban hunting, he used to get quite a few of my death threats.

"Once I paid his pension - it came out of my salary."

Foster has never met the donor Michael Foster. An Owen Smith supporter, he admits "part of me" would have been pleased if he had managed to block Corbyn from the ballot paper, but believes it could have caused problems down the line.

He does however have a warning for Corbyn supporters: "If Jeremy wins, a place like Worcester will never have a Labour MP.

"I say that having years of working in the constituency. And Worcester has to be won by Labour as part of that tranche of seats to enable it to form a government."