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.

Photo: Getty Images
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What do Labour's lost voters make of the Labour leadership candidates?

What does Newsnight's focus group make of the Labour leadership candidates?

Tonight on Newsnight, an IpsosMori focus group of former Labour voters talks about the four Labour leadership candidates. What did they make of the four candidates?

On Andy Burnham:

“He’s the old guard, with Yvette Cooper”

“It’s the same message they were trying to portray right up to the election”​

“I thought that he acknowledged the fact that they didn’t say sorry during the time of the election, and how can you expect people to vote for you when you’re not actually acknowledging that you were part of the problem”​

“Strongish leader, and at least he’s acknowledging and saying let’s move on from here as opposed to wishy washy”

“I was surprised how long he’d been in politics if he was talking about Tony Blair years – he doesn’t look old enough”

On Jeremy Corbyn:

"“He’s the older guy with the grey hair who’s got all the policies straight out of the sixties and is a bit of a hippy as well is what he comes across as” 

“I agree with most of what he said, I must admit, but I don’t think as a country we can afford his principles”

“He was just going to be the opposite of Conservatives, but there might be policies on the Conservative side that, y’know, might be good policies”

“I’ve heard in the paper he’s the favourite to win the Labour leadership. Well, if that was him, then I won’t be voting for Labour, put it that way”

“I think he’s a very good politician but he’s unelectable as a Prime Minister”

On Yvette Cooper

“She sounds quite positive doesn’t she – for families and their everyday issues”

“Bedroom tax, working tax credits, mainly mum things as well”

“We had Margaret Thatcher obviously years ago, and then I’ve always thought about it being a man, I wanted a man, thinking they were stronger…  she was very strong and decisive as well”

“She was very clear – more so than the other guy [Burnham]”

“I think she’s trying to play down her economics background to sort of distance herself from her husband… I think she’s dumbing herself down”

On Liz Kendall

“None of it came from the heart”

“She just sounds like someone’s told her to say something, it’s not coming from the heart, she needs passion”

“Rather than saying what she’s going to do, she’s attacking”

“She reminded me of a headteacher when she was standing there, and she was quite boring. She just didn’t seem to have any sort of personality, and you can’t imagine her being a leader of a party”

“With Liz Kendall and Andy Burnham there’s a lot of rhetoric but there doesn’t seem to be a lot of direction behind what they’re saying. There seems to be a lot of words but no action.”

And, finally, a piece of advice for all four candidates, should they win the leadership election:

“Get down on your hands and knees and start praying”

Stephen Bush is editor of the Staggers, the New Statesman’s political blog.