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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If the SNP truly want another referendum, the clock is ticking

At party conference in Glasgow, I heard Scotland’s governing party demand a future distinctly different from the one being sketched out in Westminster. 

Nicola Sturgeon described Glasgow as the “dear green city” in her opening address to the SNP party conference, which may surprise anyone raised on a diet of Ken Loach films. In fact, if you’re a fan of faded grandeur and nostalgic parks, there are few places to beat it. My morning walk to conference took me past chipped sandstone tenements, over a bridge across the mysterious, twisting River Kelvin, and through a long avenue of autumnal trees in Kelvingrove Park. In the evenings, the skyline bristled with Victorian Gothic university buildings and church spires, and the hipster bars turned on their lights.

In between these two walks, I heard Scotland’s governing party demand a future distinctly different from the one being sketched out in Westminster. Glasgow’s claim to being the UK’s second city expired long ago but I wonder if, post-Brexit, there might be a case for reviving it.



Scottish politics may never have looked more interesting, but at least one Glasgow taxi driver is already over it. All he hears in the back of his cab is “politics, fitba and religion”, he complained when he picked me up from the station. The message didn’t seem to have reached SNP delegates at the conference centre on the Clyde, who cheered any mention of another referendum.

The First Minister, though, seems to have sensed the nation’s weariness. Support for independence has fallen from 47 per cent in June (Survation) to 39 per cent in October (BMG Research). Sturgeon made headlines with the announcement of a draft referendum bill, but read her speeches carefully and nothing is off the table. SNP politicians made the same demands again and again – devolved control of immigration and access to the single market. None ruled out these happening while remaining in the UK.

If Sturgeon does want a soft Brexit deal, though, she must secure it fast. Most experts agree that it would be far easier for an independent Scotland to inherit Britain’s EU membership than for it to reapply. Once Article 50 is triggered, the SNP will be in a race against the clock.


The hare and the tortoise

If anyone is still in doubt about the SNP’s position, look who won the deputy leadership race. Angus Robertson, the gradualist leader of the party in the Commons, saw off a referendum-minded challenger, Tommy Sheppard, with 52.5 per cent of the vote.

Conference would be nothing without an independence rally, and on the final day supporters gathered for one outside. A stall sold “Indyref 2” T-shirts but the grass-roots members I spoke to were patient, at least for now. William Prowse, resplendent in a kilt and a waistcoat covered in pro-indy
badges, remains supportive of Sturgeon. “The reason she has not called an Indy 2 vote
is we need to have the right numbers,” he told me. “She’s playing the right game.”

Jordi McArthur, a member for 30 years, stood nearby waving a flagpole with the Scottish, Welsh and Catalan flags side by side. “We’re happy to wait until we know what is happening with Brexit,” he said. “But at the same time, we want a referendum. It won’t be Nicola’s choice. It will be the grass roots’ choice.”


No Gerrymandering

Party leaders may come and go, but SNP members can rely on one thing at conference – the stage invasions of the pensioner Gerry Fisher. A legendary dissenter, Fisher refused this year to play along with the party’s embrace of the EU. Clutching the
lectern stubbornly, he told members: “Don’t tell me that you can be independent and a member of the EU. It’s factually rubbish.” In the press room, where conference proceedings were shown unrelentingly on a big screen, hacks stopped what they were doing to cheer him on.


Back to black

No SNP conference would be complete without a glimpse of Mhairi Black, the straight-talking slayer of Douglas Alexander and Westminster’s Baby of the House. She is a celebrity among my millennial friends – a video of her maiden Commons speech has been watched more than 700,000 times – and her relative silence in recent months is making them anxious.

I was determined to track her down, so I set my alarm for an unearthly hour and joined a queue of middle-aged women at an early-morning fringe event. The SNP has taken up the cause of the Waspi (Women Against State Pension Inequality) campaign, run by a group of women born in the 1950s whose retirement age has been delayed and are demanding compensation. Black, who is 22, has become their most ­articulate spokeswoman.

The event started but her chair remained unfilled. When she did arrive, halfway through the session, it was straight from the airport. She gave a rip-roaring speech that momentarily convinced even Waspi sceptics like me, and then dashed off to her next appointment.


Family stories

Woven through the SNP conference was an argument about the benefits of immigration (currently controlled by Westminster). This culminated in an appearance by the Brain family, whose attempt to resist deportation back to Australia has made them a national cause célèbre. (Their young son has learned to speak Gaelic.) Yet for me, the most emotional moment of the conference was when another family, the Chhokars, stepped on stage. Surjit Singh Chhokar was murdered in 1998, but it took 17 years of campaigning and a change in double jeopardy laws before his killer could be brought to justice.

As Aamer Anwar, the family’s solicitor, told the story of “Scotland’s Stephen Lawrence”, Chhokar’s mother and sister stood listening silently, still stricken with grief. After he finished, the delegates gave the family a standing ovation.

Julia Rampen is the editor of The Staggers, the New Statesman’s politics blog

Julia Rampen is the editor of The Staggers, The New Statesman's online rolling politics blog. She was previously deputy editor at Mirror Money Online and has worked as a financial journalist for several trade magazines. 

This article first appeared in the 20 October 2016 issue of the New Statesman, Brothers in blood