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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Let's face it: supporting Spurs is basically a form of charity

Now, for my biggest donation yet . . .

I gazed in awe at the new stadium, the future home of Spurs, wondering where my treasures will go. It is going to be one of the architectural wonders of the modern world (football stadia division), yet at the same time it seems ancient, archaic, a Roman ruin, very much like an amphitheatre I once saw in Croatia. It’s at the stage in a new construction when you can see all the bones and none of the flesh, with huge tiers soaring up into the sky. You can’t tell if it’s going or coming, a past perfect ruin or a perfect future model.

It has been so annoying at White Hart Lane this past year or so, having to walk round walkways and under awnings and dodge fences and hoardings, losing all sense of direction. Millions of pounds were being poured into what appeared to be a hole in the ground. The new stadium will replace part of one end of the present one, which was built in 1898. It has been hard not to be unaware of what’s going on, continually asking ourselves, as we take our seats: did the earth move for you?

Now, at long last, you can see what will be there, when it emerges from the scaffolding in another year. Awesome, of course. And, har, har, it will hold more people than Arsenal’s new home by 1,000 (61,000, as opposed to the puny Emirates, with only 60,000). At each home game, I am thinking about the future, wondering how my treasures will fare: will they be happy there?

No, I don’t mean Harry Kane, Danny Rose and Kyle Walker – local as well as national treasures. Not many Prem teams these days can boast quite as many English persons in their ranks. I mean my treasures, stuff wot I have been collecting these past 50 years.

About ten years ago, I went to a shareholders’ meeting at White Hart Lane when the embryonic plans for the new stadium were being announced. I stood up when questions were called for and asked the chairman, Daniel Levy, about having a museum in the new stadium. I told him that Man United had made £1m the previous year from their museum. Surely Spurs should make room for one in the brave new mega-stadium – to show off our long and proud history, delight the fans and all those interested in football history and make a few bob.

He mumbled something – fluent enough, as he did go to Cambridge – but gave nothing away, like the PM caught at Prime Minister’s Questions with an unexpected question.

But now it is going to happen. The people who are designing the museum are coming from Manchester to look at my treasures. They asked for a list but I said, “No chance.” I must have 2,000 items of Spurs memorabilia. I could be dead by the time I finish listing them. They’ll have to see them, in the flesh, and then they’ll be free to take away whatever they might consider worth having in the new museum.

I’m awfully kind that way, partly because I have always looked on supporting Spurs as a form of charity. You don’t expect any reward. Nor could you expect a great deal of pleasure, these past few decades, and certainly not the other day at Liverpool when they were shite. But you do want to help them, poor things.

I have been downsizing since my wife died, and since we sold our Loweswater house, and I’m now clearing out some of my treasures. I’ve donated a very rare Wordsworth book to Dove Cottage, five letters from Beatrix Potter to the Armitt Library in Ambleside, and handwritten Beatles lyrics to the British Library. If Beckham and I don’t get a knighthood in the next honours list, I will be spitting.

My Spurs stuff includes programmes going back to 1910, plus recent stuff like the Opus book, that monster publication, about the size of a black cab. Limited editions cost £8,000 a copy in 2007. I got mine free, as I did the introduction and loaned them photographs. I will be glad to get rid of it. It’s blocking the light in my room.

Perhaps, depending on what they want, and they might take nothing, I will ask for a small pourboire in return. Two free tickets in the new stadium. For life. Or longer . . . 

Hunter Davies is a journalist, broadcaster and profilic author perhaps best known for writing about the Beatles. He is an ardent Tottenham fan and writes a regular column on football for the New Statesman.

This article first appeared in the 16 February 2017 issue of the New Statesman, The New Times