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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What David Hockney has to tell us about football

Why the sudden glut of blond footballers? A conversation I had with the artist back in 1966 gave me a clue. . .

In 1966, I went to interview David Hockney at a rather run-down flat in Bayswater, central London. He was 28 and had just won a gold medal at the Royal College of Art.

In his lavatory, I noticed a cut-out photograph from a newspaper of Denis Law scoring a goal. I asked if he was a football fan. He said no, he just liked Denis Law’s thighs.

The sub-editors cut that remark out of the story, to save any gossip or legal problems. In 1966 homosexual activity could still be an offence.

Hockney and a friend had recently been in the United States and had been watching an advert on TV that said “Blondes have more fun”. At two o’clock in the morning, slightly drunk, they both went out, bought some hair dye and became blond. Hockney decided to remain blond from then on, though he has naturally dark hair.

Is it true that blonds have more fun? Lionel Messi presumably thinks so, otherwise why has he greeted this brand-new season with that weird blond hair? We look at his face, his figure, his posture and we know it’s him – then we blink, thinking what the heck, does he realise some joker has been pouring stuff on his head?

He has always been such a staid, old-fashioned-looking lad, never messing around with his hair till now. Neymar, beside him, has gone even blonder, but somehow we expect it of him. He had foony hair even before he left Brazil.

Over here, blonds are popping up all over the shop. Most teams now have a born-again blondie. It must take a fortune for Marouane Fellaini of Man United to brighten up his hair, as he has so much. But it’s already fading. Cheapskate.

Mesut Özil of Arsenal held back, not going the full head, just bits of it, which I suspect is a clue to his wavering, hesitant personality. His colleague Aaron Ramsey has almost the full blond monty. Paul Pogba of Man United has a sort of blond streak, more like a marker pen than a makeover. His colleague Phil Jones has appeared blond, but he seems to have disappeared from the team sheet. Samir Nasri of Man City went startlingly blond, but is on loan to Seville, so we’re not able to enjoy his locks. And Didier Ndong of Sunderland is a striking blond, thanks to gallons of bleach.

Remember the Romanians in the 1998 World Cup? They suddenly appeared blond, every one of them. God, that was brilliant. One of my all-time best World Cup moments, and I was at Wembley in 1966.

So, why do they do it? Well, Hockney was right, in a sense. Not to have more fun – meaning more sex – because top footballers are more than well supplied, but because their normal working lives are on the whole devoid of fun.

They can’t stuff their faces with fast food, drink themselves stupid, stay up all night, take a few silly pills – which is what many of our healthy 25-year-old lads consider a reasonably fun evening. Nor can they spend all their millions on fun hols, such as skiing in the winter, a safari in the spring, or hang-gliding at the weekend. Prem players have to be so boringly sensible these days, or their foreign managers will be screaming at them in their funny foreign accents.

While not on the pitch, or training, which takes up only a few hours a day, the boredom is appalling, endlessly on planes or coaches or in some hotel that could be anywhere.

The only bright spot in the long days is to look in the mirror and think: “Hmm, I wonder what highlights would look like? I’ve done the beard and the tattoos. Now let’s go for blond. Wow, gorgeous.”

They influence each other, being simple souls, so when one dyes his hair, depending on where he is in the macho pecking order, others follow. They put in the day by looking at themselves. Harmless fun. Bless ’em.

But I expect all the faux blonds to have gone by Christmas. Along with Mourinho. I said that to myself the moment he arrived in Manchester, smirking away. Pep will see him off. OK then, let’s say Easter at the latest . . . 

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 22 September 2016 issue of the New Statesman, The New Times