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.

Getty
Show Hide image

Manchester united: "A minority of absolute idiots are trying to break us apart"

At the vigil, one man's T-shirt read: "The only thing that's allowed to be separated by colour is the laundry."

A day after one of the worst atrocities in the history of the city, Manchester's people were keen to show the world the resilience of the Mancunian spirit.

Dom's, an Italian restaurant, is in walking distance from Manchester Arena, where 22 people lost their lives to a suicide bomber the night before. On Tuesday, the staff were giving out free coffee, tea and pizza to anyone who needed it. On a table outside, there was a condolences book, and teary passersby left RIP messages to those who perished. Under a bright blue sky, the community seemed more united than ever, the goodwill pouring out of everyone I met. But the general mood was sombre. 

"We need to make space for healing and for building up our community again, and just getting people to feel comfortable in their own city," the Dean of Manchester, Rogers Govendor, told me.

The terrorist has been named as Salman Ramadan Abedi, a 22-year-old Mancunian of Libyan descent. But with a population of 600,000, Manchester is a cosmopolitan hub, and proud of it. Throughout the day I encountered people of all skin shades and religions. On one of the roads off Albert Square, a couple of Orthodox Jewish boys set up a little stand, where people could grab a bottle of water and, if they so desired, hold hands and pray.

On the night of the tragedy, Muslim and Sikh cab drivers turned off the meter and made their way to Manchester Arena to offer free rides to anyone - many of them injured - who trying to escape the mayhem and reach safety. "It's what we do around here," my taxi driver said with a thick Arabic accent.

The dissonance between the increasingly frantic debate on social media and what was discussed on the streets was stark. I spoke, on and off the record, with about two dozen residents, eavesdropped on a number of conversations, and not once did I hear anyone speaking out against the cultural melting pot that Manchester is today. If anything, people were more eager than ever to highlight it. 

"Manchester has always been hugely multicultural, and people always pull together at times of trouble and need," said Andrew Hicklin. "They are not going to change our society and who we are as people. We live free lives."

It was also a day where political divisions were put aside. Theresa May and Jeremy Corbyn agreed to suspend their campaigns. For the next few days there will be no Labour vs Tory, no Brexiteer vs Remainer, at least not in this part of the country. This city has closed ranks and nothing will be allowed to come between that cohesion.

"I don't demonise anyone," said Dennis Bolster, who stopped by to sign the condolences book outside Dom's. "I just know a small minority of absolute idiots, driven by whatever they think they are driven by, are the people who are trying to break us apart."

Later in the day, as people were getting off work, thousands flocked to Albert Square to show their respects to the victims. Members of the Sikh community entered the square carrying "I love MCR" signs. The crowd promptly applauded. A middle-aged man wore a T-shirt which said: "The only thing that's allowed to be separated by colour is the laundry." A moment of silent was observed. It was eerie, at times overwhelmingly sad. But it was also moving and inspiring.

Local poet Tony Walsh brought brief respite from the pain when he recited "This is the Place", his ode to the city and its people. The first verse went:

This is the place In the north-west of England. It’s ace, it’s the best

And the songs that we sing from the stands, from our bands

Set the whole planet shaking.

Our inventions are legends. There’s nowt we can’t make, and so we make brilliant music

We make brilliant bands

We make goals that make souls leap from seats in the stands

On stage, everyday political foes became temporary allies. Labour leader Jeremy Corbyn, home secretary Amber Rudd, Lib Dem leader Tim Farron, Mayor of Greater Manchester Andy Burnham and house speaker John Bercow all brushed shoulders. Their message was clear: "we are Manchester too."

The vigil lasted a little over half an hour. On other occasions, a crowd this size in the centre of Manchester would give authorities reason for concern. But not this time. Everyone was in their best behaviour. Only a few were drinking. 

As Mancunians made their way home, I went over to a family that had been standing not far from me during the vigil. The two children, a boy and a girl, both not older than 10, were clutching their parents' hands the whole time. I asked dad if he will give them a few extra hugs and kisses as he tucks them in tonight. "Oh, absolutely," he said. "Some parents whose children went to the concert last night won't ever get to do that again. It's heartbreaking."

Felipe Araujo is a freelance journalist based in London. He writes about race, culture and sports. He covered the Rio Olympics and Paralympics on the ground for the New Statesman. He tweets @felipethejourno.

0800 7318496