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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Jeremy Corbyn's Labour conference speech shows how he's grown

The leader's confident address will have impressed even his fiercest foes. 

It is not just Jeremy Corbyn’s mandate that has been improved by his re-election. The Labour leader’s conference speech was, by some distance, the best he has delivered. He spoke with far greater confidence, clarity and energy than previously. From its self-deprecating opening onwards ("Virgin Trains assure me there are 800 empty seats") we saw a leader improved in almost every respect. 

Even Corbyn’s firecest foes will have found less to take issue with than they may have anticipated. He avoided picking a fight on Trident (unlike last year), delivered his most forceful condemnation of anti-Semitism (“an evil”) and, with the exception of the Iraq war, avoided attacks on New Labour’s record. The video which preceded his arrival, and highlighted achievements from the Blair-Brown years, was another olive branch. But deselection, which Corbyn again refused to denounce, will remain a running sore (MPs alleged that Hillsborough campaigner Sheila Coleman, who introduced Corbyn, is seeking to deselect Louise Ellman and backed the rival TUSC last May).

Corbyn is frequently charged with lacking policies. But his lengthy address contained several new ones: the removal of the cap on council borrowing (allowing an extra 60,000 houses to be built), a ban on arms sales to abusive regimes and an arts pupil premium in every primary school.

On policy, Corbyn frequently resembles Ed Miliband in his more radical moments, unrestrained by Ed Balls and other shadow cabinet members. He promised £500bn of infrastructure investment (spread over a decade with £150bn from the private sector), “a real living wage”, the renationalisation of the railways, rent controls and a ban on zero-hours contracts.

Labour’s greatest divisions are not over policy but rules, strategy and culture. Corbyn’s opponents will charge him with doing far too little to appeal to the unconverted - Conservative voters most of all. But he spoke with greater conviction than before of preparing for a general election (acknowledging that Labour faced an arithmetical “mountain”) and successfully delivered the attack lines he has often shunned.

“Even Theresa May gets it, that people want change,” he said. “That’s why she stood on the steps of Downing Street and talked about the inequalities and burning injustices in today’s Britain. She promised a country: ‘that works not for a privileged few but for every one of us’. But even if she manages to talk the talk, she can’t walk the walk. This isn’t a new government, it’s David Cameron’s government repackaged with progressive slogans but with a new harsh right-wing edge, taking the country backwards and dithering before the historic challenges of Brexit.”

After a second landslide victory, Corbyn is, for now, unassailable. Many MPs, having voted no confidence in him, will never serve on the frontbench. But an increasing number, recognising Corbyn’s immovability, speak once again of seeking to “make it work”. For all the ructions of this summer, Corbyn’s speech will have helped to persuade them that they can.

George Eaton is political editor of the New Statesman.