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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Brexit is teaching the UK that it needs immigrants

Finally forced to confront the economic consequences of low migration, ministers are abandoning the easy rhetoric of the past.

Why did the UK vote to leave the EU? For conservatives, Brexit was about regaining parliamentary sovereignty. For socialists it was about escaping the single market. For still more it was a chance to punish David Cameron and George Osborne. But supreme among the causes was the desire to reduce immigration.

For years, as the government repeatedly missed its target to limit net migration to "tens of thousands", the EU provided a convenient scapegoat. The free movement of people allegedly made this ambition unachievable (even as non-European migration oustripped that from the continent). When Cameron, the author of the target, was later forced to argue that the price of leaving the EU was nevertheless too great, voters were unsurprisingly unconvinced.

But though the Leave campaign vowed to gain "control" of immigration, it was careful never to set a formal target. As many of its senior figures knew, reducing net migration to "tens of thousands" a year would come at an economic price (immigrants make a net fiscal contribution of £7bn a year). An OBR study found that with zero net migration, public sector debt would rise to 145 per cent of GDP by 2062-63, while with high net migration it would fall to 73 per cent. For the UK, with its poor productivity and sub-par infrastructure, immigration has long been an economic boon. 

When Theresa May became Prime Minister, some cabinet members hoped that she would abolish the net migration target in a "Nixon goes to China" moment. But rather than retreating, the former Home Secretary doubled down. She regards the target as essential on both political and policy grounds (and has rejected pleas to exempt foreign students). But though the same goal endures, Brexit is forcing ministers to reveal a rarely spoken truth: Britain needs immigrants.

Those who boasted during the referendum of their desire to reduce the number of newcomers have been forced to qualify their remarks. On last night's Question Time, Brexit secretary David Davis conceded that immigration woud not invariably fall following Brexit. "I cannot imagine that the policy will be anything other than that which is in the national interest, which means that from time to time we’ll need more, from time to time we’ll need less migrants."

Though Davis insisted that the government would eventually meet its "tens of thousands" target (while sounding rather unconvinced), he added: "The simple truth is that we have to manage this problem. You’ve got industry dependent on migrants. You’ve got social welfare, the national health service. You have to make sure they continue to work."

As my colleague Julia Rampen has charted, Davis's colleagues have inserted similar caveats. Andrea Leadsom, the Environment Secretary, who warned during the referendum that EU immigration could “overwhelm” Britain, has told farmers that she recognises “how important seasonal labour from the EU is to the everyday running of your businesses”. Others, such as the Health Secretary, Jeremy Hunt, the Business Secretary, Greg Clark, and the Communities Secretary, Sajid Javid, have issued similar guarantees to employers. Brexit is fuelling immigration nimbyism: “Fewer migrants, please, but not in my sector.”

The UK’s vote to leave the EU – and May’s decision to pursue a "hard Brexit" – has deprived the government of a convenient alibi for high immigration. Finally forced to confront the economic consequences of low migration, ministers are abandoning the easy rhetoric of the past. Brexit may have been caused by the supposed costs of immigration but it is becoming an education in its benefits.

George Eaton is political editor of the New Statesman.