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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Will Euroscepticism prove an unbeatable advantage in the Conservative leadership race?

Conservative members who are eager for Brexit are still searching for a heavyweight champion - and they could yet inherit the earth.

Put your money on Liam Fox? The former Defence Secretary has been given a boost by the news that ConservativeHome’s rolling survey of party members preferences for the next Conservative leader. Jeremy Wilson at BusinessInsider and James Millar at the Sunday Post have both tipped Fox for the top job.

Are they right? The expectation among Conservative MPs is that there will be several candidates from the Tory right: Dominic Raab, Priti Patel and potentially Owen Paterson could all be candidates, while Boris Johnson, in the words of one: “rides both horses – is he the candidate of the left, of the right, or both?”

MPs will whittle down the field of candidates to a top two, who will then be voted on by the membership.  (As Graham Brady, chair of the 1922 Committee, notes in his interview with my colleague George Eaton, Conservative MPs could choose to offer a wider field if they so desired, but would be unlikely to surrender more power to party activists.)

The extreme likelihood is that that contest will be between two candidates: George Osborne and not-George Osborne.  “We know that the Chancellor has a bye to the final,” one minister observes, “But once you’re in the final – well, then it’s anyone’s game.”

Could “not-George Osborne” be Liam Fox? Well, the difficulty, as one MP observes, is we don’t really know what the Conservative leadership election is about:

“We don’t even know what the questions are to which the candidates will attempt to present themselves as the answer. Usually, that question would be: who can win us the election? But now that Labour have Corbyn, that question is taken care of.”

So what’s the question that MPs will be asking? We simply don’t know – and it may be that they come to a very different conclusion to their members, just as in 2001, when Ken Clarke won among MPs – before being defeated in a landslide by Conservative activists.

Much depends not only on the outcome of the European referendum, but also on its conduct. If the contest is particularly bruising, it may be that MPs are looking for a candidate who will “heal and settle”, in the words of one. That would disadvantage Fox, who will likely be a combative presence in the European referendum, and could benefit Boris Johnson, who, as one MP put it, “rides both horses” and will be less intimately linked with the referendum and its outcome than Osborne.

But equally, it could be that Euroscepticism proves to be a less powerful card than we currently expect. Ignoring the not inconsiderable organisational hurdles that have to be cleared to beat Theresa May, Boris Johnson, and potentially any or all of the “next generation” of Sajid Javid, Nicky Morgan or Stephen Crabb, we simply don’t know what the reaction of Conservative members to the In-Out referendum will be.

Firstly, there’s a non-trivial possibility that Leave could still win, despite its difficulties at centre-forward. The incentive to “reward” an Outer will be smaller. But if Britain votes to Remain – and if that vote is seen by Conservative members as the result of “dirty tricks” by the Conservative leadership – it could be that many members, far from sticking around for another three to four years to vote in the election, simply decide to leave. The last time that Cameron went against the dearest instincts of many of his party grassroots, the result was victory for the Prime Minister – and an activist base that, as the result of defections to Ukip and cancelled membership fees, is more socially liberal and more sympathetic to Cameron than it was before. Don’t forget that, for all the worry about “entryism” in the Labour leadership, it was “exitism” – of Labour members who supported David Miliband and liked the New Labour years  - that shifted that party towards Jeremy Corbyn.

It could be that if – as Brady predicts in this week’s New Statesman – the final two is an Inner and an Outer, the Eurosceptic candidate finds that the members who might have backed them are simply no longer around.

It comes back to the biggest known unknown in the race to succeed Cameron: Conservative members. For the first time in British political history, a Prime Minister will be chosen, not by MPs with an electoral mandate of their own or by voters at a general election but by an entirelyself-selecting group: party members. And we simply don't know enough about what they feel - yet. 

Stephen Bush is editor of the Staggers, the New Statesman’s political blog. He usually writes about politics.