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.

Photo: Getty Images
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Why are boundary changes bad for Labour?

New boundaries, a smaller House of Commons and the shift to individual electoral registration all tilt the electoral battlefield further towards the Conservatives. Why?

The government has confirmed it will push ahead with plans to reduce the House of Commons to 600 seats from 650.  Why is that such bad news for the Labour Party? 

The damage is twofold. The switch to individual electoral registration will hurt Labour more than its rivals. . Constituency boundaries in Britain are drawn on registered electors, not by population - the average seat has around 70,000 voters but a population of 90,000, although there are significant variations within that. On the whole, at present, Labour MPs tend to have seats with fewer voters than their Conservative counterparts. These changes were halted by the Liberal Democrats in the coalition years but are now back on course.

The new, 600-member constituencies will all but eliminate those variations on mainland Britain, although the Isle of Wight, and the Scottish island constituencies will remain special cases. The net effect will be to reduce the number of Labour seats - and to make the remaining seats more marginal. (Of the 50 seats that would have been eradicated had the 2013 review taken place, 35 were held by Labour, including deputy leader Tom Watson's seat of West Bromwich East.)

Why will Labour seats become more marginal? For the most part, as seats expand, they will take on increasing numbers of suburban and rural voters, who tend to vote Conservative. The city of Leicester is a good example: currently the city sends three Labour MPs to Westminster, each with large majorities. Under boundary changes, all three could become more marginal as they take on more wards from the surrounding county. Liz Kendall's Leicester West seat is likely to have a particularly large influx of Tory voters, turning the seat - a Labour stronghold since 1945 - into a marginal. 

The pattern is fairly consistent throughout the United Kingdom - Labour safe seats either vanishing or becoming marginal or even Tory seats. On Merseyside, three seats - Frank Field's Birkenhead, a Labour seat since 1950, and two marginal Labour held seats, Wirral South and Wirral West - will become two: a safe Labour seat, and a safe Conservative seat on the Wirral. Lillian Greenwood, the Shadow Transport Secretary, would see her Nottingham seat take more of the Nottinghamshire countryside, becoming a Conservative-held marginal. 

The traffic - at least in the 2013 review - was not entirely one-way. Jane Ellison, the Tory MP for Battersea, would find herself fighting a seat with a notional Labour majority of just under 3,000, as opposed to her current majority of close to 8,000. 

But the net effect of the boundary review and the shrinking of the size of the House of Commons would be to the advantage of the Conservatives. If the 2015 election had been held using the 2013 boundaries, the Tories would have a majority of 22 – and Labour would have just 216 seats against 232 now.

It may be, however, that Labour dodges a bullet – because while the boundary changes would have given the Conservatives a bigger majority, they would have significantly fewer MPs – down to 311 from 330, a loss of 19 members of Parliament. Although the whips are attempting to steady the nerves of backbenchers about the potential loss of their seats, that the number of Conservative MPs who face involuntary retirement due to boundary changes is bigger than the party’s parliamentary majority may force a U-Turn.

That said, Labour’s relatively weak electoral showing may calm jittery Tory MPs. Two months into Ed Miliband’s leadership, Labour averaged 39 per cent in the polls. They got 31 per cent of the vote in 2015. Two months into Tony Blair’s leadership, Labour were on 53 per cent of the vote. They got 43 per cent of the vote. A month and a half into Jeremy Corbyn’s leadership, Labour is on 31 per cent of the vote.  A Blair-style drop of ten points would see the Tories net 388 seats under the new boundaries, with Labour on 131. A smaller Miliband-style drop would give the Conservatives 364, and leave Labour with 153 MPs.  

On Labour’s current trajectory, Tory MPs who lose out due to boundary changes may feel comfortable in their chances of picking up a seat elsewhere. 

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