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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The Fire Brigades Union reaffiliates to Labour - what does it mean?

Any union rejoining Labour will be welcomed by most in the party - but the impact on the party's internal politics will be smaller than you think.

The Fire Brigades Union (FBU) has voted to reaffiliate to the Labour party, in what is seen as a boost to Jeremy Corbyn. What does it mean for Labour’s internal politics?

Firstly, technically, the FBU has never affliated before as they are notionally part of the civil service - however, following the firefighters' strike in 2004, they decisively broke with Labour.

The main impact will be felt on the floor of Labour party conference. Although the FBU’s membership – at around 38,000 – is too small to have a material effect on the outcome of votes themselves, it will change the tenor of the motions put before party conference.

The FBU’s leadership is not only to the left of most unions in the Trades Union Congress (TUC), it is more inclined to bring motions relating to foreign affairs than other unions with similar politics (it is more internationalist in focus than, say, the PCS, another union that may affiliate due to Corbyn’s leadership). Motions on Israel/Palestine, the nuclear deterrent, and other issues, will find more support from FBU delegates than it has from other affiliated trade unions.

In terms of the balance of power between the affiliated unions themselves, the FBU’s re-entry into Labour politics is unlikely to be much of a gamechanger. Trade union positions, elected by trade union delegates at conference, are unlikely to be moved leftwards by the reaffiliation of the FBU. Unite, the GMB, Unison and Usdaw are all large enough to all-but-guarantee themselves a seat around the NEC. Community, a small centrist union, has already lost its place on the NEC in favour of the bakers’ union, which is more aligned to Tom Watson than Jeremy Corbyn.

Matt Wrack, the FBU’s General Secretary, will be a genuine ally to Corbyn and John McDonnell. Len McCluskey and Dave Prentis were both bounced into endorsing Corbyn by their executives and did so less than wholeheartedly. Tim Roache, the newly-elected General Secretary of the GMB, has publicly supported Corbyn but is seen as a more moderate voice at the TUC. Only Dave Ward of the Communication Workers’ Union, who lent staff and resources to both Corbyn’s campaign team and to the parliamentary staff of Corbyn and McDonnell, is truly on side.

The impact of reaffiliation may be felt more keenly in local parties. The FBU’s membership looks small in real terms compared Unite and Unison have memberships of over a million, while the GMB and Usdaw are around the half-a-million mark, but is much more impressive when you consider that there are just 48,000 firefighters in Britain. This may make them more likely to participate in internal elections than other affiliated trade unionists, just 60,000 of whom voted in the Labour leadership election in 2015. However, it is worth noting that it is statistically unlikely most firefighters are Corbynites - those that are will mostly have already joined themselves. The affiliation, while a morale boost for many in the Labour party, is unlikely to prove as significant to the direction of the party as the outcome of Unison’s general secretary election or the struggle for power at the top of Unite in 2018. 

Stephen Bush is editor of the Staggers, the New Statesman’s political blog.