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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Labour's establishment suspects a Momentum conspiracy - they're right

Bernie Sanders-style organisers are determined to rewire the party's machine.  

If you wanted to understand the basic dynamics of this year’s Labour leadership contest, Brighton and Hove District Labour Party is a good microcosm. On Saturday 9 July, a day before Angela Eagle was to announce her leadership bid, hundreds of members flooded into its AGM. Despite the room having a capacity of over 250, the meeting had to be held in three batches, with members forming an orderly queue. The result of the massive turnout was clear in political terms – pro-Corbyn candidates won every position on the local executive committee. 

Many in the room hailed the turnout and the result. But others claimed that some in the crowd had engaged in abuse and harassment.The national party decided that, rather than first investigate individuals, it would suspend Brighton and Hove. Add this to the national ban on local meetings and events during the leadership election, and it is easy to see why Labour seems to have an uneasy relationship with mass politics. To put it a less neutral way, the party machine is in a state of open warfare against Corbyn and his supporters.

Brighton and Hove illustrates how local activists have continued to organise – in an even more innovative and effective way than before. On Thursday 21 July, the week following the CLP’s suspension, the local Momentum group organised a mass meeting. More than 200 people showed up, with the mood defiant and pumped up.  Rather than listen to speeches, the room then became a road test for a new "campaign meetup", a more modestly titled version of the "barnstorms" used by the Bernie Sanders campaign. Activists broke up into small groups to discuss the strategy of the campaign and then even smaller groups to organise action on a very local level. By the end of the night, 20 phonebanking sessions had been planned at a branch level over the following week. 

In the past, organising inside the Labour Party was seen as a slightly cloak and dagger affair. When the Labour Party bureaucracy expelled leftwing activists in past decades, many on went further underground, organising in semi-secrecy. Now, Momentum is doing the exact opposite. 

The emphasis of the Corbyn campaign is on making its strategy, volunteer hubs and events listings as open and accessible as possible. Interactive maps will allow local activists to advertise hundreds of events, and then contact people in their area. When they gather to phonebank in they will be using a custom-built web app which will enable tens of thousands of callers to ring hundreds of thousands of numbers, from wherever they are.

As Momentum has learned to its cost, there is a trade-off between a campaign’s openness and its ability to stage manage events. But in the new politics of the Labour party, in which both the numbers of interested people and the capacity to connect with them directly are increasing exponentially, there is simply no contest. In order to win the next general election, Labour will have to master these tactics on a much bigger scale. The leadership election is the road test. 

Even many moderates seem to accept that the days of simply triangulating towards the centre and getting cozy with the Murdoch press are over. Labour needs to reach people and communities directly with an ambitious digital strategy and an army of self-organising activists. It is this kind of mass politics that delivered a "no" vote in Greece’s referendum on the terms of the Eurozone bailout last summer – defying pretty much the whole of the media, business and political establishment. 

The problem for Corbyn's challenger, Owen Smith, is that many of his backers have an open problem with this type of mass politics. Rather than investigate allegations of abuse, they have supported the suspension of CLPs. Rather than seeing the heightened emotions that come with mass mobilisations as side-effects which needs to be controlled, they have sought to joins unconnected acts of harassment, in order to smear Jeremy Corbyn. The MP Ben Bradshaw has even seemed to accuse Momentum of organising a conspiracy to physically attack Labour MPs.

The real conspiracy is much bigger than that. Hundreds of thousands of people are arriving, enthusiastic and determined, into the Labour party. These people, and their ability to convince the communities of which they are a part, threaten Britain’s political equilibrium, both the Conservatives and the Labour establishment. When the greatest hope for Labour becomes your greatest nightmare, you have good call to feel alarmed.