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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How Labour risks becoming a party without a country

Without establishing the role of Labour in modern Britain, the party is unlikely ever to govern again.

“In my time of dying, want nobody to mourn

All I want for you to do is take my body home”

- Blind Willie Johnson

The Conservative Party is preparing itself for a bloody civil war. Conservative MPs will tell anyone who wants to know (Labour MPs and journalists included) that there are 100 Conservative MPs sitting on letters calling for a leadership contest. When? Whenever they want to. This impending war has many reasons: ancient feuds, bad blood, personal spite and enmity, thwarted ambition, and of course, the European Union.

Fundamentally, at the heart of the Tory war over the European Union is the vexed question of ‘What is Britain’s place in the World?’ That this question remains unanswered a quarter of a century after it first decimated the Conservative Party is not a sign that the Party is incapable of answering the question, but that it has no settled view on what the correct answer should be.

The war persists because the truth is that there is no compromise solution. The two competing answers are binary opposites: internationalist or insular nationalist, co-habitation is an impossibility.

The Tories, in any event, are prepared to keep on asking this question, seemingly to the point of destruction. For the most part, Labour has answered this question: Britain will succeed as an outward looking, internationalist state. The equally important question facing the Labour Party is ‘What is the place of the Labour Party in modern Britain?’ Without answering this question, Labour is unlikely to govern ever again and in contrast to the Tories, Labour has so far refused to acknowledge that such a question is being asked of it by the people it was founded to serve. At its heart, this is a question about England and the rapidly changing nature of the United Kingdom.

In the wake of the 2016 elections, the approach that Labour needs to take with regard to the ‘English question’ is more important than ever before. With Scotland out of reach for at least a generation (assuming it remains within the United Kingdom) and with Labour’s share of the vote falling back in Wales in the face of strong challenges from Plaid Cymru and UKIP, Labour will need to rely upon winning vast swathes of England if we are to form a government in 2020.

In a new book published this week, Labour’s Identity Crisis, Tristram Hunt has brought together Labour MPs, activists and parliamentary candidates from the 2015 general election to explore the challenges facing Labour in England and how the party should address these, not purely as an electoral device, but as a matter of principle.

My contribution to the book was inspired by Led Zeppelin’s Physical Graffiti. The track list reads like the score for a musical tragedy based upon the Labour Party from 2010 onwards: In My Time of Dying, Trampled Underfoot, Sick Again, Ten Years Gone. 

Continued Labour introspection is increasingly tiresome for the political commentariat – even boring – and Labour’s Identity Crisis is a genuinely exciting attempt to swinge through this inertia. As well as exploring our most recent failure, the book attempts to chart the course towards the next Labour victory: political cartography at its most urgent.

This collection of essays represents an overdue effort to answer the question that the Party has sought to sidestep for too long.  In the run up to 2020, as the United Kingdom continues to atomise, the Labour Party must have an ambitious, compelling vision for England, or else risks becoming a party without a country.

Jamie Reed is Labour MP for Copeland.