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
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The government needs more on airports than just Chris Grayling's hunch

This disastrous plan to expand Heathrow will fail, vows Tom Brake. 

I ought to stop being surprised by Theresa May’s decision making. After all, in her short time as Prime Minister she has made a series of terrible decisions. First, we had Chief Buffoon, Boris Johnson appointed as Foreign Secretary to represent the United Kingdom around the world. Then May, announced full steam ahead with the most extreme version of Brexit, causing mass economic uncertainty before we’ve even begun negotiations with the EU. And now we have the announcement that expansion of Heathrow Airport, in the form of a third runway, will go ahead: a colossally expensive, environmentally disastrous, and ill-advised decision.

In the House of Commons on Tuesday, I asked Transport Secretary Chris Grayling why the government is “disregarding widespread hostility and bulldozing through a third runway, which will inflict crippling noise, significant climate change effects, health-damaging air pollution and catastrophic congestion on a million Londoners.” His response was nothing more than “because we don’t believe it’s going to do those things.”

I find this astonishing. It appears that the government is proceeding with a multi-billion pound project with Grayling’s beliefs as evidence. Why does the government believe that a country of our size should focus on one major airport in an already overcrowded South East? Germany has multiple major airports, Spain three, the French, Italians, and Japanese have at least two. And I find it astonishing that the government is paying such little heed to our legal and moral environmental obligations.

One of my first acts as an MP nineteen years ago was to set out the Liberal Democrat opposition to the expansion of Heathrow or any airport in southeast England. The United Kingdom has a huge imbalance between the London and the South East, and the rest of the country. This imbalance is a serious issue which our government must get to work remedying. Unfortunately, the expansion of Heathrow does just the opposite - it further concentrates government spending and private investment on this overcrowded corner of the country.

Transport for London estimates that to make the necessary upgrades to transport links around Heathrow will be £10-£20 billion pounds. Heathrow airport is reportedly willing to pay only £1billion of those costs. Without upgrades to the Tube and rail links, the impact on London’s already clogged roads will be substantial. Any diversion of investment from improving TfL’s wider network to lines serving Heathrow would be catastrophic for the capital. And it will not be welcomed by Londoners who already face a daily ordeal of crowded tubes and traffic-delayed buses. In the unlikely event that the government agrees to fund this shortfall, this would be salt in the wound for the South-West, the North, and other parts of the country already deprived of funding for improved rail and road links.

Increased congestion in the capital will not only raise the collective blood pressure of Londoners, but will have severe detrimental effects on our already dire levels of air pollution. During each of the last ten years, air pollution levels have been breached at multiple sites around Heathrow. While a large proportion of this air pollution is caused by surface transport serving Heathrow, a third more planes arriving and departing adds yet more particulates to the air. Even without expansion, it is imperative that we work out how to clean this toxic air. Barrelling ahead without doing so is irresponsible, doing nothing but harm our planet and shorten the lives of those living in west London.

We need an innovative, forward-looking strategy. We need to make transferring to a train to Cardiff after a flight from Dubai as straightforward and simple as transferring to another flight is now. We need to invest in better rail links so travelling by train to the centre of Glasgow or Edinburgh is quicker than flying. Expanding Heathrow means missing our climate change targets is a certainty; it makes life a misery for those who live around the airport and it diverts precious Government spending from other more worthy projects.

The Prime Minister would be wise to heed her own advice to the 2008 government and “recognise widespread hostility to Heathrow expansion.” The decision to build a third runway at Heathrow is the wrong one and if she refuses to U-turn she will soon discover the true extent of the opposition to these plans.

Tom Brake is the Liberal Democrat MP for Carshalton & Wallington.