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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The biggest divide in politics is not left against right, but liberals against authoritarians

My week, including a Lib Dem membership rise, The Avalanches, and why I'm putting pressure on Theresa May over child refugees.

It is a boost for us that Nick Clegg has agreed to return to the front line and be our Brexit spokesperson. I hadn’t even had a chance at our meeting to make him the offer when he said: “Before we start, I’ve been thinking about this and want to take on the fight over Europe.”

With Labour apparently willing to give the Tories a free pass to take us out of Europe, the Liberal Democrats are the only UK-wide party that will go into the next election campaigning to maintain our membership of the EU. The stage is remarkably clear for us to remind Theresa May precisely what she would be risking if we abandon free trade, free movement, environmental protection, workers’ rights and cross-border security co-operation. More than a month on from the referendum, all we have heard from the Tories is that “Brexit means Brexit” – but they have given us no clue that they understand what that means.

 

Premature obituaries

Not long ago, the received wisdom was that all political parties were dying – but lately the supposed corpses have twitched into life. True, many who have joined Labour’s ranks are so hard left that they don’t see winning elections as a primary (or even a desirable) purpose of a party, and opening up Labour to those with a very different agenda could ultimately destroy it.

Our experience has been happier: 20,000 people joined the Liberal Democrat fightback in the wake of the 2015 general election result, and 17,000 more have joined since the referendum. We now have more members than at any time this century.

 

Breaking up is hard to do

Journalists have been asking repeatedly if I want to see the break-up of the Labour Party, with moderates defecting to the Liberal Democrats. I have been clear that I am not a home-wrecker and it is for Labour to determine its own future, just as I focus on advancing the Liberal Democrat cause. Yet I have also been clear that I am happy for my party to be a home for liberals of whatever hue. I enjoyed campaigning in the referendum with a variety of progressive figures, just as moderates from different parties shared platforms in 1975. It struck me that far more unites us than divides us.

That said, not all “moderate” Labour figures could be described as “liberal”, as John Reid demonstrated as Labour home secretary. The modern political divide is less left v right than authoritarian v liberal. Both left and right are looking increasingly authoritarian and outright nasty, with fewer voices prepared to stand up for liberal values.

 

What I did on my holidays

Time off has been virtually non-existent, but I am reading A Wilderness of Mirrors by Mark Meynell (about loss of trust in politics, the media and just about everything). I’m also obsessively listening to Wildflower by the Avalanches, their second album, 16 years after their first. It’s outstanding – almost 60 minutes of intelligently crafted dialogue, samples and epic production.

During the political maelstrom, I have been thinking back to the idyllic few days I spent over half-term on the Scottish island of Colonsay: swimming in the sea with the kids (very cold but strangely exhilarating ­after a decent jog), running and walking. An added bonus is that Colonsay is the smallest island in the world to have its own brewery. I can now heartily recommend it.

 

Preparing for the next fight

The odds are weirdly long on an early general election, but I refuse to be complacent – and not merely because the bookies were so wrong about Brexit. If we have learned one truth about Theresa May as Prime Minister so far, it is that she is utterly ruthless. After her savage cabinet sackings, this is, in effect, a new government. She has refused to go to the country, even though she lectured Gordon Brown on the need to gain the endorsement of the electorate when he replaced Tony Blair. Perhaps she doesn’t care much about legitimacy, but she cares about power.

You can be sure that she will be keeping half an eye on Labour’s leadership election. With Jeremy Corbyn potentially reconfirmed as leader in September against the wishes of three-quarters of his MPs, Mrs May might conclude that she will never have a better chance to increase her narrow majority. Throw in the possibility that the economy worsens next year as Brexit starts to bite, and I rule nothing out.

So, we are already selecting candidates. It is vital that they dig in early. As we are the only party prepared to make the positive case for Europe, such an election would present us with an amazing opportunity.

 

Sitting Priti

David Cameron pledged to take an unspecified number of unaccompanied children from camps across the Continent. I am putting pressure on Theresa May to turn that vague commitment into a proper plan. Having visited such camps, I have been fighting for Britain to give sanctuary to a minimum of 3,000 unaccompanied children, who are currently open to the worst kinds of exploitation. We have heard nothing but silence from the government, with underfunded councils reporting that they are not receiving the help they need from Whitehall.

Meanwhile, it remains government policy to send refugees to Turkey – whose increasingly authoritarian government has just suspended human rights protection.

As if all of this were not grim enough, we have a new Secretary of State for International Development, Priti Patel, who has said that she thinks aid should be used largely to promote trade. As someone who wants our country to be respected around the world, I find this plain embarrassing. Actually, it’s worse. It’s shaming. As with Europe, so with the world: the ­Conservative government is hauling up the drawbridge just when we need more than ever to engage with people beyond our shores.

Tim Farron is the leader of the Liberal Democrats. To join the party, visit: libdems.org.uk/join

Tim Farron is leader of the Liberal Democrats.

This article first appeared in the 28 July 2016 issue of the New Statesman, Summer Double Issue