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 rise of the green mayor – Sadiq Khan and the politics of clean energy

At an event at Tate Modern, Sadiq Khan pledged to clean up London's act.

On Thursday night, deep in the bowls of Tate Modern’s turbine hall, London Mayor Sadiq Khan renewed his promise to make the capital a world leader in clean energy and air. Yet his focus was as much on people as power plants – in particular, the need for local authorities to lead where central governments will not.

Khan was there to introduce the screening of a new documentary, From the Ashes, about the demise of the American coal industry. As he noted, Britain continues to battle against the legacy of fossil fuels: “In London today we burn very little coal but we are facing new air pollution challenges brought about for different reasons." 

At a time when the world's leaders are struggling to keep international agreements on climate change afloat, what can mayors do? Khan has pledged to buy only hybrid and zero-emissions buses from next year, and is working towards London becoming a zero carbon city.

Khan has, of course, also gained heroic status for being a bête noire of climate-change-denier-in-chief Donald Trump. On the US president's withdrawal from the Paris Agreement, Khan quipped: “If only he had withdrawn from Twitter.” He had more favourable things to say about the former mayor of New York and climate change activist Michael Bloomberg, who Khan said hailed from “the second greatest city in the world.”

Yet behind his humour was a serious point. Local authorities are having to pick up where both countries' central governments are leaving a void – in improving our air and supporting renewable technology and jobs. Most concerning of all, perhaps, is the way that interest groups representing business are slashing away at the regulations which protect public health, and claiming it as a virtue.

In the UK, documents leaked to Greenpeace’s energy desk show that a government-backed initiative considered proposals for reducing EU rules on fire-safety on the very day of the Grenfell Tower fire. The director of this Red Tape Initiative, Nick Tyrone, told the Guardian that these proposals were rejected. Yet government attempts to water down other EU regulations, such as the energy efficiency directive, still stand.

In America, this blame-game is even more highly charged. Republicans have sworn to replace what they describe as Obama’s “war on coal” with a war on regulation. “I am taking historic steps to lift the restrictions on American energy, to reverse government intrusion, and to cancel job-killing regulations,” Trump announced in March. While he has vowed “to promote clean air and clear water,” he has almost simultaneously signed an order to unravel the Clean Water Rule.

This rhetoric is hurting the very people it claims to protect: miners. From the Ashes shows the many ways that the industry harms wider public health, from water contamination, to air pollution. It also makes a strong case that the American coal industry is in terminal decline, regardless of possibile interventions from government or carbon capture.

Charities like Bloomberg can only do so much to pick up the pieces. The foundation, which helped fund the film, now not only helps support job training programs in coal communities after the Trump administration pulled their funding, but in recent weeks it also promised $15m to UN efforts to tackle climate change – again to help cover Trump's withdrawal from Paris Agreement. “I'm a bit worried about how many cards we're going to have to keep adding to the end of the film”, joked Antha Williams, a Bloomberg representative at the screening, with gallows humour.

Hope also lies with local governments and mayors. The publication of the mayor’s own environment strategy is coming “soon”. Speaking in panel discussion after the film, his deputy mayor for environment and energy, Shirley Rodrigues, described the move to a cleaner future as "an inevitable transition".

Confronting the troubled legacies of our fossil fuel past will not be easy. "We have our own experiences here of our coal mining communities being devastated by the closure of their mines," said Khan. But clean air begins with clean politics; maintaining old ways at the price of health is not one any government must pay. 

'From The Ashes' will premiere on National Geograhpic in the United Kingdom at 9pm on Tuesday, June 27th.

India Bourke is an environment writer and editorial assistant at the New Statesman.

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