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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Theresa May gambles that the EU will blink first

In her Brexit speech, the Prime Minister raised the stakes by declaring that "no deal for Britain is better than a bad deal for Britain". 

It was at Lancaster House in 1988 that Margaret Thatcher delivered a speech heralding British membership of the single market. Twenty eight years later, at the same venue, Theresa May confirmed the UK’s retreat.

As had been clear ever since her Brexit speech in October, May recognises that her primary objective of controlling immigration is incompatible with continued membership. Inside the single market, she noted, the UK would still have to accept free movement and the rulings of the European Court of Justice (ECJ). “It would to all intents and purposes mean not leaving the EU at all,” May surmised.

The Prime Minister also confirmed, as anticipated, that the UK would no longer remain a full member of the Customs Union. “We want to get out into the wider world, to trade and do business all around the globe,” May declared.

But she also recognises that a substantial proportion of this will continue to be with Europe (the destination for half of current UK exports). Her ambition, she declared, was “a new, comprehensive, bold and ambitious Free Trade Agreement”. May added that she wanted either “a completely new customs agreement” or associate membership of the Customs Union.

Though the Prime Minister has long ruled out free movement and the acceptance of ECJ jurisdiction, she has not pledged to end budget contributions. But in her speech she diminished this potential concession, warning that the days when the UK provided “vast” amounts were over.

Having signalled what she wanted to take from the EU, what did May have to give? She struck a notably more conciliatory tone, emphasising that it was “overwhelmingly and compellingly in Britain’s national interest that the EU should succeed”. The day after Donald Trump gleefully predicted the institution’s demise, her words were in marked contrast to those of the president-elect.

In an age of Isis and Russian revanchism, May also emphasised the UK’s “unique intelligence capabilities” which would help to keep “people in Europe safe from terrorism”. She added: “At a time when there is growing concern about European security, Britain’s servicemen and women, based in European countries including Estonia, Poland and Romania, will continue to do their duty. We are leaving the European Union, but we are not leaving Europe.”

The EU’s defining political objective is to ensure that others do not follow the UK out of the club. The rise of nationalists such as Marine Le Pen, Alternative für Deutschland and the Dutch Partij voor de Vrijheid (Party for Freedom) has made Europe less, rather than more, amenable to British demands. In this hazardous climate, the UK cannot be seen to enjoy a cost-free Brexit.

May’s wager is that the price will not be excessive. She warned that a “punitive deal that punishes Britain” would be “an act of calamitous self-harm”. But as Greece can testify, economic self-interest does not always trump politics.

Unlike David Cameron, however, who merely stated that he “ruled nothing out” during his EU renegotiation, May signalled that she was prepared to walk away. “No deal for Britain is better than a bad deal for Britain,” she declared. Such an outcome would prove economically calamitous for the UK, forcing it to accept punitively high tariffs. But in this face-off, May’s gamble is that Brussels will blink first.

George Eaton is political editor of the New Statesman.