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.

Ukip's Nigel Farage and Paul Nuttall. Photo: Getty
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Is the general election 2017 the end of Ukip?

Ukip led the way to Brexit, but now the party is on less than 10 per cent in the polls. 

Ukip could be finished. Ukip has only ever had two MPs, but it held an outside influence on politics: without it, we’d probably never have had the EU referendum. But Brexit has turned Ukip into a single-issue party without an issue. Ukip’s sole remaining MP, Douglas Carswell, left the party in March 2017, and told Sky News’ Adam Boulton that there was “no point” to the party anymore. 

Not everyone in Ukip has given up, though: Nigel Farage told Peston on Sunday that Ukip “will survive”, and current leader Paul Nuttall will be contesting a seat this year. But Ukip is standing in fewer constituencies than last time thanks to a shortage of both money and people. Who benefits if Ukip is finished? It’s likely to be the Tories. 

Is Ukip finished? 

What are Ukip's poll ratings?

Ukip’s poll ratings peaked in June 2016 at 16 per cent. Since the leave campaign’s success, that has steadily declined so that Ukip is going into the 2017 general election on 4 per cent, according to the latest polls. If the polls can be trusted, that’s a serious collapse.

Can Ukip get anymore MPs?

In the 2015 general election Ukip contested nearly every seat and got 13 per cent of the vote, making it the third biggest party (although is only returned one MP). Now Ukip is reportedly struggling to find candidates and could stand in as few as 100 seats. Ukip leader Paul Nuttall will stand in Boston and Skegness, but both ex-leader Nigel Farage and donor Arron Banks have ruled themselves out of running this time.

How many members does Ukip have?

Ukip’s membership declined from 45,994 at the 2015 general election to 39,000 in 2016. That’s a worrying sign for any political party, which relies on grassroots memberships to put in the campaigning legwork.

What does Ukip's decline mean for Labour and the Conservatives? 

The rise of Ukip took votes from both the Conservatives and Labour, with a nationalist message that appealed to disaffected voters from both right and left. But the decline of Ukip only seems to be helping the Conservatives. Stephen Bush has written about how in Wales voting Ukip seems to have been a gateway drug for traditional Labour voters who are now backing the mainstream right; so the voters Ukip took from the Conservatives are reverting to the Conservatives, and the ones they took from Labour are transferring to the Conservatives too.

Ukip might be finished as an electoral force, but its influence on the rest of British politics will be felt for many years yet. 

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