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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Will Jeremy Corbyn stand down if Labour loses the general election?

Defeat at the polls might not be the end of Corbyn’s leadership.

The latest polls suggest that Labour is headed for heavy defeat in the June general election. Usually a general election loss would be the trigger for a leader to quit: Michael Foot, Gordon Brown and Ed Miliband all stood down after their first defeat, although Neil Kinnock saw out two losses before resigning in 1992.

It’s possible, if unlikely, that Corbyn could become prime minister. If that prospect doesn’t materialise, however, the question is: will Corbyn follow the majority of his predecessors and resign, or will he hang on in office?

Will Corbyn stand down? The rules

There is no formal process for the parliamentary Labour party to oust its leader, as it discovered in the 2016 leadership challenge. Even after a majority of his MPs had voted no confidence in him, Corbyn stayed on, ultimately winning his second leadership contest after it was decided that the current leader should be automatically included on the ballot.

This year’s conference will vote on to reform the leadership selection process that would make it easier for a left-wing candidate to get on the ballot (nicknamed the “McDonnell amendment” by centrists): Corbyn could be waiting for this motion to pass before he resigns.

Will Corbyn stand down? The membership

Corbyn’s support in the membership is still strong. Without an equally compelling candidate to put before the party, Corbyn’s opponents in the PLP are unlikely to initiate another leadership battle they’re likely to lose.

That said, a general election loss could change that. Polling from March suggests that half of Labour members wanted Corbyn to stand down either immediately or before the general election.

Will Corbyn stand down? The rumours

Sources close to Corbyn have said that he might not stand down, even if he leads Labour to a crushing defeat this June. They mention Kinnock’s survival after the 1987 general election as a precedent (although at the 1987 election, Labour did gain seats).

Will Corbyn stand down? The verdict

Given his struggles to manage his own MPs and the example of other leaders, it would be remarkable if Corbyn did not stand down should Labour lose the general election. However, staying on after a vote of no-confidence in 2016 was also remarkable, and the mooted changes to the leadership election process give him a reason to hold on until September in order to secure a left-wing succession.

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