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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How the Lib Dems learned to love all-women shortlists

Yes, the sitting Lib Dem MPs are mostly white, middle-aged middle class men. But the party's not taking any chances. 

I can’t tell you who’ll be the Lib Dem candidate in Southport on 8 June, but I do know one thing about them. As they’re replacing a sitting Lib Dem (John Pugh is retiring) - they’ll be female.

The same is true in many of our top 20 target seats, including places like Lewes (Kelly-Marie Blundell), Yeovil (Daisy Benson), Thornbury and Yate (Clare Young), and Sutton and Cheam (Amna Ahmad). There was air punching in Lib Dem offices all over the country on Tuesday when it was announced Jo Swinson was standing again in East Dunbartonshire.

And while every current Lib Dem constituency MP will get showered with love and attention in the campaign, one will get rather more attention than most - it’s no coincidence that Tim Farron’s first stop of the campaign was in Richmond Park, standing side by side with Sarah Olney.

How so?

Because the party membership took a long look at itself after the 2015 election - and a rather longer look at the eight white, middle-aged middle class men (sorry chaps) who now formed the Parliamentary party and said - "we’ve really got to sort this out".

And so after decades of prevarication, we put a policy in place to deliberately increase the diversity of candidates.

Quietly, over the last two years, the Liberal Democrats have been putting candidates into place in key target constituencies . There were more than 300 in total before this week’s general election call, and many of them have been there for a year or more. And they’ve been selected under new procedures adopted at Lib Dem Spring Conference in 2016, designed to deliberately promote the diversity of candidates in winnable seats

This includes mandating all-women shortlists when selecting candidates who are replacing sitting MPs, similar rules in our strongest electoral regions. In our top 10 per cent of constituencies, there is a requirement that at least two candidates are shortlisted from underrepresented groups on every list. We became the first party to reserve spaces on the shortlists of winnable seats for underrepresented candidates including women, BAME, LGBT+ and disabled candidates

It’s not going to be perfect - the hugely welcome return of Lib Dem grandees like Vince Cable, Ed Davey and Julian Huppert to their old stomping grounds will strengthen the party but not our gender imbalance. But excluding those former MPs coming back to the fray, every top 20 target constituency bar one has to date selected a female candidate.

Equality (together with liberty and community) is one of the three key values framed in the preamble to the Lib Dem constitution. It’s a relief that after this election, the Liberal Democratic party in the Commons will reflect that aspiration rather better than it has done in the past.

Richard Morris blogs at A View From Ham Common, which was named Best New Blog at the 2011 Lib Dem Conference

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