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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“I felt very lonely”: addressing the untold story of isolation among young mothers

With one in five young mothers lonely “all the time”, it’s time for employers and services to step up.

“Despite having my child with me all the time, I felt very lonely,” says Laura Davies. A member of an advisory panel for the Young Women’s Trust, she had her son age 20. Now, with a new report suggesting that one in five young mums “feels lonely all the time”, she’s sharing her story.

Polling commissioned by the Young Women’s Trust has highlighted the isolation that young motherhood can bring. Of course, getting out and about the same as you did before is never easy once there’s a young child in the picture. For young mothers, however, the situation can be particularly difficult.

According to the report, over a quarter of young mothers leave the house just once a week or less, with some leaving just once a month.

Aside from all the usual challenges – like wrestling a colicky infant into their jacket, or pumping milk for the trip with one hand while making sure no-one is crawling into anything dangerous with the other – young mothers are more likely to suffer from a lack of support network, or to lack the confidence to approach mother-baby groups and other organisations designed to help. In fact, some 68 per cent of young mothers said they had felt unwelcome in a parent and toddler group.

Davies paints what research suggests is a common picture.

“Motherhood had alienated me from my past. While all my friends were off forging a future for themselves, I was under a mountain of baby clothes trying to navigate my new life. Our schedules were different and it became hard to find the time.”

“No one ever tells you that when you have a child you will feel an overwhelming sense of love that you cannot describe, but also an overwhelming sense of loneliness when you realise that your life won’t be the same again.

More than half of 16 to 24-year-olds surveyed said that they felt lonelier since becoming a mother, with more than two-thirds saying they had fewer friends than before. Yet making new friends can be hard, too, especially given the judgement young mothers can face. In fact, 73 per cent of young mothers polled said they’d experienced rudeness or unpleasant behaviour when out with their children in public.

As Davies puts it, “Trying to find mum friends when your self-confidence is at rock bottom is daunting. I found it easier to reach out for support online than meet people face to face. Knowing they couldn’t judge me on my age gave me comfort.”

While online support can help, however, loneliness can still become a problem without friends to visit or a workplace to go to. Many young mothers said they would be pleased to go back to work – and would prefer to earn money rather than rely on benefits. After all, typing some invoices, or getting back on the tills, doesn’t just mean a paycheck – it’s also a change to speak to someone old enough to understand the words “type”, “invoice” and “till”.

As Young Women’s Trust chief executive Dr Carole Easton explains, “More support is needed for young mothers who want to work. This could include mentoring to help ease women’s move back into education or employment.”

But mothers going back to work don’t only have to grapple with childcare arrangements, time management and their own self-confidence – they also have to negotiate with employers. Although the 2003 Employment Act introduced the right for parents of young children to apply to work flexibly, there is no obligation for their employer to agree. (Even though 83 per cent of women surveyed by the Young Women’s Trust said flexible hours would help them find secure work, 26 per cent said they had had a request turned down.)

Dr Easton concludes: “The report recommends access to affordable childcare, better support for young women at job centres and advertising jobs on a flexible, part-time or job share basis by default.”

Stephanie Boland is digital assistant at the New Statesman. She tweets at @stephanieboland