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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Air pollution: 5 steps to vanquishing an invisible killer

A new report looks at the economics of air pollution. 

110, 150, 520... These chilling statistics are the number of deaths attributable to particulate air pollution for the cities of Southampton, Nottingham and Birmingham in 2010 respectively. Or how about 40,000 - that is the total number of UK deaths per year that are attributable the combined effects of particulate matter (PM2.5) and Nitrogen Oxides (NOx).

This situation sucks, to say the very least. But while there are no dramatic images to stir up action, these deaths are preventable and we know their cause. Road traffic is the worst culprit. Traffic is responsible for 80 per cent of NOx on high pollution roads, with diesel engines contributing the bulk of the problem.

Now a new report by ResPublica has compiled a list of ways that city councils around the UK can help. The report argues that: “The onus is on cities to create plans that can meet the health and economic challenge within a short time-frame, and identify what they need from national government to do so.”

This is a diplomatic way of saying that current government action on the subject does not go far enough – and that cities must help prod them into gear. That includes poking holes in the government’s proposed plans for new “Clean Air Zones”.

Here are just five of the ways the report suggests letting the light in and the pollution out:

1. Clean up the draft Clean Air Zones framework

Last October, the government set out its draft plans for new Clean Air Zones in the UK’s five most polluted cities, Birmingham, Derby, Leeds, Nottingham and Southampton (excluding London - where other plans are afoot). These zones will charge “polluting” vehicles to enter and can be implemented with varying levels of intensity, with three options that include cars and one that does not.

But the report argues that there is still too much potential for polluters to play dirty with the rules. Car-charging zones must be mandatory for all cities that breach the current EU standards, the report argues (not just the suggested five). Otherwise national operators who own fleets of vehicles could simply relocate outdated buses or taxis to places where they don’t have to pay.  

Different vehicles should fall under the same rules, the report added. Otherwise, taking your car rather than the bus could suddenly seem like the cost-saving option.

2. Vouchers to vouch-safe the project’s success

The government is exploring a scrappage scheme for diesel cars, to help get the worst and oldest polluting vehicles off the road. But as the report points out, blanket scrappage could simply put a whole load of new fossil-fuel cars on the road.

Instead, ResPublica suggests using the revenue from the Clean Air Zone charges, plus hiked vehicle registration fees, to create “Pollution Reduction Vouchers”.

Low-income households with older cars, that would be liable to charging, could then use the vouchers to help secure alternative transport, buy a new and compliant car, or retrofit their existing vehicle with new technology.

3. Extend Vehicle Excise Duty

Vehicle Excise Duty is currently only tiered by how much CO2 pollution a car creates for the first year. After that it becomes a flat rate for all cars under £40,000. The report suggests changing this so that the most polluting vehicles for CO2, NOx and PM2.5 continue to pay higher rates throughout their life span.

For ClientEarth CEO James Thornton, changes to vehicle excise duty are key to moving people onto cleaner modes of transport: “We need a network of clean air zones to keep the most polluting diesel vehicles from the most polluted parts of our towns and cities and incentives such as a targeted scrappage scheme and changes to vehicle excise duty to move people onto cleaner modes of transport.”

4. Repurposed car parks

You would think city bosses would want less cars in the centre of town. But while less cars is good news for oxygen-breathers, it is bad news for city budgets reliant on parking charges. But using car parks to tap into new revenue from property development and joint ventures could help cities reverse this thinking.

5. Prioritise public awareness

Charge zones can be understandably unpopular. In 2008, a referendum in Manchester defeated the idea of congestion charging. So a big effort is needed to raise public awareness of the health crisis our roads have caused. Metro mayors should outline pollution plans in their manifestos, the report suggests. And cities can take advantage of their existing assets. For example in London there are plans to use electronics in the Underground to update travellers on the air pollution levels.

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Change is already in the air. Southampton has used money from the Local Sustainable Travel Fund to run a successful messaging campaign. And in 2011 Nottingham City Council became the first city to implement a Workplace Parking levy – a scheme which has raised £35.3m to help extend its tram system, upgrade the station and purchase electric buses.

But many more “air necessities” are needed before we can forget about pollution’s worry and its strife.  

 

India Bourke is an environment writer and editorial assistant at the New Statesman.