Science and technology

Benford law works for most naturally occurring numbers.The surface area of lakes, population sizes in the world's cities and the total sums of a company's receipts all have one thing in common.

If you look at these, and other naturally occurring numbers, you will find that they start with the digit one much more often than could be intuitively expected. This phenomenon is called Benford's law.

Our common sense tells us that there would be roughly the same number of cities' population sizes beginning with each different digit between 1 and 9. The little statistician in us estimates that one in nine numbers will begin with one, meaning that the likelihood of its occurring at the beginning of a number stands at 11.1 per cent.

In reality, in up to 30.1 per cent of the cases the population size begins with 1, with small digits from 1 to 3 occurring much more frequently than larger digits.

Why do numbers follow Benford's law?

Risto Hiltunen, a senior analyst at the Finnish Tax Administration, explains in simple terms why small digits are so common.

"Take a stock that has a value of under a euro to begin with but goes up by 20 per cent a year."

During the first year, the stock is worth 1.0 euros, 1.20 euros the second year and 1.44 euros the third year.

"It takes five years for the stock price to go up from one euro to two, but then only four years to jump from two euros to three," Hiltunen explains.

As the years go by, the price development accelerates.

A number starting with the digit nine will never be reached as in a geometric sequence the price jumps directly from eight euros to ten.

Like stock prices, many other naturally occurring numbers arise as a result of a growth process and follow Benford's pattern.

The Tax Administration has applied Benford's law to the detection of tax frauds. Professor Theodore Hill, who was the first to propose a mathematically satisfactory explanation for the principle behind Benford's law, was also interested in catching cheats.

Taking a chance

Hill was in the habit of taking his students at the Georgia Institute of Technology by surprise by giving them an unusual homework assignment. The professor invited the students to flip a coin 200 times and write down the results – or to forge the records.

At the next lecture, Hill would spot the made-up results at a glance.

Hill's rapid analysis was based on a simple statistical principle.

When a coin is thrown 200 times it is highly likely that a run of six heads or tails occurs in the series, even though this goes against the gut feeling of most people. Students who faked their results avoided such long runs – giving themselves away in the process.

The Benford's test used by auditors works on the same principle.

Not many crooked entrepreneurs forging receipts have the nous to check their figures for Benford's law to see that the numbers often start with the digit one.

"The Benford's test is a standard feature in the software we use for auditing," says Hiltunen.

Hiltunen, who has studied how the Benford test can be used as a tool in auditing, presents the test results for company accounts that have been tested for the first two digits.

The distribution shows a sudden peak at 32 euros, which straightaway catches the auditor's attention. The explanation is found in a product with a retail price of 32.02 euros, which has been flying off the shelf.

Deviations before the EU

Besides garden-variety fraudsters, the Benford test has helped reveal scams at a national level.

When Greece wanted to join the EU in the 1990s, the strict terms set for the country's economy led to problems.

Published in 2011, a German research report Fact and Fiction in EU-Governmental Economic Data revealed that the Greek economic data showed the greatest deviations from Benford's law among the EU member states.

To boot, the deviations were at their greatest in the statistics for 2000, just before Greece joined the monetary union.

The results did not come as a total surprise as the European statistical office Eurostat and the European Commission have both previously stated that the Greek debt figures had been fiddled.

The test run by the researchers showed that the Benford test is effective in revealing discrepancies in financial records.

Benford law works for most naturally occurring numbers but there is no point in applying it to lottery numbers, which all have an equal likelihood of popping up in the draw.

Petja Partanen – HS
Niina Woolley – HT
Image: Hans Eiskonen