Taxing numbers take the law into their own hands

by Burkard Polster and Marty Ross

The Age, 20 August 2007

Last week, we called the dob-in number of the Australian Tax Department. It took them more than half an hour to answer the phone. Possible explanations for this long waiting time: a) our fellow citizens new favorite pastime is to dob each other in; b) there is just one person answering the phone; c) …

Whatever the explanation, it would appear that the tax department is not overly concerned that a dobber may lose interest and hang up. Why? Well, maybe they know about Benford’s Law, one of the new mathematical super-weapons used by tax departments throughout the world.

Here is the idea of how Benford’s Law is used. An elaborate tax return arrives at the tax department, the data is entered into a computer, and everything adds up. But then they run some fraud detection software over the data, and one check is to determine what percentage of numbers in the tax return start with a 1 in the leftmost spot. The numbers arise from a variety of sources, and therefore should be fairly random. So what would you expect the percentage of numbers starting with any given digit to be? Well there are ten digits, so 10% would be a fair guess, right?

Actually, the digit 0 should probably never appear in the leftmost spot. So let’s adjust our guess to 100/9=11.1%. Now, here is the surprise. If the tax return has been filled in correctly then Benford’s Law states that the expected percentage of numbers starting with a 1 is actually about 30.1%. And, those starting with a 2 is about 17.6%, those with a 3 about 12.5%, etc., down to a mere 4.6% of numbers starting with a 9. It applies to many different statistical data: tax returns, stock prices, population rates, lengths of rivers, and many more. The law was first discovered (not by Benford) more than a hundred years ago, but it is only recently that a satisfying mathematical explanation has been given.

The proof of Benford’s Law is complicated, but here is a simple argument which shows why the percentages should not be equal. If there were equal percentages over all tax data, then these percentages should be scale-invariant. That is, if you convert the dollar amounts into another currency, then the percentages should remain equal. But this is impossible --- just imagine a conversion in which you have to multiply all dollar amounts by 2: then all numbers which started with a 5, 6, 7, 8, and 9 would convert into numbers starting with a 1, and so in the new currency the numbers starting with 1 would now dominate. So, equal percentages are impossible!

For a long time Benford’s Law was thought of as an interesting but useless curiosity. However, recently Benford’s law has found such interesting applications, whenever it comes to check whether an applicable set of data has been tampered with. In the case of our tax return, this means that if the percentages are sufficiently off, the tax man knows to take a closer look.

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