Of logarithms, fraudsters and exoplanets

Pristine log tables. Well-thumbed would be better.

Suppose you need to multiply 263.4 by 351.2 you would nowadays use a calculator, but when I was at school in the 1960’s we used logarithms.  Logs to their friends.  In a book of log tables we would look up the log of 263.4 and the log of 361.2, add the two logs together, then look up the result in an antilog table, and bobs your uncle. Not as quick as a calculator but easier than multiplying.  We each carried around a well-thumbed book of log tables, yet one thing we all failed to notice was that the pages for numbers beginning with 1 were more well-thumbed than pages for numbers beginning with 9.  Or if we noticed we never asked ourselves why. But a character called Newcomb did, in 1881. For he it was that discovered Benford’s law and in the process proved Stigler’s law (Stigler’s law being that in science, laws are always named after the second person to discover them; and in this case the second person to discover the law was Benford.) 

Benford's law states that in most lists of data, the first digits of the numbers follow a pattern of probability, where 1 is the commonest first digit, and 9 the least common.  Take for example a list giving the heights of the tallest buildings. Almost one third of the buildings in the list will have a height whose leading digit is 1. Next most frequent in the leading position is the digit 2. And so on, till you come to 9 which is likely to be found as the leading digit in only 4.6 per cent of buildings.

Or rivers. Look at Wikipedia’s list of world rivers longer than 1000 km. You'll find a table giving length in km, length in miles, drainage area in km², and average discharge in m³/s. At my rough count, it contains five hundred numbers, of which only 18 start with a 9.

Fraudsters

Benford's law also applies to financial data, a fact unknown to most fraudsters, who tend to suppose that the best way to insert phoney entries into a list of expenses or transactions, is to make up numbers at random, with as many starting with 9 as starting with 1. But Benford’s law soon finds them out. And the spooky thing is, it matters not whether the transactions are in pounds, euros or dollars.

Exoplanet with two suns and an exomoon. But is it real?

I've read the Wikipedia article on Benford’s law which purports to explain why this should be so,  and I can't follow it as it involves high level maths.  But no matter, all this is by way of working out how many exoplanets have been discovered.

I recently mentioned that the Kepler mission found over two thousand planets orbiting other stars. Now it's important to note that these haven't actually been seen in the usual sense of the word. They have been seen in strings of data, indicating very slight periodic dimmings in the brightness of a star. From this data scientists have inferred size of a planet, distance from the star, and other factors.

But inferences can be wrong. In some cases the data could perhaps result from another phenomenon entirely, and have nothing to do with a planet at all.

And according to this week’s New Scientist, that’s where Benford’s law comes in. 

Thomas Hair at Florida Gulf Coast University wondered if Benford's law would hold true even beyond the solar system, and examined data from an online catalogue that lists 755 confirmed exoplanets and nearly 3500 planet candidates. Planet masses are given in multiples of Earth's or Jupiter's mass. He found that whichever of these two units is used, the figures closely fit Benford's law, making it highly likely the supposed planets really are out there. "The close fit with Benford's law gives a confirmation to experts' belief that most of the candidates are valid," he says. 

I wish I still had my log tables so I could check that well-thumbed business for myself. I looked in vain on the web for an image of a used copy to put at the top of this post. For fun, I've spent the last half hour reminding myself how to use logs. If you too last used them in the 1960’s take a look.