# Lucky numbers and Benford’s law

Happy lunar new year to you if you’re celebrating! Here in Melbourne, the festivities are in full swing – you can hear the lion dances during the evenings, and there are lots of decorations in Chinatown.

I just walked past a display for Chinese zodiac signs. Here’s mine:

As you can see, my lucky numbers for the year are 8 and 9. These lucky numbers seem to be as lucky as any you might choose, but they are particularly *unlucky* in certain situations.

Here’s an example.

After I read my Saturday newspaper, I like to look at the Friday closing stock prices in the finance section. There are hundreds of companies, and the game I like to play is to count up how many times each digit appears as the first number of a stock price. For example, last Saturday, the closing price for Apple was \$137.87, so I add a tally mark for 1.

You might expect that every digit between 1 and 9 appears about the same number of times, but this is not the case!

Across all the weeks that I’ve done this exercise, the number 1 appears about 30% of the time, while 8 and 9 appear about 5% of the time each. How unlucky for my lucky numbers!

Well, this actually doesn’t have a lot to do with luck – it’s a phenomenon known as Benson ‘s Law. It’s named after physicist Frank Benford who was one of the first to observe it in 1938, although there is evidence that Simon Newcomb discovered it much earlier in 1881.

The explanation for Benford’s Law is that a lot of datasets (particularly those that span several orders of magnitude) are arranged in a lognormal distribution. That is, the (base 10) logarithms of the values in the data set are normally distributed. Then, on a logarithmic scale, numbers starting with 1 constitute ~30.1% of the interval between two powers of 10. This percentage decreases as the digit increases, with numbers starting with 8 or 9 constituting about 5% of this interval each.

Generally, any data that is from a source that fluctuates in a multiplicative way (e.g., stocks increase or decrease by percentages, populations of towns have a particular birth rate etc.) tends towards obeying Benford’s Law.

There are also examples of Benford’s Law being used to uncover fraud, when data sets that should obey Benson’s Law obviously do not. It’s still a bit unclear whether we can rely on Benson’s Law in this way, but it is nevertheless an interesting tool.

Here’s hoping my lucky numbers work well in other ways!