Published June 14, 2015

In short, for most naturally occuring numbers, a lot more of them should start with the digit 1 than 9. There should be more initial 2s than 8s, but the difference is not as big as for 1s and 9s, and by the time you get to 5s and 6s, there are only slightly more 5s than 6s (on average). A fraudster who makes up the numbers might be revealed if they don’t obey Benford’s Law about the frequency of the leading digits. In “Looking out for number one” by Jon Walthoe (maths.org), the most relevant part is under “Tracking Down Fraud With Benford”.

On this 15 x 15 multiplication table, I counted 75 numbers starting with 1 (not counting the headers across the top and down the left). That’s out of 225 numbers, and 75/225 = 1/3, or 33.3333 … %, which is above the 30% expected by Benford’s Law. I only counted 13 numbers starting with 9, and 13/225 = 0.057777 … or about 5.77778%, compared to 4.57575% expected by the law.

My own simple but woolly theory is that Benford’s Law applies when the scale of the numbers changes by multiplication rather than addition. That seems to be undermined if you look at a small addition table such as this 15×15 table. However, if you start with 1 and keep adding 1 until you get to 999, the numbers you get include 1, the numbers 10 to 19, and the numbers 100 to 199. There are the same amount of initial 2s, as in 2, the numbers 20 to 29, and the numbers 200 to 299. Altogether there are 1 + 10 + 100 = 111 numbers for each initial digit (1, 2, 3, 4, 5, 6, 7, 8 and 9). It’s more complicated if you start with 2 and keep adding 5, for example, which gets 2, 7, 12, 17, 22, 27, 32, 37 …, but there’s no systematic bias towards numbers starting with the digit 1, or any other digit, except for where you start and where you stop (if you count from 1 to 111, you’ll get more numbers starting with 1). In contrast, if you start with 1.00 and keep increasing by a constant percentage, which is a case of successive multiplication, the results should conform to Benford’s Law. For some percentages you might need to include some randomness to avoid the quirks of number theory that could sabotage the law, and the naturally occuring numbers that the law applies to are not the deterministic result of applying a fixed percentage growth or any other simple rule. I don’t expect anyone to take my woolly theory very seriously, and the invariance explained in the link is much more elegant.

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