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Let's shoot down that "mystery" about the first-digit law.

Date/Time Permalink: 05/12/07 06:46:08 am
Category: LINKS and Lists

OK, I'm sick of this one going around and around the Internet. To start by quoting:

"Does your house address start with a 1? According to a strange mathematical law, about 1/3 of house numbers have 1 as their first digit. The same holds true for many other areas that have almost nothing in common: the Dow Jones index history, size of files stored on a PC, the length of the world’s rivers, the numbers in newspapers’ front page headlines, and many more.

Their paper also includes useful applications and explains that no one has been able to provide an underlying reason for the consistent frequencies."

number frequency chart by first digit.

The first thing that pops into my mind is "What about lottery numbers?"

And sure enough, this chart thought to include them too. Note the diamond shapes hovering over each number. Now, not to keep you in suspense a moment longer, against this article's claim there is in fact a very crystal-clear explanation of Benford's law. It is about as mysterious as what happens to food after we eat it.

Now, I'm no mathematician, but let me try to break it down in simple English for the one or two of you who just followed that Wikipedia link and stopped dead at the equation.

The lottery numbers in the above chart show that random numbers picked blindly don't follow the pattern. That is because they are not constrained by being natural.

However, natural, tangible things that we count would of course be most likely to start with one. That is because we're dealing with the sets of finite items which can range through the positive integers starting from zero and going to infinity, but for practical reasons are bound by an upper limit.

There are no countries with a trillion people, no women who have given birth to a thousand babies, no countries with a billion square miles of land mass, rivers several light-years in length, etc. So these statistical figures are the equivalent of saying "Start counting and I'll stop you at a random place.", then noting how many of each of the numbers you spoke start with each digit.

If I stopped you at a number starting with the digit 8, then we can be assured there were an equal number of numbers starting with the digits 1, 2, 3, 4, 5, 6, 7, and 8, but fewer starting with 9. If I stopped you on a number starting with 5, we have the same number of numbers starting with 1, 2, 3, 4, and 5, but fewer starting with 6, 7, 8, and 9. So, measure a river, and its length went on and on and on and then had to stop - because it had to end eventually. On and on, until we measure all the rivers and compile our results. Whatever our longest river was, the set of all the rivers that are shorter than it will have a strong showing of numbers beginning with one.

Over time, we get that logarithmic distribution of numbers. Note the pattern in the image in that last link, then check out this slide rule. See the bands of marks that get denser and denser until they jump up by an order of magnitude and thin out again?

And that is why these stories of the "mystery" of the first digit are starting to circulate on the Internet; slide rules have gone out of fashion, and we no longer have that visual reminder hanging around.

I'm just dying to hear what A.K. Dewdney would say about this supposed "mystery" that "scientists can't explain"!

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