[btilly]

I'm sorry, but your mathematical reasoning is very muddled, and the pattern you predict is wrong.

According to Benford's law the odds of a leading 1 are 1.709511291351... times the odds of a leading 2. This isn't the factor of 2 you thought it should be. The odds of a leading 1 are 2.409420839653... times the odds of a leading 3. This isn't the factor of 3 you thought it should be.

Yes, I know that it is fun to try to figure things out for yourself. But it is essential to learn when you're headed down the wrong path. That lets you correct your misconceptions before they cement and lead to severely wrong impressions of how to do things. Your whole desert/rock analogy? That's a wrong path.

[breck]

Thank you, very interesting contributions to the conversation.

It looks to me though, that my line of reasoning(note I said more precisely 2.6x, I used 3x initially to simplify it) more closely matches the data than the numbers you provided.

[btilly]

I gave you exact numbers, not approximations from a small data set. Unless you match the numbers that I gave exactly, the numbers you give won't match what will be found in large datasets.

[breck]

I understand that the law is a formula that can generate exact numbers. However, as experience shows, almost nothing generates these exact numbers. Almost nothing follows the formula exactly.

I think my explanation is about what is it that causes the law to occur. Not what is benfords , but rather, why does it occur. I know about the law scale And the picture on Wikipedia. It's neat, but I don't think it reveals the underlying cause. I think the cause could be simply that it is about 2x easier to find 1 unit than 2 units, 10 units than 20 units, 1000 units than 2000 units.

[btilly]

What can I say? Your explanation is wrong. It generates wrong numbers. And gives little to no insight as to why Benford's law works.

Benford's law will hold approximately for any set of numbers with the property that they are distributed over many orders of magnitude, from a distribution which doesn't change much if you multiply by a random number in some range.

An example of such a set of numbers is the set of numbers that come up in intermediate calculations involving a lot of different numbers. (This explains the logarithm books where the phenomena was first noticed.)

Another example are the numbers you see coming out of any sort of self-similar phenomena. As fractals show, self-similar behavior is ubiquitous. As a result numbers like the length of rivers, the height of hills, and the size of cities all tend to follow Benford's law.

For any particular source of numbers, the explanation for why they fall into a category that matches Benford's law will differ. Benford's law is a property that mathematical models tend to have, rather than being a rigorous mathematical theorem.

(FYI Benford's law is something that I've known about, and thought about off and on, for close to 20 years ago now.)

[breck]

Really good comments here. I'll try and refine my position which I did a poor job of explaining and post something in the next few weeks or months.