TIL Benford’s Law (from Mike Cannon Brooks on Twitter)

Why does Benford's Law work?

TIL Benford’s Law (from Mike Cannon Brooks on Twitter) Why does Benford's Law work? There is a youtube video by James Grime that explains it really well. https://www.youtube.com/watch?v=vIsDjbhbADY  To get an intuition for it: We can tell what the leading digit of any given number is by the fractional part of the log of our base. (The reason is that if we scale by a factor of our base we are just shifting digits and `[;log_b(b*x) = log_b(x) + 1;]` So in the case of base ten we find the first digit of a number x by taking `[; d(x) = \lfloor 10^{frac(log_{10}(x))}\rfloor ;]`.  So what this means is, that we can look at what `[;log_{10}(x);]` is to see what digit we will get. Since we are only interested in the fractional part we only need to look at values between 0 and 1.  So we just look at the log of our digits.   We get a 1 for log(1)<x<log(2) <-> 0 < x < 0.301...   We get a 2 for log(2)<x<log(3) <-> 0.301... < x < 0.477...   and so on.  For why if such a law exists it has to be benfords law I highly recommend James' video. https://www.reddit.com/r/math/comments/4smdbf/why_does_benfords_law_work/

Chris Nheu

I just thought it was interesting that such a pattern exists. But it completely makes sense intuitively that any sufficiently natural set of numbers would feature more 1s than 9s

With few exceptions (constrained, random generated, etc) makes a lot of sense. Clever way to detect fraud, but as this knowledge gains awareness, also easy to evade?

I guess I need to google logarithmic scale at some point

Nikolas Kong

This is the same thing other comments here are saying, but very concisely:

Numbers in the real world tend to be distributed uniformly in the logarithmic scale, not the linear scale.

If that makes sense to you, it should be pretty clear and should explain a lot.

Benford's law seems weird and non-intuitive, because humans train each other to think linearly and not logarithmically. But humans are naturally predisposed to the log scale - it's our own fault for de-training our minds.

This is the same thing other comments here are saying, but very concisely:  Numbers in the real world tend to be distributed uniformly in the logarithmic scale, not the linear scale.  If that makes sense to you, it should be pretty clear and should explain a lot.  Benford's law seems weird and non-intuitive, because humans train each other to think linearly and not logarithmically. But humans are naturally predisposed to the log scale - it's our own fault for de-training our minds.

Chris Nheu

A group of likeminded people in The Most Important Thing are talking about this.