[susam]

The numbers in the range 0-300 do not obey Benford's law. In base 10, a set of numbers that Benford's law if the leading significant digit d (0 < d < 10) occurs with probability log10(1 + 1/d). This isn't the case for the set of numbers between 1 and 300, inclusive.

[EGreg]

Your assertion that for large ranges every digit has the same chance of appearing is very wrong. Your empirical test is rigged by choosing a very rare max, literally the only one where it would “prove” your assertion.

Benford’s law appears when the max of your range is uniformly distributed

[foo101]

If you present a weird distribution to begin with, it should not be surprising that every digit does not have the same chance of appearing. That's not the point. We are not talking about weird distributions here.

If we are going to argue like this, I might as well present a set of two numbers S = {1, 2} and claim that when we choose numbers from uniform distribution, the probability of 3 occurring as the first digit is 0. Other commenters are not assuming weird distributions like this because this kind of discussion does not provide any new insights and is just a waste of time.

[EGreg]

You can create all the strawmen you want. I am going to quote from Wikipedia:

The law states that in many naturally occurring collections of numbers, the leading significant digit is likely to be small.

I have explained why that happens for the vast majority of UNIFORMLY DISTRIBUTED VARIABLES.

The vast majority. That implies that there is a collection of all possible uniformly distributed variables, and in particular those that are sampled from real world processes.

As long as they are uniformly distributed, with 0 as the minimum and M as the maximum, the first digit will appear more commonly.

I explained it several times. Why are you still insisting that statements about MAJORITY of uniform distributions are weird?

Yes statements about collections of uniform distributions are not statements about ONE SPECIFIC uniform distribution. And?

[foo101]

Can you provide an example range of uniformly distributed integers that obeys Benford's law?

[EGreg]

* The law states that in many naturally occurring collections of numbers, the leading significant digit is likely to be small.*

Pretty much all of them.

[foo101]

You are quoting from Wikipedia and that quote is an oversimplification.

If you redefine the law like that, then sure, I agree that there are many uniform distributions too where the 1st digit is likely to be small. Here is another simple example: Consider the distribution of positive integers from 1 to 2. If we pick a number at random from {1, 2} then the 1st digit is likely to be small. This kind of analysis is boring.

But (fortunately!) that's not what Benford's law says. Benford's law provides a specific formula. Check https://en.wikipedia.org/wiki/Benford%27s_law#Definition to see the specific formula that must hold good for a set of numbers to be said to obey Benford's law. That's what makes Benford's law so interesting whereas your example ranges are degenerative cases where nothing new, surprising, or interesting is going on.