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Comment by mathduck | Hacker News Reader

You are artificially constructing a very limited range (upper bound and lower bound) in which of course the likelihood of any digit being the first digit won't be equal. This is nothing new. This does not contradict what other commenters have said about uniform distribution. With large ranges, even if you exclude a power of 10 in the upper bound, it does not change the 11.11% chance of each digit being the first digit.

But let us accept your very limited range for a moment and go along with it. Then you say that the numbers in this range follow Benford's law. But clearly, it doesn't. None of the probabilities in this range obey the probabilities in Benford's law.

Someone needs to revert the dubious edit (https://en.wikipedia.org/w/index.php?title=Benford%27s_law&d...) you have made in Wikipedia.

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This is simply wrong, and creating a new green account and downvoting me isn’t going to change that.

It is trivial to see that literally any range with min = 0 and max = any number other than a power of 10 makes it LESS likely that a 9 will come up as the first digit. For example the range 0-300 has 1 and 2 come up as the first digit way more than the rest. Don’t you think the same is true of 0-30000 and 0-300000000000000000000000? The size of the range doesn’t make your assertion any more true, that for large ranges every leading digit begins to have an equal chance of appearing.

My point is that, given a uniform distribution from 0 to a max, it has to have a max somewhere. If we assume that max itself is uniformly distributed then we derive the proportions you find in Benford’s law.

Look to put it another way, Benford’s law comes from the numbers which are the same number of digits as the max. The rest are evenly distributed but those numbers are the most numerous at that point and they contribute the phenomenon. Ok?

Are you convinced?

PS: There has got to be someone who figured this out before 2020. Come on. Someone post a link to this derivation.

> creating a new green account and downvoting me isn’t going to change that.

It is impossible on Hacker News for a new green account with less than 500 points to downvote someone else.

Do you admit this statement is wrong?

With large ranges, even if you exclude a power of 10 in the upper bound, it does not change the 11.11% chance of each digit being the first digit.

The empirical test is cherrypicked also.

If you don’t admit this then there is really no point to continue.

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