Can you provide a concrete example of a range of numbers that you think obeys Benford's law?
I wrote: > The leading digits of a uniform distribution does not follow Benford's law.
And @EGreg wrote: > I’m sorry to tell you this, but you inadvertently misled people with that empirical test. This just goes to show that we have to check our assumptions, as scientists or mathematicians trying to prove a statement. (Even with empirical tests :)
So, what specific range of the uniform distribution yields leading digits that follows Benford's law?
For example 0-300
One third of numbers are evenly distributed: 0-100
One third starts with 1: 100-200
One third starts with 2: 200-300
Do you understand?
Bensford's law is 1:30.1%, 2:17.6% 3:12.5% etc.
When the max is uniformly distributed then Benford’s law emerges. I mean, all you have to do is read the link - where I derive it.
What exactly is the law — please don’t handwave. If the law is those exact point values mentioned in the article then I just showed you how we arrived at them.
What does "max is uniformly distributed" even mean? If you think that the Benford's law holds good for a set of uniformly distributed numbers, why not simply provide that set? It would be so easy to prove your claim if you just provide an example set of numbers that obeys Benford's law.
All sets of numbers you have presented so far (0-300, 0-30000, 0-300000000000000000000000) do not follow Benford's law. It is very simple to show. In all these sets, the probability of first digit as 1 is equal to the probability of first digit as 2 which contradicts Benford's law.
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.*
That is JUST FALSE ok? For for pretty much any distribution you choose for the max, other than 100% chance it is a power of 10 and 0% chance other numbers, you’ll get that the digit 1 comes up way more than 2, which would come up more than 3, etc. How much more? This comes from the fact that there are just as many numbers 100-200 as there are 0-100. Ok? And that’s all 1s. Then you hit the 2s, and so on.
If the max happens to be anywhere in the range 100-1000 with equal probability, you get that result. Benford’s law. If the max is distributed as some sort of continuous distribution — and not that ridiculous distribution of ONLY ever being powers of 10 — then you likely get something similar.
What are you arguing about?? If you are saying it’s mysterious why the lower digits come up more than higher ones, well the mystery is over. If you want an EXACT fit to the numbers in the article then I think they come out whenever the max is uniformly distributed between 10^n and 10^(n+1). But they may also have a sort of “law of large numbers” thing where pretty much any continuous distribution of the max leads to this law. That part I can’t tell you. What I can tell you is OBVIOUSLY the lower digits will come out more frequently.
(Btw it is NOT true for eg dictionary words for example, an initial A doesnt appear more than B. That should tell you something!)
And to understand the reason we just have to look at the family of uniform distributions, and see that for almost all of them, this proportion holds. Sure, for some of them, the 1,2,3 may be even MORE prevalent relative to 4-9 because the maximum value was 400 or 4000 or 40000. Ok? You can see this. For a uniformly distributed process that happens to have that as the maximum, Benford’s law will have the same proportions between 1,2,3 but then drop for 4-9 since they didn’t get that “boost”.
But if you keep sampling and this maximum keeps growing by some continuous distribution that’s not perfectly synced with the metric system, then it’s as likely to be in the range 100-200 as it is to be in 200-300. And then as likely to be in 1000-2000 as in 2000-3000. Given that, we get something like Benford’s law.
Now, perhaps it is ALSO TRUE for other distributions. I just explained why it’s true for uniform ones.
Benford’s law appears when the max of your range is uniformly distributed
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.
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?
Pretty much all of them.
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.
And once again, this is because of a simple analysis. Let me state it a DIFFERENT way, maybe this is something that you will take note of: the only time all possible digits have equal probability of being leading digits is when we have a uniform distribution with a max that is a power of 10. Literally every other distribution starts to exhibit that phenomenon. Now you can quibble as to what distributions lead to that EXACT curve fit. And there can be explanations for why power law distributions do. But other distributions exhibit this same PHENOMENON, while not necessarily converging to that exact proportion. As I said before, any other uniform distribution would not have those exact proportions, but would exhibit the phenomenon.
Basically, every continuous distribution is highly unlikely to have a cliff at a power of 10. It is going to go down gradually, and therefore if it includes the range 8000-9000 then it will probably include numbers above 10000. And even a discontinuous one with a uniform distribution (with a cliff at the end) will exhibit the phenomenon. OK? So if you have the range 8000-9000 in there, that means 1s and 2s were a lot more prevalent, and if you have a continuous distribution then 10,000+ numbers will be there, but perhaps not numbers 90,000+.
Do you at least get the intuition behind this? As soon as you get numbers close to a power of 10, your distribution probably includes numbers in the next order of magnitude, i.e. a lot of leading 1s. The more numbers you get starting with 9, the more it is highly unlikely you're right at the max of your distribution, with a cliff. Unless that happens to be the contrived "empirical test" that was linked to as "proof" that uniform distributions lead to equal changes for every digit to be leading.
The intuition is what matters. Now, maybe for UNIFORM distributions, or POWER distributions, that exact curve fit can be worked out. Perhaps you can show there is a "large" family of distributions for which the curve fits. Kind of like the law of large numbers. But I didn't do anything quite as ambitious. I simply showed why it's not just true for normally distributed processes, but others which you would think are uniformly distributed. Because chances are, that process has more 1s than 2s, and 2s than 3s, in exactly the proportion that a uniform distribution with some max would have, and the chances are the max isn't exactly a power of 10.
In practice, all this means is that the distribution may look like Benford's law for 1, 2, 3 and then drop to equally small for 4, 5, 6, 7, 8, 9. The 1 is going to be 10x more prevalent than the 9. The 2 would 5x more prevalent or whatever, UNLESS the distribution had a cutoff right before 200 or 2000. Understand? And this DOES HAPPEN.
There is a distribution of leading digits that looks like:
d P(d)
1 30.1%
2 17.6%
3 12.5%
4 9.7%
5 7.9%
6 6.7%
7 5.8%
8 5.1%
9 4.6%
Neat! For each of those data sets you get the same distribution. Now, someone (I won't say who), says that it also is true for the uniform distribution.
But it isn't.
It simply isn't.
And I said as much when I said, "The leading digits of a uniform distribution does not follow Benford's law."
And your counter example is if you take a uniform distribution from 0-300, the leading digits go to something like:
d P(d)
1 36.7%
2 36.7%
3 3.7%
4 3.7%
5 3.7%
6 3.7%
7 3.7%
8 3.7%
9 3.7%
> "The leading digits of a uniform distribution does not follow Benford's law." -- me
And you, directly disagreeing with that correct statement:
> This just goes to show that we have to check our assumptions, as scientists or mathematicians trying to prove a statement. -- EGreg
Indeed.