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班佛定律的解釋 (班佛定律奧妙的續集) (英) : Vi 與 Sal 談班佛定律背後的概念 (part 2)
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- So where we left off in the last video, Vi and myself had posed a mystery to you. We had talked about
- Benford's law, and we asked, "What is up with Benford's law?"
- This idea that if you took just random countries and took their population and took the most significant
- figures of their population and plotted
- the numbers, the numbers of countries that their most significant figures were a 1 versus a 2 versus
- a 3, you just had - it was much more likely that it was a 1.
- Or that, if you took physical constants of the universe that they're most likely to have 1
- as their most significant digit.
- And that - I wish we had more graphs, because graphs are FUN -
- but if you look at information from the stock market or ANYTHING
- and it seemed to all follow this curve. And what was EXTREMELY mysterious -
- and this is where we finished off the last video -
- if you look at PURE, I would say, compounding phenomenon
- like, say, the Fibbonaci sequence or powers of 2
- that exactly fits the Benford distribution.
- It exactly fits this. If you take all the powers of 2 exactly 30%, or I don't know if a little bit over 30%
- of those powers of 2, all of those powers of 2 have 1 as their most significant digit.
- A little bit - what is this? 17? Roughly 17% of all of them have 2 as their most significant digit.
- Although in this case, there was an infinite number in every test case, so it was harder to graph.
- But if you wanted to try it out, you could take the first million powers of 2 and then find the percentage
- and that would probably give you a pretty good approximation of things.
- So to me that's, like, less mysterious when you're looking at - I mean -
- on the one hand, wow, this fits exactly with mathematics but that also gives you a really good handle
- 'cuz you realize, alright, there's something here I can actually take a look at.
- You can take a look at and it becomes something you can dig deeper in.
- We said in the last video we wanted you to pause it and think deeper
- because frankly, we had to do the same thing,
- and a big clue for us was when we looked at a logarithmic scale
- and we're looking at one right over here, and just to be clear:
- what's going on in this logarithmic scale is, you see equal spaces on the scale are powers of 10
- so on a linear scale this would be a 1 and maybe this would be a 2 and then a 3
- Or if we wanted to say this is a 2, you'd say this is a 1, this is a 10, this would be a 20 and a 30,
- and so on and so forth, but on a logarithmic scale equal distances are times 10
- or in this case if we're taking powers of 10
- so this is 1:10 then 10:100 then 100:1000 and you see how the numbers in between fall out
- so the space between 1 and 2 is pretty big and the space between 2 and 3 is still pretty big
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