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Post by jandl100 on Oct 1, 2020 12:31:36 GMT
Numbers, wonderful things. But they can be a bit, well ..... strange. I came across Benford's Law today in an article about astronomy, and was intrigued and delved further. Benford's law states that in many naturally occurring collections of numbers, the leading digit is likely to be small. For example, in sets that obey the law, the number 1 appears as the leading significant digit about 30% of the time, while 9 appears as the leading significant digit less than 5% of the time. If the digits were distributed uniformly, they would each occur about 11.1% of the time. It has been shown that this result applies to a wide variety of data sets, including electricity bills, street addresses, stock prices, house prices, population numbers, death rates, lengths of rivers, and physical and mathematical constants. Here's a plot of Benford's Law, showing the % likelihood of the first digit of any number from a naturally occurring distribution.  Seriously weird and counterintuitive. Apparently used to detect financial fraud - made up numbers don't follow Benford's Law while real accounts do. Wow. Used by the EU to deduce that the economic figures presented by Greece when it was joining the Eurozone had been manipulated. en.wikipedia.org/wiki/Benford's_law |
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Post by MartinT on Oct 1, 2020 12:47:30 GMT
I had that law completely straight in my mind 19 times before I realised I was in error.
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Post by brian2957 on Oct 1, 2020 13:06:29 GMT
Och ...I use that all the time in ma cookin recipes  |
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Post by MikeMusic on Oct 1, 2020 15:05:43 GMT
Something else I may understand day when I reach enlightenment  |
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Post by jandl100 on Oct 1, 2020 15:26:07 GMT
It just doesn't seem to make any sense to me.
Why don't all the numbers have equal probability?
What would the world be like if they did?
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Post by MartinT on Oct 1, 2020 18:03:16 GMT
Nicer?  |
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Post by jandl100 on Oct 1, 2020 18:28:52 GMT
Some people have no curiosity.  I shall see if I can delve deeper, this has me fascinated. It seems a crazy thing to me. |
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Post by MartinT on Oct 1, 2020 19:35:09 GMT
Well, I was going to suggest it could be a facet of base 10 arithmetic, but that's already been rejected. I'm curious, alright, but I have no working hypothesis right now!
EDIT: come on, it can't be weirder than superposition.
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Post by brian2957 on Oct 1, 2020 19:53:44 GMT
Sometimes I wish I had listened at school |
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Post by jandl100 on Oct 1, 2020 19:57:53 GMT
Superposition.
Well, yes, but that's quantum mechanics and is not supposed to be sensible.
Benford's Law is everyday number distribution, nothing quantum about it.
A further curious thing about it is that the distribution of subsequent significant digits (2nd,3rd etc) becomes increasingly linear and rapidly gets to near equal probability of each number occuring.
Stranger and stranger!
There's a recent paper from Imperial College that can be found with a search engine (it's a pdf file and I can't seem to get a link) that goes into it in great depth - including as well as interesting text a whole bunch of equations which frankly are quite beyond me - and no formal proof has been found to explain it.
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Post by jandl100 on Oct 1, 2020 19:58:27 GMT
Sometimes I wish I had listened at school I don't think that would have helped! |
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Post by brian2957 on Oct 1, 2020 20:08:30 GMT
Hahaha you're probably right Jerry |
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Post by ChrisB on Oct 1, 2020 20:11:06 GMT
Well, as Benford's Law concerns probability, what are the chances of this then...?
I was talking to my Dad on the phone just now and asked him if he had heard of it. The reason being, that he is fascinated by maths theories and runs a study group on the subject. He told me that, just yesterday, Benford's Law was the subject of their latest study.
He recommended a video from Gresham college of a lecture from 2012
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Post by jandl100 on Oct 1, 2020 20:13:27 GMT
Wow, that's a coincidence weird enough to be quantum!
I'll have a look at that video later.
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Post by ChrisB on Oct 1, 2020 20:15:45 GMT
Brilliant initial observation that it was spotted by noticing that the early pages of a well used set of log tables become worn faster than the later pages!!
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Post by jandl100 on Oct 1, 2020 20:21:05 GMT
Interesting that both discoveries of the Law, independent and 50 years apart, were prompted by wear in logarithm tables! Imagine the important discoveries that are being missed now that we have computers and don't use log tables any more. |
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Post by edward on Oct 1, 2020 20:23:15 GMT
Its a bit like string theory, confusing and counter intuitive yet almost 'graspable'. I'm gonna ask a mate who did mathematics/philosophy at degree level. Is this the paper you wanted to link to Jerry? |
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Post by jandl100 on Oct 1, 2020 21:51:41 GMT
Thanks for the video, Chris.
I have to confess that I rather lost the plot about half way through! Too late in the evening, I suspect.
But the initial raffle ticket idea working out to 30.1% has certainly given me food for thought.
It's not clear to me atm how applicable that result is to the generality of the distribution of numbers.
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Post by user211 on Oct 2, 2020 9:36:28 GMT
So why does it apply to physical constants, which they don't seem to be sure of? Well, it's because the universe is discrete and we are living in a computer simulation i.e. it's binary. Blinking obvious, really! |
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