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Current time:0:00Total duration:6:27

Video transcript

so where we left off in the last video vai 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 digit in their population and plotted the numbers the numbers of countries that they're most significant digit is a one versus a two versus a three you just had it was much more likely there would be a one or that she took physical constants of the universe that they're more most likely to have one as their most significant digit and that I wish we had more graphs because a graphs are fun yes but if you look at information from the stock market or yes and then and yes and it seemed to all follow this curve and what was extremely mysterious and this is where we finished off the last video was if you look at pure I would say compounding phenomenon like like for example the Fibonacci sequence or powers of two that exactly fits the Benford distribution it exactly fits this if you take all the powers of two exactly 30 or I don't know a little bit over thirty percent of those powers of two all of the powers of two have one as their most significant digit a little bit what is this like seventeen roughly seventeen percent of all of them have two as their most significant although in this case there's an infinite number in every yes it's hard to graph but if you want it if you want it if you wanted to try it out you could take the first million powers of two and then find the percentage and that'll probably give you a pretty good approximation of things so that's the Tsarina so that's like less mysterious when you're looking at I mean on the one hand wow this fits exactly with mathematics but this also gives you a really good handle because you're you realize all right there's something here I can actually take a look you can take a look at it starts to become something you can dig deeper in and we said in the last video we wanted you to pause it and and think about why this is happening because frankly we had to do that the same thing and 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 our powers of 10 so in 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 that this is a 2 you would say this is a 1 this is a 10 this would be a 20 and it would be a 30 so on and so forth but in a logarithmic scale equal distances are our times 10 or in this case if we're taking powers of 10 so this is 1 to 10 then 10 to 100 then 100 to 1000 and you see how the numbers in between fall out that the space between 1 and 2 is pretty big and then 2 & 3 is still pretty big but a little bit smaller than 3 & 4 you get smaller and smaller smaller until you get to 10 and that's a pretty big clue about what's going on with Benford's law yeah it seems to match up somehow so there's a connection here and it actually turns out and this is actually a very big clue that this if you take this area if you take this area right here as a percentage of this entire area as a percentage of this entire area it's exactly this percentage it's exactly that percentage there and if you take this area as a percentage of that entire it's exactly this percentage that roughly 17 percent or whatever that number is right over there so that's a huge clue yeah or at least 4 4 powers of 2 or if in a key sequence thing for Paris it definitely makes sense yes for any powers and so and and so the logic is and this is now our biggest clue is to actually plot the powers of 2 on a logarithmic scale like this let's see where they fall all right let's try it out so 2 to the zeroth power is 1/2 to the first power is 2 then you get to 4 then you get to 8 then you get to 16 which is going to be someplace around here then you want to go to 32 which is going to be someplace around there that's 36 32 then you want to go to 64 and so this is 40 50 60 64 is going to be right right over there and so what you see is when you plot the powers of 2 on this logarithmic scale they're equal distance apart so you keep you keep stepping along if you were plotting on a linear scale to get further and further apart yeah actually twice as far apart every time but if on this scale right over here they are equally spaced so what's happening is you have something that's just equally stepping along you could imagine even just like walking along this and if you're sidewalk is shaped like this logarithmic scale you're just much more the probability on any given step as you do many many steps or as you count all the steps you can have many many more steps that fall into the block that's between 1 & 2 or between 10 and 20 then you will for example to block this between 9 and 10 if you just take a random point along here you're more likely to fall in a area starting with 1 right 1 of these areas or you know between exactly starting with 1 so between 1 & 2 or 10 and 20 or 100 and and that's exactly so taking equal steps is going to give you that distribution unless your steps happen to cuz of special cases right so where people walk logarithmically if you walk well if you walk from 1 to 10 if your steps are 10 yes in the special cases yes if you're if your other steps are 10 long exactly right but if you're anything any slight variation away from that exact thing and then you'll get the distribution all over the place so Benford's distribution that first distribution but this is this even though I think we now understand why it's still fascinating yeah well this this explains it for these number series yes it's not X so now we have to somehow figure how to connect that right and and and real world it for me the general idea will so for populations and we read up a little bodies and Benford's distribution tends to work for things that grow exponentially like powers obtain powers of two like powers of two and populations grow exponentially yeah and in finance a lot of things also grow exponentially yes or the client expedite or but but it tends to operate exponential you keep growing by 10% every year that's an exponential of a path what's fascinating is physical constants and we actually aren't 100% sure what no this is still crazy to me this is this is we only have theories here and the general idea because you know physical constants are dependent on the units you're dealing with the depending on a whole bunch of things but but they are actually I you know I'll I have a few very loose theories uh but but I'll let y'all think about that more okay all right uh and so hopefully I'll enjoyed this