Number 1 and Benford's Law - Numberphile

Key Takeaways

well this is benford's law uh and it's about numbers but it's about the leading digit for example uh you could look at the populations of all the countries in the world and look at the leading digits of all those so for example if it was 1,269 then the leading digit in that case is is the one benis law works on a distribution of numbers uh if that distribution spans uh quite a few orders of magnitude and the brilliant thing about populations of countries is it actually goes from tens up to billions if you were to think about that okay uh what are the distribution of leading digits so um some of the populations will start with a one some will start with a two 3 four 5 6 s eight or nine and so there are nine possible leading digits and you might imagine that uh each one of those possible leading digits are equally likely to appear so that's at 1 in N 11% and if I was to plot that on on a graph you might expect that to fluctuate around um uh 11% so it's going to sort of you know go like that so what actually happens is that a third of the time that's up here right a third of the time the number you choose will start with a one and it will hardly ever start with a nine so Nine's kind of like down here you know tiny number and then you get this This brilliant curve that goes up like that is that crazy I know you talk about this sometimes and in talks and things you do what's the reaction to that normally when you tell people this the reaction okay the no noise is sort of like this uh and there's a certain amount of disbelief sometimes as well and the way we do it actually in the show is that we get people to uh tweet numbers to us so we're collecting numbers and I try to make it I try to give them ideas so maybe like uh take the distance from the venue to where they live and convert that into some strange units or something like that the interesting thing is it uh like I was saying it works so long as the distribution you're you're choosing from uh spans uh uh loads of orders of magnitude but if you're picking numbers from lots of different distributions the individual distributions don't have to span lots of orders of magnitude the the sort of the meta distribution of individual things picked from different distributions follows benford's law anyway uh so so so so it works it works brilliantly well what clump of numbers will this not work for um uh human height in meters so humans are between uh you know 1 meter and 3 met tall so it doesn't work for that you get you know you get a massive load around around there and and you know no one's 9 MERS toall anything that has that kind of short distribution um it doesn't work for um but it does work for several distributions put together that don't necessarily individually uh follow the rule so I did it I did it for populations I did it for areas of countries in kilom squared if you take one number and convert it to loads of different units uh that will tend to follow bis as well you can do it for the uh the financial times so look at all the numbers on the front cover of the financial times they will tend to follow bis L as well just a quick interjection you can also apply this to the number of times you watch number file videos or leave comments Underneath more information at the end of the video so the explanation is to do with scale inv variance which I'm just getting my head round now but there are a couple of intuitive ways of understanding it one of them is to use the idea of a raffle to begin with it's a very small raffle okay so there are only two tickets in this raffle what are the chances of the winning ticket in this raffle having a leading digit of one well that's this one so it's one and two it's 50% uh but then if you increase the size of the raffle so there are now three tickets uh in the raffle the chances now are one and three or about 33% if you add a fourth ticket then the probability of the leading digit um of the winning ticket being a one is now uh 25% and then 20% uh and so on and so on until you have a raffle with nine tickets in it and then the probability of the winning ticket having leading digit of one is one in N it's 11% which was the kind of intuitive thing that you might think but then you add your 10th ticket and now there are two tiet tickets that start with a one so now the probability is 2 in 10 or 1 in five so it go back up to 20% the probability will go up and up and up as you add more tickets that start with a with a one and once you have a raffle with 19 tickets in it you're up to something like 58% and then you add the 20th ticket and so the probability goes down again so the probability of the the winning ticket having leading digit of one will go down and down and down through the 20s uh it will go down through the 40s down through the 50s 67s 80s '90s until you add the 100th ticket and then the probability will start to go up again and the probability will go up and up and up and up all the way through the hundreds and then you get to the 200s and it goes down and down and down through all the 200 300 400 500 600 700 800s 900s and you'll be back to uh 11% again then and then you add the thousandth ticket and the probability will start to go up again so the probability goes up and up and up through the thousands uh and then down through 2000 3,00 and then you get to 10,000 it goes up and so basically the probability of the winning ticket having leading digit of one fluctuates as the size of the raffle increases um and so so this is a log scale of the uh raffle increasing in size so you might have you know uh 10 100 uh a th000 10,000 and so on uh and then this is the probability here of the ticket having lean digit of one it will it will sort of go uh uh it sort of goes like that what Frank Benford realized was that if you pick a number from a distribution that spans loads of orders of magnitude or if you pick a number from the world and and uh you you don't necessarily know what the distribution is in advance then it's like picking a ticket from a raffle when you don't know the size of the raffle so you have to take the average of this wiggly line which is what he did so that's the average there and it's around 30% there's a formula for it which is the probability of picking a number with a particular leading digit of D is equal to uh log uh to base 10 of 1 + 1 D like that and so that's how you do it um and if you plug one into there then it's log to base 10 of two and it ends up being about 30% the beauty is that you can do it in any base so uh this doesn't have to be base 10 it could be base five you know base 16 whatever you want to do and so you can apply benis off different bases um this is a Formula that they use a forensic accountant would use so tax for or something like that if you're making up numbers in your accounts and the numbers you make up don't follow benford's law then that's a clue that you might be cheating so this is a number you need to this is a Formula you need to remember if you're going to cheat on your tax return a lot of a lot of things that sort of mathematically incline people like your tell me when I hear about them sort of seem counterintuitive and then you cleverly explain why it works the way it works this is one of the few things that when I've heard about it this just seems logical to me like when someone says one will come up more often yeah to me that just seemed like of course that would happen yeah it's funny isn't it uh some people are like you I would say you're in the minority of people that go well yeah see and I wonder if there is another sort of intuitive way of looking at it that you've tapped into which is that if you imagine something like uh the NASDAQ index or something like that um and I don't know what the NASDAQ index is size-wise but imagine that it's uh uh the NASDAQ index is at 1,000 to change that to 2,000 you would have to double it so the NASDAQ index would have to increase by 100% to get from something that starts with a one to something that starts with a two so that's quite a big change but if the NASDAQ index was on 9,000 and you wanted to increase it to 10,000 then that's uh an 11% increase so it's hardly anything so basically you don't really hang around the nines as things are growing and shrinking you don't hang around whereas you do hang around the ones and and maybe that's intuitive to you and so you're like yeah obviously if you'd like to see even more about benford's law we've done bit of a statistical analysis to find out whether or not your viewing habits and the number of times you comment on number file videos is following the Benford curve the link is below this video or here on the screen so why don't you check it out