Benford's law (with Vi Hart, 1 of 2)

I'm very excited to have vihart visiting the office over here and we were just having a very mathematical conversation earlier today and she mentioned something that is fascinating yeah I was just telling Sal about a cool thing called Benford's law Benford's law and and what is Benford's law it's this weird phenomenon that you get when you're looking at numbers in the real world so for example we've got some graphs here if you take the populations of all the countries and you say alright what is the first digit of the population of the country whether it's 1 million or 1 thousand or a hundred thousand we'll say okay that starts with 1 so we'll count up all of the countries that start with 1 and I guess here we've got 27 of them yes about 27 so literally like like anything that starts with a 1 here so it could be a country that has a population of 1 a population has a population of 17 or a population of 1 billion 300 million ba-ba-ba-ba they would all fall into this bucket right over here alright and then if you start with 2 you fall in the second bucket and so on and so forth better color oh yeah go ahead yeah better definitely better better color yeah yeah that's the one better contrast so the question is alright you think you'd have kind of random numbers here yes for that first digit like it's kind of random yeah I mean there's huge differences in populations of countries some have Riley ins and and some have I don't I don't know the smallest population in yes it's like like Montenegro or something like that yeah um so I wanted Negroes not a country what am I thinking I'm thinking of what was the one that's on the French Riviera we candid that out ok I don't I would read your city to four populations of states and everything and the Vatican is the smallest country I believe yeah does that still count I think the Vatican House they have their own yes I don't know exactly what the requirement okay to include the Vatican which I think they like the thousands yeah and so so why would this happen why would you see more ones than twos like yeah it's not some small some small chance I mean we were talking also about the idea that it is more likely to have it is more likely to have odd-numbered addresses than even-numbered you were talking about I just learned about that reason yes but that's not a season that makes sense because every house will have a number one on it ever change if your street start you know with house number one house number two house number three right or yes then your street has more odd numbers exactly and if you have an even number you have the same amount right right but that's starting with one which is odd whereas here populations don't start with one exactly and and that phenomena that we're talking about with the street numbers it's not an extreme phenomenon it's like 50 points some oh it's a slightly more you have a slightly higher probability of having an odd numbered house or or I guess a one house and then everything is kind of exactly what you would expect it's exactly you expect but here here it's a significantly higher probability of a random country's population that its first digit is a one versus its first digit isn't is an eight or a nine I mean it seems it seems a little bit a little bit strange and this isn't just in country yes you see this if you're looking at a lot of financial stuff like yes which money does a company make yes it the ones just show up as a first digit much more frequently much more frequently yeah and here we have another fancy graph which is like completely crazy it's the first digit of physical constants so what what would be example so some physical like I I'm assuming that there and we don't we don't we weren't able to figure out exactly what they applied here but I'm assuming as things like the gravitational constant Planck's constant and this seems kind of crazy to me because it depends on the units that you're using it depends on a whole bunch of you know things that you have to assume about it but even when you do these these kind of arbitrary physical constants which I'm assuming they're doing here the first the most significant digit in these in these physical constants is still much more likely to be one it almost exactly follows Benford's law it's it's it's it's it kind of gives you goose pimples it's yeah it's a challenge here is to oh by the way Benford it is is the guy with the glass oh yes yes you might be wondering we know these aren't yes these are these these are not been frayed pre pretty shave and a post shave no no this right here this right here is Benford and obviously it was named after him Benford's law but but we put this gentleman who didn't shave the cool guy with the beard is Simon Newcomb Simon Newcomb is not Duke Nukem not Duke Nukem and we put him here because he's actually the first person who's dated Benford's law he obviously did not call it better yeah the better beard and and he had yes the most more he was overall more imposing character at least to me this guy looks a little bit like Harry Truman maybe this is Harry guy has had enough that's a maybe I've got the wrong picture anyway let's just look so the question what is kind of the pure case of this I mean when you got this random data you see some fluctuations like in our country right there there arms pretty close so it's pretty close is pretty good and our sample size is pretty low right right right right right we've got 200 countries or something like that yeah it went up by 150 you have to the Soviet Union fell and all that but yeah it's not a huge number of countries and even physical constants I don't know how many physical constants they randomly sampled over here but it is shockingly close to Binford's Locker but but there's there's kind of a more pure way of studying it right so um when we look at this other graph this is kind of like the pure Benford's law here our Benford's law so that's this curve that all these were kind of fitting to that other more rough data and and what's amazing here is that if we take kind of pure mathematical constructs like the powers of two or our the Fibonacci series if you just you'd think that in the Fibonacci series are adding all this stuff up and why and you just take the first digit and put them in these piles you it would actually exactly match Benford's law like not no deviation it is exactly mathematically mathematically so let's just because this is this is fascinating so if you were to take the powers of 2 so you get 1 2 4 8 16 32 64 I want to go pretty high so you can start to see how we're doing this 128 256 512 and you just keep going on and on forever with every power of 2 and you say okay how many of these start with a 1 and you would go in and you would say well this starts with a 1 this starts or the most significant digit is 1 that start the one that starts the one you would just find the percentage that start with the one and then you would plot it on on you would give the ones that credit for that percentage and then you say okay the ones that start with the two and you say okay that starts with the two and that starts with two but we want to keep going on this and we could probably do this with a computer program or something with really high powers of two and then you would and then you would say what percentage of all of these powers of two start with a two you would you would say get this percentage right over here this problem the number that start with a nine you would get this and you'd perfectly match Benford's law yeah this seems magical if it does seem pretty much include fibbin and this isn't just for powers of two this is powers of any number almost any number there's potential cases oh yeah one one would know this right powers of ten powers of ten would not work as well yes but every other numbers that kind of mix it up a little bit yes the every night you are correct every number that would mix it up a little bit yeah mixing it up a little right right but every yes they would do an exhibit Benford's distribution and so we want to challenge you to think about why that is and maybe you could even put your own explanations in our little message board on either YouTube or our page if you're curious but we'll challenge you to think why that is and then we'll offer at least a decent shot at an explanation maybe we'll have other explanations so actually we'll leave you there in this video and the next video will explain why why we think an intuitive reason why it works