Benford's law (with Vi Hart, 1 of 2)

SAL KHAN: I am very excited to have Vi Hart visiting the office over here. And we were just having a very mathematical conversation earlier today. And she mentioned something that is fascinating. VI HART: Yeah, I was just telling Sal about a cool thing called Benford's Law. SAL KHAN: Benford's Law. And what is Benford's law? VI HART: 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 all right, what is the first digit of the population of the country? Whether it's 1 million or 1,000 or 100,000, we'll say, OK, 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. SAL KHAN: Yes, about 27. So literally anything that starts with a 1 here. So it could be a country that has a population of 1, a population of 17, or a population of 1 billion, 300 million, blah, blah, blah. They would all fall into this bucket right over here. VI HART: Right. And then if you start with 2, you fall in the second bucket, and so on and so forth. SAL KHAN: Better color. Oh yeah, go ahead. VI HART: Yeah, definitely better color. SAL KHAN: Better color. VI HART: Oh, that's great. SAL KHAN: Yeah, that's the blue, better contrast. VI HART: So the question is, all right, you'd think you'd have kind of random numbers here. SAL KHAN: Yes. For that first digit, it's kind of random. VI HART: Yeah. I mean, there's huge differences in populations of countries. Some have billions and some have-- I don't know what the smallest population is. SAL KHAN: Yes. It's, like, Montenegro or something like that. VI HART: Yeah. So-- SAL KHAN: Montenegro is not a country. What am I thinking? I'm thinking of-- what's the one that's on the French Riviera? Anyway, we can edit that out. [LAUGHTER] VI HART: I don't know. I would be interested to see for populations of states and everything. SAL KHAN: The Vatican is the smallest country, I believe. VI HART: Yeah, it is. Does that still count? I guess-- SAL KHAN: I think the Vatican counts. They have their own-- yes. VI HART: I don't know exactly what the requirements were to be. [INAUDIBLE]. SAL KHAN: But it would include the Vatican, which I think would be in, like, the thousands. VI HART: Yeah. And so why would this happen? Why would you see more 1's than 2's? Like, what is going on? SAL KHAN: Yeah. And it's not some small chance. I mean, we were talking also about the idea that it is more likely to have odd-numbered addresses than even-numbered. We were talking about that earlier. VI HART: Yeah, I just learned about that recently. And that makes sense. SAL KHAN: That makes sense, because every house will have a number 1 on it, or a 10. VI HART: Right, every street, if your street starts, you know, with house number 1, house number 2, house number 3, if you have an odd number of houses, then your street has more odd numbers than even numbers. SAL KHAN: Exactly. And if you have an even number, you have the same amount. VI HART: Right. SAL KHAN: Right. VI HART: But that's starting with 1, which is odd. Whereas here, populations don't start with 1. SAL KHAN: Exactly. And that phenomenon that we're talking about the street numbers, it's not an extreme phenomenon. It's like, 50 point some 0-- you have a slightly higher probability of having an odd-numbered house, or I guess a 1 house, than everything else. VI HART: Yeah, it's kind of exactly what you would expect. SAL KHAN: It's exactly what you'd expect. But here, it's a significantly higher probability, of a random country's population, that its first digit is a 1 versus its first digit as an 8 or a 9. I mean, it seems a little bit strange. VI HART: Yeah. And this isn't just in countries. You see this if you're looking at a lot of financial stuff. Like, how much money does a company make. SAL KHAN: Yes. The 1's just show up as a first digit much more frequently. VI HART: Much more frequently, yeah. And here, we have another fancy graph, which is completely crazy. It's the first digit of physical constants. So what would be examples of some physical constants? SAL KHAN: I'm assuming that-- and we weren't able to figure out exactly what they applied here-- but I'm assuming it's 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 things that you have to assume about it. But even when you use these kind of arbitrary physical constants, which I'm assuming they're doing here, the most significant digit in these physical constants is still much more likely to be 1. It almost exactly follows Benford's Law. And it kind of gives you goose pimples. VI HART: Yeah. So the challenge here is to-- oh, by the way, Benford is the guy with the glasses. SAL KHAN: Oh, yes. Yes, you might be wondering. These aren't-- yes. These are-- VI HART: They're not both Benford. SAL KHAN: These are not Benford pre-shave and post-shave. No. This right there is Benford. And obviously it was named after him, Benford's Law. But we put this gentleman, who didn't shave-- VI HART: Yeah, the cool guy with the beard is Simon Newcomb. SAL KHAN: Simon Newcomb. VI HART: Not Duke Nukem. SAL KHAN: Not Duke Nukem. And we put him here because he's actually the first person who stated Benford's Law. He obviously did not call it Benford's Law. VI HART: And he had the better beard. SAL KHAN: And he had, yes, the most-- he was overall a more imposing character. At least to me. This guy looks a little bit like Harry Truman. Maybe this is Harry. I don't know. Yeah, maybe I've got the wrong picture. Anyway, let's just-- VI HART: So the question, what is kind of a pure case of this? I mean, when you've this random data, you see some fluctuations. Like, in a country, right, there are more-- SAL KHAN: It's pretty close, though. It's pretty close to that curve. VI HART: Yeah, and our sample size is pretty low. I mean-- SAL KHAN: Right. VI HART: --when you've only got-- SAL KHAN: There's, like, 200 countries or something like that. VI HART: Yeah. SAL KHAN: It went up by, like, 50-- VI HART: It doesn't perfectly follow it. SAL KHAN: --after 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 Benford's law curve. But there's kind of a more pure way of studying it. VI HART: Right. So when we look at this other graph, this is kind of like the pure Benford's Law. SAL KHAN: Pure Benford's Law. So that's this curve that we're kind of fitting to that other, more rough data. And what's amazing here is that if we take kind of pure mathematical constructs, like the powers of 2, or-- VI HART: Or the Fibonacci series. You'd think that in the Fibonacci series, you're adding all this stuff up, and why-- SAL KHAN: And then you just take the first digit-- VI HART: Just the first digit. SAL KHAN: --and put them in these piles, it would actually exactly match Benford's Law. Like, no deviation. It is exactly-- VI HART: Right, mathematically [INAUDIBLE]. SAL KHAN: Mathematically. So let's just be clear, because this is fascinating. So if you were to take the powers of 2's, you get 1, 2, 4, 8, 16, 32, 64. I'm gonna 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, OK, how many of these start with a 1? And you would go and you would say, well, this starts with a 1, this starts with-- or the most significant digit is 1. That starts with a 1, that starts with a 1. And you would just find the percentage that start with a 1. And you would plot it on-- you would give the 1's that credit for that percentage. And then you'd say, OK, the ones that start with a 2. And you'd say, OK, that starts with a 2, and that starts with 2. But we want to keep going on this. And we could probably do this with a computer program or something, where we'd go really high powers of 2. And then you would say, what percentage of all of these powers of 2 start with a 2? You would, say, get this percentage right over here. The numbers that start with a 9, you would get this. And you would perfectly match Benford's Law. This seems magical. VI HART: It does seem pretty magical. SAL KHAN: And this isn't just for powers of 2. This is powers of any number. VI HART: Almost any number. There's special cases. SAL KHAN: I believe it might be every number. Oh, yes. No. VI HART: Well, powers of 1. SAL KHAN: 1 would not work, right. VI HART: And then there's, like, powers of 10. SAL KHAN: Powers of 10 would not work as well, yes. But every other-- numbers that kind of mix it up a little bit. VI HART: Powers of 0. SAL KHAN: Yes. You are correct. Every number that would mix it up a little bit. VI HART: Yeah. It'd have to have a certain amount of mixing it up a little bit. SAL KHAN: Right, right. But every other, yes, they would 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. VI HART: Yeah. SAL KHAN: So actually, we'll leave you there in this video. And the next video will explain why we think-- an intuitive reason why it works.