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Benford's law (with Vi Hart, 1 of 2)

Vi Hart visits Khan Academy and talks about the mysteries of Benford's Law with Sal. Created by Sal Khan.

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  • male robot hal style avatar for user Greg Kroening
    What brought Benford to his discover of this pattern? Was it just curiousity? Did Benford apply this law to any use in science, finance, or engineering?
    (50 votes)
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    • blobby green style avatar for user Roger Allen Sweeny
      Actually, it was Simon Newcomb who noticed the wear on the book and originally stated the law. Though he published in 1881, the paper didn't have much impact. The law was rediscovered by physicist Frank Benford, who published in 1936. This time people took notice, and his name is now attached to the phenomenon.
      (39 votes)
  • leafers ultimate style avatar for user Marco Merlini
    Just two minutes ago I went to a site with the first 100000 digits of pi, and jotted down the number of times my browser said it found "1" on the page, then "2" on the page, etc etc. Expecting Benford's law, this made my results much more astounding. The number of times each digit appeared in comparison to all the others was stunningly regular, as each tally was between 9800 and 10200. I know it's kinda not related to this video, but why is this happening? What is going on?
    (28 votes)
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    • piceratops ultimate style avatar for user sumo99
      Pi is suspected to be a normal number. A normal number is a number that has a equal frequency for all it's digits, so five would be as common as six. However we do not know if Pi is actually a normal number.
      (33 votes)
  • leaf blue style avatar for user Matthew Zeitgeist Bishop
    Would this work in other number systems like octal, hexadecimal, binary. base 7 etc.?
    (21 votes)
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  • leaf green style avatar for user Justin Gold
    Montenegro IS a country! It is on the Adriatic in Eastern Europe. But he was thinking of Monaco, which is the little country on the eastern coast of France.
    (27 votes)
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    • starky ultimate style avatar for user Rohan
      Yes,it is Monaco, which has the feat of the world's most densely populated country in the world,with 36,000 residents in about one sq. mile.

      Montenegro is a country,but it was made in 2006, and was part of Serbia and Montenegro,so it may have not yet have become a country when Sal and Vi made the video.
      Good catch!
      (4 votes)
  • starky ultimate style avatar for user Arthur pan(多多or潘一乐)
    Who is Vi hart?
    (10 votes)
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  • leafers ultimate style avatar for user Shriya Nevatia
    In a more general sense, isn't this because every time you cross a numerical "threshold" the numbers all start with 1? as we go from 1's to 10's to 100's to 1000's, wherever these numbers stop there will be more numbers in the earlier part of that group than the later ones. (so for example, if you just took some random sample of a lot of numbers from 1 to 60, obviously more numbers would start with 1, 2, 3, 4, 5, or 6. if you did it with 1 to 3000, a lot would start with 1, 2, or 3)
    (7 votes)
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    • blobby green style avatar for user Dale Anderson
      I am thinking it is probably a property of exponential functions. Most of the examples given are exponential in nature. If you look at an exponential function it is constantly getting stepper. Hence within any given range the smaller numbers will take a larger amount of the x axis. So, if you pick a random assortment of numbers from the x axis and then find the exponential value, for any given magnitude, it is more likely to be from the lower end (ie closest to 0).
      (7 votes)
  • male robot hal style avatar for user Drew Summitt
    What is the distribution of Benford's law in base two? How would one go about finding that?
    (4 votes)
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  • mr pants teal style avatar for user joyshi456
    About the odds and evens thing, doesn't that make the most sense? because there are only 4 evens besides 0 and 5 odds.
    (4 votes)
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  • old spice man green style avatar for user iamsoBLAKE
    I wonder if Bernard's law behaves the same within other count systems, say like the Mayan or Roman count systems?
    (2 votes)
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    • male robot hal style avatar for user Ben Willetts
      Yes; the effect is independent of the counting system you use. If you count in base 60 like the Mayans then you will still see a 1 crop up as the most common first digit. The Roman example is slightly different because the Romans counted in base 10 like we do; they just wrote their numbers differently. In Roman numerals the concept of the "first digit" has no meaning because the numbers are written according to absolute value, not place value. So Benford's Law has nothing to apply to in the case of Roman numerals.
      (2 votes)
  • leaf red style avatar for user Jack
    Surely this couldn't be telling us something fundamentally about the universe though? isn't it just a side effect of the fact that we use a base ten number system? i mean would this still be true if you used a base Sexagesimal like the Babylonians?
    (2 votes)
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    • old spice man green style avatar for user Ivo van der Hoeven
      This is independent of what base number you use. The effect is observed in exponential functions, because those show linear distribution on a logarithmic scale (because they are linear on a logarithmic scale). If you've ever seen logarithmic graph paper, you will understand why ones are more likely to pop up.
      (2 votes)

Video transcript

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.