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  • STEVE MOULD: This is Benford's Law.

  • And it's about numbers, but it's about the leading digit.

  • For example, 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 the one.

  • Benford's Law works on a distribution of numbers if

  • that distribution spans quite a few orders of magnitude.

  • And the brilliant thing about populations of countries is

  • that it actually goes from tens up to billions.

  • If you were to think about that, OK, what are the

  • distribution of leading digits.

  • So some of the populations will start with the one, some

  • will start with two, three, four, five, six,

  • seven, eight, or nine.

  • And so there are nine possible leading digits.

  • And you might imagine that each one of those possible

  • leading digits are equally likely to appear.

  • So that's one in nine--

  • 11%.

  • And if I was to plot that on a graph, you might expect that

  • to fluctuate around 11%.

  • So it's going to go like that.

  • So what actually happens is that a third of the time,

  • that's up here.

  • 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 is down here--

  • tiny number.

  • And then you get this brilliant curve that

  • goes up like that.

  • Isn't that crazy?

  • BRADY HARAN: I know you talk about this sometimes in talks

  • and things you do.

  • What's the reaction to that normally when

  • you tell people this?

  • STEVE MOULD: The reaction?

  • The noise is sort of like this--

  • ohh.

  • 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 tweet numbers to us.

  • So we're collecting numbers, and I try to give them ideas.

  • So maybe like, 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, like I was saying, it works so long

  • as the distribution you're choosing from spans 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 meta-distribution of individual things picked from

  • different distributions follows Benford's Law anyway.

  • So it works brilliantly well.

  • BRADY HARAN: What clump of numbers will

  • this not work for?

  • STEVE MOULD: Human height in meters.

  • So humans are between one meter and three meters.

  • So it doesn't work for that.

  • You get a massive load around there.

  • And no one's nine meters tall.

  • Anything that has that short distribution, it

  • doesn't work for.

  • But it does work for several distributions put together

  • that don't necessarily individually follow the rule.

  • So I did it for populations.

  • I did it for areas of countries

  • in kilometers squared.

  • If you take one number and convert it to loads of

  • different units, that will tend to follow

  • Benford's Law as well.

  • You can do it for the Financial Times.

  • Look at all the numbers on the front cover of

  • the Financial Times.

  • They will tend to follow Benford's Law as well.

  • BRADY HARAN: Just a quick interjection--

  • you can also apply this to the number of times you watch

  • Numberphile videos or leave comments underneath.

  • More information at the end of the video.

  • STEVE MOULD: So the explanation is to do with

  • scale invariance, which I'm just getting

  • my head around 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.

  • 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 in two.

  • It's 50%.

  • But then if you increase the size of the raffle, so there

  • are now three tickets in the raffle, the chance now are one

  • in three or about 33%.

  • If you add a fourth ticket, then the probability of the

  • leading digit of the winning ticket being a one is now 25%,

  • and then 20%, 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 a

  • leading digit of one is one in nine.

  • It's 11%, which was the intuitive thing

  • that you might think.

  • But then you add your tenth ticket.

  • And now there are two tickets that start with a one.

  • So now the probability is 2 in 10 or 1 in 5.

  • So it would go back up to 20%.

  • The probability will go up, and up, and up as you add more

  • tickets that start 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 winning ticket having a

  • leading digit of one will go down, and down, and down

  • through the 20s.

  • It will go down through the 40s, down through the 50s,

  • 60s, 70s, 80s, 90s, until you add the hundredth ticket.

  • And then the probability will start to go up again.

  • And then the probability will go up, and up, and up, all the

  • way through the 100s.

  • And then you get to the 200s, and it goes down, and down,

  • and down through all the 200s, 300s, 400s, 500s, 600s, 700s,

  • 800s, 900s.

  • And you'll be back to 11% again then.

  • 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 and then down through the 2000s, 3000s,

  • blah, blah, blah.

  • And then you get to 10,000 and it goes up.

  • And so basically the probability of the winning

  • ticket having any digit of one fluctuates as the size of the

  • raffle increases.

  • And so this is a log scale of the raffle increasing in size.

  • So you might have a 10, 100, 1,000, 10,000, and so on.

  • And then this is the probability here of having a

  • leading digit of one.

  • It 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 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 log to base 10 of 1 plus 1/d, like that.

  • And so that's how you do it.

  • And if you plug one into there, then it's

  • log base 10 of two.

  • And it ends up being about 30%.

  • The beauty is that you can do it in any base.

  • So this doesn't have to be base 10.

  • It could be base five, base 16, whatever you want to do.

  • You can apply Benford's Law to different bases.

  • This is a formula that a forensic accountant would use

  • as a tax formula of 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 formula you need to remember if you're going to

  • cheat on your tax return.

  • BRADY HARAN: A lot of things that mathematically inclined

  • people like yourself tell me when I hear about them seem

  • counter-intuitive.

  • 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.

  • When someone says one will come up more often, to me that

  • just seems like, of course that would happen.

  • STEVE MOULD: Yes.

  • Funny isn't it?

  • Some people are like you.

  • I would say you're in the minority of people that go,

  • well, yeah.

  • And I wonder if there is another intuitive way of

  • looking at it that you've tapped into, which is that if

  • you imagine something like the NASDAQ index or

  • something like that--

  • and I don't know what the NASDAQ index is size-wise--

  • but imagine that the NASDAQ index is at 1,000.

  • To change that to 2,000, you'd 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 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 maybe that's intuitive to you.

  • So you're like, yeah obviously.

  • BRADY HARAN: If you'd like to see even more about Benford's

  • Law, we've done a bit of a statistical analysis to find

  • out whether or not your viewing habits and the number

  • of times you comment on Numberphile 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?

STEVE MOULD: This is Benford's Law.

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B1 raffle probability ticket digit leading distribution

Number 1 and Benford's Law - Numberphile

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    fisher posted on 2013/04/09
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