 ## Subtitles section Play video

• STEVE MOULD: This is Benford's Law.

• 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

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

• 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

• So nine is down here--

• tiny number.

• And then you get this brilliant curve that

• goes up like that.

• Isn't that crazy?

• 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.

• STEVE MOULD: So the explanation is to do with

• scale invariance, which I'm just getting

• 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.

• 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

• 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

• 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.

Subtitles and vocabulary

Operation of videos Adjust the video here to display the subtitles

B1 raffle probability ticket digit leading distribution

# Number 1 and Benford's Law - Numberphile

• 39 0
fisher posted on 2013/04/09
Video vocabulary