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?