Humans aren’t as smart as Pokemon, but we might be in 7.7 million years

Co Written with Eric Bakan

So someone on my facebook posted that according to the Pokedex that, my favorite type of pokemon, Alakazam has an IQ of 5000. Now, for humans, IQ follows a distribution that is normal with mean 100, and standard deviation of 15. So that means that Alakazam is about 327 standard deviations above normal! Now, 327 std deviations above the normal is pretty rare, but exactly how rare?

So we wanted to ask, how smart has the smartest human ever been? And how many humans would there need to match Alakazam’s staggering intellect?

So time for statistics!

First things first, let’s normalize our average person, so that on average they have an normalized IQ of 0, with standard deviation 1. This means that instead of thinking about 5000, we care about (5000–100)/15 ~= 327. [So instead of thinking of the IQ we are thinking of the z-score]. And if we denote Xi to be the normalized IQ of the ith person we have

And we care about

More specifically we care about the expected value of Y, and we care about it as a function of N (so right now, about 100 billion people have ever lived, so N = 10¹¹ )

Okay, so now we claim that Y is always (well almost always) positive for a large N.

The way to see this is that a standard normal random variable is symmetric about zero, so there is a 50/50 chance it will be positive or negative (and a 0 chance it will be zero!)

So then the only way for Y to be negative is if all the X’s are negative, since Y is just the maximum of all the X’s. You can think of this like a coin flip, where for Y to be negative all the coin flips have to be heads. Each X has a 1/2 chance of that, so the chance they are all negative is (1/2)^N . And so we know that

Which goes to 1 as N goes to infinity.

So the reason we want to use is because of this nifty identity for positive random variables

Details here

So we know that

Now we just have to figure out that probability. This can be done with simple trick about maxes (also works for mins)[in general these things are called the order statistics, but order statistics that aren’t the max and min are hard to work with].

The trick is this: the maximum of a bunch of iid random variables is less than some number if and only if each random variable was less than that number.

So we can actually write it out explicitly as

Where Phi is the CDF for a standard normal. So now we have

So if we plug this into a numerical integral estimator with N = 100 billion, we get that E[Y] = 6.78, so the smartest person who has ever lived is expected to have an IQ of 100 + 15*6.78 = 202.

So now we want to figure out how many humans will need to have lived for us to expect someone 327 std deviations away from the mean, which means solving for N

Here is when things start to get tricky — because if you start raising numbers to the 10⁸⁰ power and such, they will underflow and after a certain point any N will give you the same number. So what we did was switch to an arbitrary precision arithmetic library in python to try and figure out the population size needed.

Since these integral takes a bit to compute, and you have to keep increasing the precision (which makes it take longer and longer), we decided to regress on

Getting the graph, and estimating F as follows

X-axis is log(N), and Y axis is number of standard deviations

From this we can estimate F(N) = 327, which solves to N = 10³⁸⁰⁶³, which is quite a lot!!!

For some perspective, there are ~10⁸⁰ atoms in the universe. So we need more people than atoms by about 10³⁸⁰⁰⁰. Now, if humans kept growing at our current rate, into different universes, how many years would it take us to get there?

Well, if we assume that humanity is growing at our current rate of 1.13% we get

Which solves to around T = 7.7 million years.

So in conclusion

• The smartest person to ever live up until now had an IQ of around 202
• We will need about 10³⁸⁰⁰⁰ people to get to the 5000 IQ level
• It will take 7.7 million years to match up with Alakazam