41–50 of 133 posts
Sorry, does the article actually give reasons why the bell curve is "everywhere"? For simplicity, take N identically distributed random variables that are uniform on the interval from [-1/2,1/2], so the probability distribution function, f(x), on the interval from [-1/2,1/2] is 1. The Fourier transform of f(x), F(w), is essentially sin(w)/w. Taking only the first few terms of the Taylor expansion, ignoring constants,…
This is one of my favorite philosophical questions to ponder. I always ask it in interviews as a warmup to get their thoughts. I’ve noticed that interviewees often curl up, thinking it’s a technical question, so I’ve been modifying the question one after the other to make it less scary. The interviews are for data scientist roles.
Hot take: bell curves are everywhere exactly because the math is simple. The causal chain is: the math is simple -> teachers teach simple things -> students learn what they're taught -> we see the world in terms of concepts we've learned. The central limit theorem generalizes beyond simple math to hard math: Levy alpha stable distributions when variance is not finite, the Fisher-Tippett-Gnedenko theorem and Gumbel/Fr…
Most things aren't infinite or extreme, though. Almost by definition, most phenomena aren't extreme phenomena.
In practice when modeling you are almost always better not assuming normality, and you want to test models that allow the possibility of heavy tails. The CLT is an approximation, and modern robust methods or Bayesian methods that don't assume Gaussian priors are almost always better models. But this of course brings into question the very universality of the CLT (i.e. it is natural in math, but not really in nature).
This is one of my favorite philosophical questions to ponder. I always ask it in interviews as a warmup to get their thoughts. I’ve noticed that interviewees often curl up, thinking it’s a technical question, so I’ve been modifying the question one after the other to make it less scary. The interviews are for data scientist roles.
All summation roads lead to normal curves. (There might be an exception for weird probability distributions that do not have a mean; I was surprised when I learned these exist.)
Life is full of sums. Height? That's a sum of genetics and nutrition, and both of those can be broken down into other sums. How long the treads last on a tire? That's a sum of all the times the tire has been driven, and all of those times driving are just sums of every turn and acceleration.
I'm not a data scientist. I'm just a programmer that works with piles of poorly designed business logic.
How did I do in my interview? (I am looking for a job.)
> suppose that a large sample of observations is obtained, each observation being randomly produced in a way that does not depend on the values of the other observations, and the average (arithmetic mean) of the observed values is computed. If this procedure is performed many times, resulting in a collection of observed averages, the central limit theorem says that if the sample size is large enough, the probability distribution of these averages will closely approximate a normal distribution.
Bell curves are everywhere because all distributions of any properties clump in some way at some level. The basics of any probability shows this. The result is you “seeing” bell curves everywhere. Aka clumps. This is a tautology to the extreme.
A little disappointing. All about the history of bell curves, but I don't think it does a very good job explaining why the bell curve appears or the CLT is as it is.
Bell curves are everywhere because all distributions of any properties clump in some way at some level. The basics of any probability shows this. The result is you “seeing” bell curves everywhere. Aka clumps. This is a tautology to the extreme.
Second, your "aka" is incorrect --- there is all sorts of clumping that is not a normal distribution.
100 year floods are not happening more often in most cases - it is just that the central limit therom teachs us the 10 year flood is almost as high water as the 100 or even 1000 year flood.
No, the central limit theorem specifically doesn't address that. It says that the sum of iid random variables is well approximated by a normal distribution near the mean; it doesn't tell you how well that approximation works in the tails. The rarer the event you're modeling is, the less relevant the normal approximation is.
Sorry, does the article actually give reasons why the bell curve is "everywhere"? For simplicity, take N identically distributed random variables that are uniform on the interval from [-1/2,1/2], so the probability distribution function, f(x), on the interval from [-1/2,1/2] is 1. The Fourier transform of f(x), F(w), is essentially sin(w)/w. Taking only the first few terms of the Taylor expansion, ignoring constants,…
It's not super hard to prove the central limit theorem, and you gave the flavor of one such proof, but it's still a bit much for the likely audience of this article, who can't be assumed to have the math background needed to appreciate the argument. And I think you're on the right track with the comment about stable distributions.
Widths of different uniform distributions along with different centers all still have a quadratic center, so the above argument only needs to be minimally changed.
The added bonus is that if the (1-w^2)^n is replaced by (1-w^a)^n, you can sort of see how to get at the Levy stable distribution (see the characteristic function definition [0]).
The point is that this gives a simple, high-level motivation as to why it's so common. Aside from seeing this flavor of proof in "An Invitation to Modern Number Theory" [1], I haven't really seen it elsewhere (though, to be fair, I'm not a mathematician). I also have never heard the connection of this method to the Levy stable distributions but for someone communicating it to me personally.
I disagree about the audience for Quanta. They tend to be exposed to higher level concepts even if they don't have a lot of in depth experience with them.
[0] https://en.wikipedia.org/wiki/Stable_distribution#Parametriz...
[1] https://www.amazon.com/Invitation-Modern-Number-Theory/dp/06...