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The math that explains why bell curves are everywhere

207 points by ibobev 8mo ago

DroneBetter 8mo ago
I hate Quanta a lot a vast amount of fluff for less than a college statistics professor would (hopefully) be able to impart with a chalkboard in 10 minutes, when Quanta has the ability to prepare animated diagrams like 3Blue1Brown but chooses not to use it they could go down myriad paths, like how it provides that random walks on square lattices are asymptotically isotropic, or give any other simple easy-to-understand applications (like getting an asymptotic on the expected # of rolls of an n-sided die before the first reoccurring face) or explain what a normal distribution is, but they only want to tell a story to convey a feeling they are a blight upon this world for not using their opportunity to further public engagement in a meaningful way
- tptacek 8mo ago
  A lot of times on HN when a math topic comes up that isn't about 3b1b, someone will jump in to say "this isn't as good as 3b1b". Last time I saw that, I was moved to comment: https://news.ycombinator.com/item?id=45800657 https://news.ycombinator.com/item?id=45800657 3b1b doesn't have the same goal as Quanta, or as introductory guides. It's actually not that great a teaching tool (it's truly great at what it is for, which is (a) appreciation and motivation, and (b) allowing people to signal how smart they are on message board threads by talking about how much people would get out of watching 3b1b). This is prose writing about math. It's something you're meant to read for enjoyment. If you don't enjoy it, fine; I don't enjoy cowboy fiction. So I don't read it. I don't so much look for opportunities to yell at how much I hate "The Ballad of Easy Breezy".
  - bmenrigh 8mo ago
    I don’t fault Quanta (or 3b1b) for being the way they are. Each is serving their goal audience pretty well. My compliant is only that there should be a dozen more just like them, each competing with each other for the best, most engaging math and science content. This would allow for more a broader audience skillevel to be reached. As it stands, we’re lucky even to have Quanta and 3b1b. I think there is hope though, quite a few new-ish creators on YouTube are following in Grant’s footsteps and producing very technically detailed and informative content at similar quality levels.
  - paulpauper 8mo ago
    there is no getting around that learning math requires actually having to buckle down and read and do math . A video will not suffice.
    - tptacek 8mo ago
      Couldn't agree more, which is why I think it's odd to suggest that a pop-sci magazine article is somehow a disservice that 3b1b would correct.
    - DroneBetter 8mo ago
      well for one who does buckle down and read and do math, the expected amount of new information brought to them by a 3B1B video as supplementary material upon a topic (with the normal distribution being one that admits a direct comparison from the article) is nonzero, by merit of it possibly having ideas to convey from outside their usual purview and formal background that may be applicable to the doing of math (as has been the case for me, someone who [does math](https://oeis.org/wiki/User:Natalia_L._Skirrow https://oeis.org/wiki/User:Natalia_L._Skirrow)), while for Quanta fluff pieces it's zero. by the metric of "if this expository piece were to be taken to a time before its subject had been considered and presented to researchers, how useful would its outline be towards reproducing the theory in its totality," Quanta's writings (on both classical and research math) mostly score 0
- deleted 8mo ago
  [deleted]
- KnuthIsGod 8mo ago
  3Blue1Brown Seems a bit like Ted Talks. Lightweight popcorn for the simple minded.
- throwaway81523 8mo ago
  Quanta used to have tons of good stuff and not much crap. Now there's enough crap that if there's still good stuff, it gets lost in the noise.
- andyjohnson0 8mo ago
  I probably don't have your mathematical sophistication - but I like and appreciate Quanta precisely because it helps people like me to understand a little bit about challenging things. This enriches my tiny life, and I hope it also makes the world a fractionally better place for us all. Perhaps you're just not in their intended audience?
mikrl 8mo ago
Great article. Personally I have been learning more about the mathematics of beyond-CLT scenarios (fat tails, infinite variance etc) The great philosophical question is why CLT applies so universally. The article explains it well as a consequence of the averaging process. Alternatively, I’ve read that natural processes tend to exhibit Gaussian behaviour because there is a tendency towards equilibrium: forces, homeostasis, central potentials and so on and this equilibrium drives the measurable into the central region. For processes such as prices in financial markets, with complicated feedback loops and reflexivity (in the Soros sense) the probability mass tends to ends up in the non central region, where the CLT does not apply.
- parpfish 8mo ago
  As to ye philosophy of “why” the CLT gives you normals, my hunch is that it’s because there’s some connection between: a) the CLT requires samples drawn from a distribution with finite mean and variance and b) the Gaussian is the maximum entropy distribution for a particular mean and variance I’d be curious about what happens if you starting making assumptions about higher order moments in the distro
  - sobellian 8mo ago
    IIRC there's a video by 3b1b that talks about that, and it is important that gaussians are closed under convolution.
    - gowld 8mo ago
      That makes it an equilibrium point in function space, but the other half is why it's an a global attractor.
      - pfortuny 8mo ago
        There must be a contractive nature in "passing to the limit". And then Brower's fixed point theorem. (I know it is very easy to do "maths" this way).
  - 8mo ago
fritzo 8mo ago
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/Fréchet/Weibull distributions regarding extreme values. Those curves are also everwhere, but we don't see them because we weren't taught them because the math is tough.
- AndrewKemendo 8mo ago
  That’s exactly the right take and the article proves it: Statisticians love averages so everywhere that could be sampled as a normal distribution will be presented as one The median is actually more descriptive and power law is equally as pervasive if not more
  - fsckboy 8mo ago
    combining repeated samples of any distribution* (any population density fuction including power law distributions) will converge to the normal distribution, that's why it appears everywhere. * excluding bizarre degenerates like constants or impulse functions
    - abetusk 8mo ago
      No, that's not correct. Sums of power law distributions can converge to power low tailed distributions, not normal distributions.
      - AndrewKemendo 8mo ago
        No use arguing with them they don’t have enough mathematical understanding to understand what they’re saying
- BobbyTables2 8mo ago
  It also took me a little while to realize “least squares” and MMSE approaches were not necessarily the “correct” way to do things but just “one thing we actually know how to do” because everything else is much harder. We can use Calculus to do so much but also so little…
gowld 8mo ago
3b1b playlist on Central Limit Theorem: https://www.youtube.com/playlist?list=PLZHQObOWTQDOMxJDswBaLu8xBMKxSTvg8 https://www.youtube.com/playlist?list=PLZHQObOWTQDOMxJDswBaL... He has several other related videos also. https://www.youtube.com/@3blue1brown/search?query=convolution https://www.youtube.com/@3blue1brown/search?query=convolutio...
bluGill 8mo ago
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.
- gowld 8mo ago
  Explain? What are "most cases"?
- thaumasiotes 8mo ago
  > 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.
tsunamifury 8mo ago
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.
- D-Machine 8mo ago
  Yup. And in general more heavy-tailed bumps are in fact better models (assuming normality tends to lead to over-confidence). Really think the universality is strictly mathematical, and actually rare in nature.
- jibal 8mo ago
  First, every mathematical theorem is a tautology ... don't conflate "tautological" with "obvious". Second, your "aka" is incorrect --- there is all sorts of clumping that is not a normal distribution.
  - thaumasiotes 8mo ago
    As I'm sure tsunamifury would agree, it is incredibly common for people to label "bell curves" by eyeball, regardless of whether they are normal curves. To most people, "clumping" in a one-dimensional spectrum is all they mean by the phrase "bell curve".
    - tsunamifury 8mo ago
      [dead]
    - D-Machine 8mo ago
      This was sort of my reading as well: I took "clumping" to mean "bump-shaped".
    - jibal 8mo ago
      This completely misses the point, which is that the central limit theorem says that it isn't just any old clumping, it's always the normal distribution. tsunamifury dismissed this strong finding as "tautology" because clumping is obvious ... but that it's always precisely a bell curve is far from obvious. Again, > your "aka" is incorrect --- there is all sorts of clumping that is not a normal distribution. That it's "incredibly common for people to label "bell curves" by eyeball, regardless of whether they are normal curves" is not just not relevant, it's anti-relevant ... the central limit theorem says that the distribution of the means is always a bell curve--a normal distribution--not merely a "bell curve". Anyway, this is covered in far more detail in other comments and material elsewhere, so this is my last contribution.
EGreg 8mo ago
Related: https://magarshak.com/blog/explaining-benfords-law/ https://magarshak.com/blog/explaining-benfords-law/
gwern 8mo ago
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.
- jibal 8mo ago
  https://en.wikipedia.org/wiki/Central_limit_theorem https://en.wikipedia.org/wiki/Central_limit_theorem
fiforpg 8mo ago
On opening the article, I was somehow expecting a mention of the large deviations formalism, which was (is?) fashionable in late 20th century, and gives a nice information theoretic view of the CLT. Or something like that. There's a ton of deep math there. So having a bio statistician say "look, the CLT is cool" is a bit underwhelming. Edit: see eg John Baez's write-up What is Entropy? about the entropy maximization principle, where gaussians make an entrance.
abetusk 8mo ago
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, gives (1-w^2). Convolution is multiplication in Fourier space, so you get (1-w^2)^n. Squinting, (1-w^2)^n ~ (1-n w^2 / n)^n ~ exp(-n w^2). The Fourier transform of a Gaussian is a Gaussian, so the result holds. Unfortunately I haven't worked it out myself but I've been told if you fiddle with the exponent of 2 (presumably choosing it to be in the range of (0,2]), this gives the motivation for Levy stable distributions, which is another way to see why fat-tailed/Levy stable distributions are so ubiquitous.
- WCSTombs 8mo ago
  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.
  - abetusk 8mo ago
    The Fourier transform of a uniform distribution is the sinc function which looks like a quadratic locally around 0. Convolution to multiplication is how the quadratic goes from downstairs to upstairs, giving the Gaussian. 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#Parametrizations https://en.wikipedia.org/wiki/Stable_distribution#Parametriz... [1] https://www.amazon.com/Invitation-Modern-Number-Theory/dp/0691120609 https://www.amazon.com/Invitation-Modern-Number-Theory/dp/06...
WCSTombs 8mo ago
It's not a bad article, but I have to point something out: > Laplace distilled this structure into a simple formula, the one that would later be known as the central limit theorem. No matter how irregular a random process is, even if it’s impossible to model, the average of many outcomes has the distribution that it describes. “It’s really powerful, because it means we don’t need to actually care what is the distribution of the things that got averaged,” Witten said. “All that matters is that the average itself is going to follow a normal distribution.” This is not really true, because the central limit theorem requires a huge assumption: that the random process has finite variance. I believe that distributions that don't satisfy that assumption, which we can call heavy-tailed distributions, are much more common in the real world than this discussion suggests. Pointing out that infinities don't exist in the real world is also missing the point, since a distribution that just has a huge but finite variance will require a correspondingly huge number of samples to start behaving like a normal distribution. Apart from the universality, the normal distribution has a pretty big advantage over others in practice, which is that it leads to mathematical models that are tractable in practice. To go into a slightly more detail, in mathematical modeling, often you define some mathematical model that approximates a real-world phenomenon, but which has some unknown parameters, and you want to determine those parameters in order to complete the model. To do that, you take measurements of the real phenomenon, and you find values for the parameters that best fit the measurements. Crucially, the measurements don't need to be exact, but the distribution of the measurement errors is important. If you assume the errors are independent and normally distributed, then you get a relatively nice optimization problem compared to most other things. This is, in my opinion, about as much responsible for the ubiquity of normal distributions in mathematical modeling as the universality from the central limit theorem. However, as most people who solve such problems realize, sometimes we have to contend with these things called "outliers," which by another name are really samples from a heavy-tailed distribution. If you don't account for them somehow, then Bad Things(TM) are likely to happen. So either we try to detect and exclude them, or we replace the normal distribution with something that matches the real data a bit better. Anyway, to connect this all back to the central limit theorem, it's probably fair to say measurement errors tend to be the combined result of many tiny unrelated effects, but the existence of outliers is pretty strong evidence that some of those effects are heavy-tailed and thus we can't rely on the central limit theorem giving us a normal distribution.
AxEy 8mo ago
I remember seeing one of these https://en.wikipedia.org/wiki/Galton_board https://en.wikipedia.org/wiki/Galton_board at the (I think) Boston Science Museum when I was a kid. They have some pretty cool videos on Youtube if you're curious.
nsnzjznzbx 8mo ago
So Abraham de Moivre was the worlds first quant?
bicepjai 8mo ago
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.
- hilliardfarmer 8mo ago
  A lot of times I can't tell if I'm the idiot or if everyone else is. Says that this isn't an interesting question at all and the article was horrible. I studied data science for a few years but I'm no expert, but it seems pretty obvious to me that if you make a series of 50/50 choices randomly, that's the shape you end up with and there's really nothing more interesting about it than that.
  - alanbernstein 8mo ago
    I don't think "obvious" is the right word here. It makes perfect sense when you understand it, but it's not a conclusion that most people could come to immediately without detailed, assisted study.
  - smcin 8mo ago
    Sampling 50/50 choices would be a binary distribution that (very crudely) approximates a normal distribution. But the counterintuitive thing about the CLT is that it applies to distributions that are not normal.
- Buttons840 8mo ago
  I haven't read the article, but my understanding is that a normal curve results from summing several samples from most common probability distributions, and also a normal curve results from summing many normal curves. 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.)
jibal 8mo ago
https://en.wikipedia.org/wiki/Central_limit_theorem https://en.wikipedia.org/wiki/Central_limit_theorem > 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.
- abetusk 8mo ago
  That sentence is flat out wrong. If the probability distribution converges, it converges to a Levy stable distribution [0]. [0] https://en.wikipedia.org/wiki/Stable_distribution https://en.wikipedia.org/wiki/Stable_distribution
throwaway81523 8mo ago
Now do power laws.
causalityltd 8mo ago
Causes mostly add up: molecular kinetic energies aggregate to temperature, collisions to pressure, imperfections to measurement errors, etc. So, normal or CLT is the attractor state for the unexceptional world. BUT for the exceptional world, causes multiply or cascade: earthquake magnitudes, network connectivity, etc. So, you get log-normal or fat-tailed.
GeoSys 8mo ago
It's in many places, but not everywhere. CLT means that samples tend towards the mean, which is neat. Unfortunately, many "researchers" blindly assume that many real life phenomena follow Gaussian, which they don't... So then their models are skewed
bandrami 8mo ago
I flinch at "everywhere", particularly when people keep asserting they are places that they aren't (and in fact can't be). Nothing with a hard zero can be normally distributed, for instance, but people will keep insisting quantities with a hard zero are.
- Epa095 8mo ago
  Is this not just a linguistic issue, where people say normal distributed but actually mean approximate or assumed normality? Its not like height is normally distributed (there is nobody 8 feet tall), but its not like the distribution bares no resemblance to the normal distribution either, and in a colloquial sense the term seems to he used more freely than the mathematical defined term.
srean 8mo ago
A result of broader applicability is that of convergence to infinitely divisible distributions, more generally the stable distributions https://en.wikipedia.org/wiki/Infinite_divisibility_(probability) https://en.wikipedia.org/wiki/Infinite_divisibility_(probabi... https://en.wikipedia.org/wiki/Stable_distribution https://en.wikipedia.org/wiki/Stable_distribution This applies even when the variance is not finite. Note independence and identical nature of distribution is not necessary for Central Limit Theorem to hold. It is a sufficient condition, not a necessary one, however, it does speed up the convergence a lot. Gaussian distribution is a special case of the infinitely divisible distribution and is the most analytically tractable one in that family. Whereas, averaging gives you Gaussian as long as the original distribution is somewhat benign, the MAX operator also has nice limiting properties. They converge to one of three forms of limiting distributions, Gumbel being one of them. The general form of the limiting distributions when you take MAX of a sufficiently large sample are the extreme value distributions https://en.wikipedia.org/wiki/Generalized_extreme_value_distribution https://en.wikipedia.org/wiki/Generalized_extreme_value_dist... Very useful for studying record values -- severest floods, world records of 100m sprints, world records of maximum rainfall in a day etc
- topaz0 8mo ago
  I think part of why we're much more likely to learn about the iid, finite-variance CLT is that it's a lot easier to prove than the more general ones.
  - srean 8mo ago
    Yes that's a big part. The proofs get hairier otherwise. But I think there is more to it, the convergence to Gaussian also gets slower. In practice, we deal with finite averaging, so speed of convergence matters. For some non-iid case, the convergence may be so slow that the distribution cannot be approximated well by a Gaussian.
srean 8mo ago
The way I understand this is that adding of random variables is a smoothening operation on their densities (more generally the distributions, but let me speak of densities only). A little more formally, additions over random variables are convolutions of their densities. Repeated additions are repeated convolutions. A single convolution can be understood as a matrix multiplication by a specific symmetric matrix. Repeated convolutions are therefore repeated matrix multiplications. Anyone familiar with linear algebra will know that repeated matrix multiplication by a non degenerate matrix reveals it's eigenvectors. The Gaussian distribution is such an eigenvector. Just like an eigenvector, it is also a fixed point -- multiplying again by the same matrix wil lead to the same vector, just scaled. The Gaussian distribution convolved is again a Gaussian distribution. The addition operation in averaging is a matrix multiplication in the distribution space and the division by the the 'total' in the averaging takes care of the scaling. Linear algebra is amazing. Pagerank is an eigenvector of the normalised web adjacency matrix. Gaussian distribution is the eigenvector of the infinite averaging matrix. Essentially the same idea.
- CoastalCoder 8mo ago
  > Anyone familiar with linear algebra will know that repeated matrix multiplication by non degenerate matrices reveals it's eigenvectors. TIL that I'm not "familiar" with linear algebra ;) But seriously, thanks for sharing that knowledge.
  - srean 8mo ago
    If you are not speaking in jest (I strongly suspect you are), knowledge of linear algebra is one of the biggest bang for buck one can get as an investment in mathematical knowledge. So humble and basic a field. So wide it's consequences and scope.
    - Sharlin 8mo ago
      Their point was that "familiarity" apparently means different things for different people :P Someone using linalg in computer graphics applications may say they're familiar with it even though they've never heard the term "eigenvector". I'm not actually sure about what you mean – how does repeated multiplication reveal eigenvectors?
wodenokoto 8mo ago
> the “steadfast order of the universe” that eventually overcame any and all deviations from the bell. I can’t believe the author wrote that without explaining why it’s called the bell curve. I find the article spends a lot of time talking about repeating games without really getting to the meat of it. If you throw a dice a million times the result is still following a uniform distribution. It isn’t until you start summing random events that the normal distribution occurs.
- api 8mo ago
  A huge amount of nonfiction writing, especially nonfiction books, feels padded to length. It’s something I’ve gotten out of AI. Summarize, please, and it’s pretty good at extracting the key ideas. If I want a story I read fiction, which is writing with a much wider set of objectives than just conveying information and ideas (though it can do that).
- xbar 8mo ago
  True and false. You don't need anywhere near a million samples to get a good approximation for your normal distribution. Far fewer than 100 is sufficient (and 14 is a fine place to start if you are really constrained on data and need to get to 90-10).
fedeb95 8mo ago
Nassim Nicholas Taleb is triggered, then calms down a bit toward the end.
sayYayToLife 8mo ago
Okay at my core I'm an inductionist. However this article is a mere tautology at best. The article doesn't explain why. It explains a bunch of cases and works backwards to show that the original premise was true. This sounds fine but the end of the article specifically mentioned that this is dangerous because the world doesn't always work like this. This is the problem with induction, it might work in 99% of cases, I've never seen a Black Swan so there must not be any black swans? Deduction has more value when it comes to math specifically... I'll admit that as an inductionist.
Projectiboga 8mo ago
A requirement is multiple independent influences. An example of what shouldn't target a normal distribution are a single course's grade outcomes, having a teacher and a defined curriculum goes against that. Yes, there is a variability of student effort and aptitude. But a top teir university selects a group of students based on some merit their student body isn't random. There are airheads who were dragged over the finish line with connections and family money and some students fall prey to substance abuse and mental illness. I argue a different distribution recognizing that a skilled teacher can get a class grade distribution centered around at least a B of not B+, A-. I feel grading on the curve and limiting A's to a fixed percent target can encourage bad test design or worse bad grading.
Heer_J 8mo ago
[dead]
bux93 8mo ago
The article doesn't share the actual math, but also not the relatively easy intuition. When you roll a pair of dice, there are more combinations that add up to 7 than any other number. Change the numbers on the dice (change the 1 to a 6, e.g.), there's again more combinations that add up to some numbers than to others. The histogram of the number of combinations that add up to different results is a bell curve. That's why it pops up everywhere you have addition of independent events. It's sad that even introductory statistics courses skip this simple intuition.
- cortesoft 8mo ago
  Anybody who has ever played Craps will know this.
- AlexCornila 8mo ago
  Yeah that is definitely not the relatively easy intuition for this. The relatively easy intuition comes from learning about the Bernoulli trials, binomial distribution and Pascal triangle. Once you understand those you understand why normal distribution is so prevalent. Or just watch this https://youtu.be/AwEaHCjgeXk?si=tV72uauquCHvzkNE https://youtu.be/AwEaHCjgeXk?si=tV72uauquCHvzkNE
  - estearum 8mo ago
    They explained the intuition in like two sentences that an 8th grader can understand and test themselves. Sounds simpler than whatever you’re talking about here
- trymas 8mo ago
  > It's sad that even introductory statistics courses skip this simple intuition. I was probably lucky. We got homework as one of the first lessons in statistics course, for exactly this case. Roll pair of dice, save the result, do it 200 (or some other bigger number) times, plot the histogram, do some maths, maybe provide any conclusions, etc. Such things then definitely stuck with you for a long time.
- srean 8mo ago
dwpdwpdwpdwpdwp 8mo ago
Francis Galton, namesake of the Galton Board,(https://en.wikipedia.org/wiki/Galton_board https://en.wikipedia.org/wiki/Galton_board) had this to say of the Central Limit Theorem: "Order in Apparent Chaos.-I know of scarcely any-, thing so apt to impress the imagination as the wonderful form of cosmic order expressed by the " Law of Frequency of Error." The law would have been personified by the Greeks and deified, if they had known of it. It reigns with serenity and • in complete self-effacement amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason." https://galton.org/cgi-bin/searchImages/galton/search/books/natural-inheritance/pages/natural-inheritance_0073.htm https://galton.org/cgi-bin/searchImages/galton/search/books/...
tom-blk 8mo ago
Good read but as others have already mentioned, I would have loved to see some actual math
FabHK 8mo ago
> Place a measuring cup in your backyard every time it rains and note the height of the water when it stops: Your data will conform to a bell curve. That strikes me as unlikely, actually: that the amount of water to fall (per area) across rain showers ("when it stops") is normally distributed. Why would the author think that? Also, not much of "the math that explains" the CLT in the article. The basic conditions are: The samples you add together must be - sufficiently independent - sufficiently well-behaved in the sense of not having huge outliers (finite variance is good enough for this) Not sure either condition holds for rainfall.
- deleted 8mo ago
  [deleted]
tsoukase 7mo ago
The Bell curve is the distribution with the maximum information entropy. It surely relates to it being the curve that all others regress to.

207 points by ibobev 8mo ago

I hate Quanta a lot a vast amount of fluff for less than a college statistics professor would (hopefully) be able to impart with a chalkboard in 10 minutes, when Quanta has the ability to prepare animated diagrams like 3Blue1Brown but chooses not to use it they could go down myriad paths, like how it provides that random walks on square lattices are asymptotically isotropic, or give any other simple easy-to-understand applications (like getting an asymptotic on the expected # of rolls of an n-sided die before the first reoccurring face) or explain what a normal distribution is, but they only want to tell a story to convey a feeling they are a blight upon this world for not using their opportunity to further public engagement in a meaningful way

tptacek 8mo ago
A lot of times on HN when a math topic comes up that isn't about 3b1b, someone will jump in to say "this isn't as good as 3b1b". Last time I saw that, I was moved to comment: https://news.ycombinator.com/item?id=45800657 https://news.ycombinator.com/item?id=45800657 3b1b doesn't have the same goal as Quanta, or as introductory guides. It's actually not that great a teaching tool (it's truly great at what it is for, which is (a) appreciation and motivation, and (b) allowing people to signal how smart they are on message board threads by talking about how much people would get out of watching 3b1b). This is prose writing about math. It's something you're meant to read for enjoyment. If you don't enjoy it, fine; I don't enjoy cowboy fiction. So I don't read it. I don't so much look for opportunities to yell at how much I hate "The Ballad of Easy Breezy".
- bmenrigh 8mo ago
  I don’t fault Quanta (or 3b1b) for being the way they are. Each is serving their goal audience pretty well. My compliant is only that there should be a dozen more just like them, each competing with each other for the best, most engaging math and science content. This would allow for more a broader audience skillevel to be reached. As it stands, we’re lucky even to have Quanta and 3b1b. I think there is hope though, quite a few new-ish creators on YouTube are following in Grant’s footsteps and producing very technically detailed and informative content at similar quality levels.
- paulpauper 8mo ago
  there is no getting around that learning math requires actually having to buckle down and read and do math . A video will not suffice.
  - tptacek 8mo ago
    Couldn't agree more, which is why I think it's odd to suggest that a pop-sci magazine article is somehow a disservice that 3b1b would correct.
  - DroneBetter 8mo ago
    well for one who does buckle down and read and do math, the expected amount of new information brought to them by a 3B1B video as supplementary material upon a topic (with the normal distribution being one that admits a direct comparison from the article) is nonzero, by merit of it possibly having ideas to convey from outside their usual purview and formal background that may be applicable to the doing of math (as has been the case for me, someone who [does math](https://oeis.org/wiki/User:Natalia_L._Skirrow https://oeis.org/wiki/User:Natalia_L._Skirrow)), while for Quanta fluff pieces it's zero. by the metric of "if this expository piece were to be taken to a time before its subject had been considered and presented to researchers, how useful would its outline be towards reproducing the theory in its totality," Quanta's writings (on both classical and research math) mostly score 0
deleted 8mo ago
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KnuthIsGod 8mo ago
3Blue1Brown Seems a bit like Ted Talks. Lightweight popcorn for the simple minded.
throwaway81523 8mo ago
Quanta used to have tons of good stuff and not much crap. Now there's enough crap that if there's still good stuff, it gets lost in the noise.
andyjohnson0 8mo ago
I probably don't have your mathematical sophistication - but I like and appreciate Quanta precisely because it helps people like me to understand a little bit about challenging things. This enriches my tiny life, and I hope it also makes the world a fractionally better place for us all. Perhaps you're just not in their intended audience?

mikrl 8mo ago

Great article. Personally I have been learning more about the mathematics of beyond-CLT scenarios (fat tails, infinite variance etc) The great philosophical question is why CLT applies so universally. The article explains it well as a consequence of the averaging process. Alternatively, I’ve read that natural processes tend to exhibit Gaussian behaviour because there is a tendency towards equilibrium: forces, homeostasis, central potentials and so on and this equilibrium drives the measurable into the central region. For processes such as prices in financial markets, with complicated feedback loops and reflexivity (in the Soros sense) the probability mass tends to ends up in the non central region, where the CLT does not apply.

parpfish 8mo ago
As to ye philosophy of “why” the CLT gives you normals, my hunch is that it’s because there’s some connection between: a) the CLT requires samples drawn from a distribution with finite mean and variance and b) the Gaussian is the maximum entropy distribution for a particular mean and variance I’d be curious about what happens if you starting making assumptions about higher order moments in the distro
- sobellian 8mo ago
  IIRC there's a video by 3b1b that talks about that, and it is important that gaussians are closed under convolution.
  - gowld 8mo ago
    That makes it an equilibrium point in function space, but the other half is why it's an a global attractor.
    - pfortuny 8mo ago
      There must be a contractive nature in "passing to the limit". And then Brower's fixed point theorem. (I know it is very easy to do "maths" this way).
- 8mo ago

fritzo 8mo ago

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/Fréchet/Weibull distributions regarding extreme values. Those curves are also everwhere, but we don't see them because we weren't taught them because the math is tough.

AndrewKemendo 8mo ago
That’s exactly the right take and the article proves it: Statisticians love averages so everywhere that could be sampled as a normal distribution will be presented as one The median is actually more descriptive and power law is equally as pervasive if not more
- fsckboy 8mo ago
  combining repeated samples of any distribution* (any population density fuction including power law distributions) will converge to the normal distribution, that's why it appears everywhere. * excluding bizarre degenerates like constants or impulse functions
  - abetusk 8mo ago
    No, that's not correct. Sums of power law distributions can converge to power low tailed distributions, not normal distributions.
    - AndrewKemendo 8mo ago
      No use arguing with them they don’t have enough mathematical understanding to understand what they’re saying
BobbyTables2 8mo ago
It also took me a little while to realize “least squares” and MMSE approaches were not necessarily the “correct” way to do things but just “one thing we actually know how to do” because everything else is much harder. We can use Calculus to do so much but also so little…

gowld 8mo ago

3b1b playlist on Central Limit Theorem: https://www.youtube.com/playlist?list=PLZHQObOWTQDOMxJDswBaLu8xBMKxSTvg8 https://www.youtube.com/playlist?list=PLZHQObOWTQDOMxJDswBaL... He has several other related videos also. https://www.youtube.com/@3blue1brown/search?query=convolution https://www.youtube.com/@3blue1brown/search?query=convolutio...

bluGill 8mo ago

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.

gowld 8mo ago
Explain? What are "most cases"?
thaumasiotes 8mo ago
> 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.

tsunamifury 8mo ago

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.

D-Machine 8mo ago
Yup. And in general more heavy-tailed bumps are in fact better models (assuming normality tends to lead to over-confidence). Really think the universality is strictly mathematical, and actually rare in nature.
jibal 8mo ago
First, every mathematical theorem is a tautology ... don't conflate "tautological" with "obvious". Second, your "aka" is incorrect --- there is all sorts of clumping that is not a normal distribution.
- thaumasiotes 8mo ago
  As I'm sure tsunamifury would agree, it is incredibly common for people to label "bell curves" by eyeball, regardless of whether they are normal curves. To most people, "clumping" in a one-dimensional spectrum is all they mean by the phrase "bell curve".
  - tsunamifury 8mo ago
    [dead]
  - D-Machine 8mo ago
    This was sort of my reading as well: I took "clumping" to mean "bump-shaped".
  - jibal 8mo ago
    This completely misses the point, which is that the central limit theorem says that it isn't just any old clumping, it's always the normal distribution. tsunamifury dismissed this strong finding as "tautology" because clumping is obvious ... but that it's always precisely a bell curve is far from obvious. Again, > your "aka" is incorrect --- there is all sorts of clumping that is not a normal distribution. That it's "incredibly common for people to label "bell curves" by eyeball, regardless of whether they are normal curves" is not just not relevant, it's anti-relevant ... the central limit theorem says that the distribution of the means is always a bell curve--a normal distribution--not merely a "bell curve". Anyway, this is covered in far more detail in other comments and material elsewhere, so this is my last contribution.

EGreg 8mo ago

Related: https://magarshak.com/blog/explaining-benfords-law/ https://magarshak.com/blog/explaining-benfords-law/

gwern 8mo ago

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.

jibal 8mo ago
https://en.wikipedia.org/wiki/Central_limit_theorem https://en.wikipedia.org/wiki/Central_limit_theorem

fiforpg 8mo ago

On opening the article, I was somehow expecting a mention of the large deviations formalism, which was (is?) fashionable in late 20th century, and gives a nice information theoretic view of the CLT. Or something like that. There's a ton of deep math there. So having a bio statistician say "look, the CLT is cool" is a bit underwhelming. Edit: see eg John Baez's write-up What is Entropy? about the entropy maximization principle, where gaussians make an entrance.

abetusk 8mo ago

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, gives (1-w^2). Convolution is multiplication in Fourier space, so you get (1-w^2)^n. Squinting, (1-w^2)^n ~ (1-n w^2 / n)^n ~ exp(-n w^2). The Fourier transform of a Gaussian is a Gaussian, so the result holds. Unfortunately I haven't worked it out myself but I've been told if you fiddle with the exponent of 2 (presumably choosing it to be in the range of (0,2]), this gives the motivation for Levy stable distributions, which is another way to see why fat-tailed/Levy stable distributions are so ubiquitous.

WCSTombs 8mo ago
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.
- abetusk 8mo ago
  The Fourier transform of a uniform distribution is the sinc function which looks like a quadratic locally around 0. Convolution to multiplication is how the quadratic goes from downstairs to upstairs, giving the Gaussian. 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#Parametrizations https://en.wikipedia.org/wiki/Stable_distribution#Parametriz... [1] https://www.amazon.com/Invitation-Modern-Number-Theory/dp/0691120609 https://www.amazon.com/Invitation-Modern-Number-Theory/dp/06...

WCSTombs 8mo ago

It's not a bad article, but I have to point something out: > Laplace distilled this structure into a simple formula, the one that would later be known as the central limit theorem. No matter how irregular a random process is, even if it’s impossible to model, the average of many outcomes has the distribution that it describes. “It’s really powerful, because it means we don’t need to actually care what is the distribution of the things that got averaged,” Witten said. “All that matters is that the average itself is going to follow a normal distribution.” This is not really true, because the central limit theorem requires a huge assumption: that the random process has finite variance. I believe that distributions that don't satisfy that assumption, which we can call heavy-tailed distributions, are much more common in the real world than this discussion suggests. Pointing out that infinities don't exist in the real world is also missing the point, since a distribution that just has a huge but finite variance will require a correspondingly huge number of samples to start behaving like a normal distribution. Apart from the universality, the normal distribution has a pretty big advantage over others in practice, which is that it leads to mathematical models that are tractable in practice. To go into a slightly more detail, in mathematical modeling, often you define some mathematical model that approximates a real-world phenomenon, but which has some unknown parameters, and you want to determine those parameters in order to complete the model. To do that, you take measurements of the real phenomenon, and you find values for the parameters that best fit the measurements. Crucially, the measurements don't need to be exact, but the distribution of the measurement errors is important. If you assume the errors are independent and normally distributed, then you get a relatively nice optimization problem compared to most other things. This is, in my opinion, about as much responsible for the ubiquity of normal distributions in mathematical modeling as the universality from the central limit theorem. However, as most people who solve such problems realize, sometimes we have to contend with these things called "outliers," which by another name are really samples from a heavy-tailed distribution. If you don't account for them somehow, then Bad Things(TM) are likely to happen. So either we try to detect and exclude them, or we replace the normal distribution with something that matches the real data a bit better. Anyway, to connect this all back to the central limit theorem, it's probably fair to say measurement errors tend to be the combined result of many tiny unrelated effects, but the existence of outliers is pretty strong evidence that some of those effects are heavy-tailed and thus we can't rely on the central limit theorem giving us a normal distribution.

AxEy 8mo ago

I remember seeing one of these https://en.wikipedia.org/wiki/Galton_board https://en.wikipedia.org/wiki/Galton_board at the (I think) Boston Science Museum when I was a kid. They have some pretty cool videos on Youtube if you're curious.

nsnzjznzbx 8mo ago

So Abraham de Moivre was the worlds first quant?

bicepjai 8mo ago

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.

hilliardfarmer 8mo ago
A lot of times I can't tell if I'm the idiot or if everyone else is. Says that this isn't an interesting question at all and the article was horrible. I studied data science for a few years but I'm no expert, but it seems pretty obvious to me that if you make a series of 50/50 choices randomly, that's the shape you end up with and there's really nothing more interesting about it than that.
- alanbernstein 8mo ago
  I don't think "obvious" is the right word here. It makes perfect sense when you understand it, but it's not a conclusion that most people could come to immediately without detailed, assisted study.
- smcin 8mo ago
  Sampling 50/50 choices would be a binary distribution that (very crudely) approximates a normal distribution. But the counterintuitive thing about the CLT is that it applies to distributions that are not normal.
Buttons840 8mo ago
I haven't read the article, but my understanding is that a normal curve results from summing several samples from most common probability distributions, and also a normal curve results from summing many normal curves. 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.)

jibal 8mo ago

https://en.wikipedia.org/wiki/Central_limit_theorem https://en.wikipedia.org/wiki/Central_limit_theorem > 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.

abetusk 8mo ago
That sentence is flat out wrong. If the probability distribution converges, it converges to a Levy stable distribution [0]. [0] https://en.wikipedia.org/wiki/Stable_distribution https://en.wikipedia.org/wiki/Stable_distribution

throwaway81523 8mo ago

Now do power laws.

causalityltd 8mo ago

Causes mostly add up: molecular kinetic energies aggregate to temperature, collisions to pressure, imperfections to measurement errors, etc. So, normal or CLT is the attractor state for the unexceptional world. BUT for the exceptional world, causes multiply or cascade: earthquake magnitudes, network connectivity, etc. So, you get log-normal or fat-tailed.

GeoSys 8mo ago

It's in many places, but not everywhere. CLT means that samples tend towards the mean, which is neat. Unfortunately, many "researchers" blindly assume that many real life phenomena follow Gaussian, which they don't... So then their models are skewed

bandrami 8mo ago

I flinch at "everywhere", particularly when people keep asserting they are places that they aren't (and in fact can't be). Nothing with a hard zero can be normally distributed, for instance, but people will keep insisting quantities with a hard zero are.

Epa095 8mo ago
Is this not just a linguistic issue, where people say normal distributed but actually mean approximate or assumed normality? Its not like height is normally distributed (there is nobody 8 feet tall), but its not like the distribution bares no resemblance to the normal distribution either, and in a colloquial sense the term seems to he used more freely than the mathematical defined term.

srean 8mo ago

A result of broader applicability is that of convergence to infinitely divisible distributions, more generally the stable distributions https://en.wikipedia.org/wiki/Infinite_divisibility_(probability) https://en.wikipedia.org/wiki/Infinite_divisibility_(probabi... https://en.wikipedia.org/wiki/Stable_distribution https://en.wikipedia.org/wiki/Stable_distribution This applies even when the variance is not finite. Note independence and identical nature of distribution is not necessary for Central Limit Theorem to hold. It is a sufficient condition, not a necessary one, however, it does speed up the convergence a lot. Gaussian distribution is a special case of the infinitely divisible distribution and is the most analytically tractable one in that family. Whereas, averaging gives you Gaussian as long as the original distribution is somewhat benign, the MAX operator also has nice limiting properties. They converge to one of three forms of limiting distributions, Gumbel being one of them. The general form of the limiting distributions when you take MAX of a sufficiently large sample are the extreme value distributions https://en.wikipedia.org/wiki/Generalized_extreme_value_distribution https://en.wikipedia.org/wiki/Generalized_extreme_value_dist... Very useful for studying record values -- severest floods, world records of 100m sprints, world records of maximum rainfall in a day etc

topaz0 8mo ago
I think part of why we're much more likely to learn about the iid, finite-variance CLT is that it's a lot easier to prove than the more general ones.
- srean 8mo ago
  Yes that's a big part. The proofs get hairier otherwise. But I think there is more to it, the convergence to Gaussian also gets slower. In practice, we deal with finite averaging, so speed of convergence matters. For some non-iid case, the convergence may be so slow that the distribution cannot be approximated well by a Gaussian.

srean 8mo ago

The way I understand this is that adding of random variables is a smoothening operation on their densities (more generally the distributions, but let me speak of densities only). A little more formally, additions over random variables are convolutions of their densities. Repeated additions are repeated convolutions. A single convolution can be understood as a matrix multiplication by a specific symmetric matrix. Repeated convolutions are therefore repeated matrix multiplications. Anyone familiar with linear algebra will know that repeated matrix multiplication by a non degenerate matrix reveals it's eigenvectors. The Gaussian distribution is such an eigenvector. Just like an eigenvector, it is also a fixed point -- multiplying again by the same matrix wil lead to the same vector, just scaled. The Gaussian distribution convolved is again a Gaussian distribution. The addition operation in averaging is a matrix multiplication in the distribution space and the division by the the 'total' in the averaging takes care of the scaling. Linear algebra is amazing. Pagerank is an eigenvector of the normalised web adjacency matrix. Gaussian distribution is the eigenvector of the infinite averaging matrix. Essentially the same idea.

CoastalCoder 8mo ago
> Anyone familiar with linear algebra will know that repeated matrix multiplication by non degenerate matrices reveals it's eigenvectors. TIL that I'm not "familiar" with linear algebra ;) But seriously, thanks for sharing that knowledge.
- srean 8mo ago
  If you are not speaking in jest (I strongly suspect you are), knowledge of linear algebra is one of the biggest bang for buck one can get as an investment in mathematical knowledge. So humble and basic a field. So wide it's consequences and scope.
  - Sharlin 8mo ago
    Their point was that "familiarity" apparently means different things for different people :P Someone using linalg in computer graphics applications may say they're familiar with it even though they've never heard the term "eigenvector". I'm not actually sure about what you mean – how does repeated multiplication reveal eigenvectors?

wodenokoto 8mo ago

> the “steadfast order of the universe” that eventually overcame any and all deviations from the bell. I can’t believe the author wrote that without explaining why it’s called the bell curve. I find the article spends a lot of time talking about repeating games without really getting to the meat of it. If you throw a dice a million times the result is still following a uniform distribution. It isn’t until you start summing random events that the normal distribution occurs.

api 8mo ago
A huge amount of nonfiction writing, especially nonfiction books, feels padded to length. It’s something I’ve gotten out of AI. Summarize, please, and it’s pretty good at extracting the key ideas. If I want a story I read fiction, which is writing with a much wider set of objectives than just conveying information and ideas (though it can do that).
xbar 8mo ago
True and false. You don't need anywhere near a million samples to get a good approximation for your normal distribution. Far fewer than 100 is sufficient (and 14 is a fine place to start if you are really constrained on data and need to get to 90-10).

fedeb95 8mo ago

Nassim Nicholas Taleb is triggered, then calms down a bit toward the end.

sayYayToLife 8mo ago

Okay at my core I'm an inductionist. However this article is a mere tautology at best. The article doesn't explain why. It explains a bunch of cases and works backwards to show that the original premise was true. This sounds fine but the end of the article specifically mentioned that this is dangerous because the world doesn't always work like this. This is the problem with induction, it might work in 99% of cases, I've never seen a Black Swan so there must not be any black swans? Deduction has more value when it comes to math specifically... I'll admit that as an inductionist.

Projectiboga 8mo ago

A requirement is multiple independent influences. An example of what shouldn't target a normal distribution are a single course's grade outcomes, having a teacher and a defined curriculum goes against that. Yes, there is a variability of student effort and aptitude. But a top teir university selects a group of students based on some merit their student body isn't random. There are airheads who were dragged over the finish line with connections and family money and some students fall prey to substance abuse and mental illness. I argue a different distribution recognizing that a skilled teacher can get a class grade distribution centered around at least a B of not B+, A-. I feel grading on the curve and limiting A's to a fixed percent target can encourage bad test design or worse bad grading.

Heer_J 8mo ago

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bux93 8mo ago

The article doesn't share the actual math, but also not the relatively easy intuition. When you roll a pair of dice, there are more combinations that add up to 7 than any other number. Change the numbers on the dice (change the 1 to a 6, e.g.), there's again more combinations that add up to some numbers than to others. The histogram of the number of combinations that add up to different results is a bell curve. That's why it pops up everywhere you have addition of independent events. It's sad that even introductory statistics courses skip this simple intuition.

cortesoft 8mo ago
Anybody who has ever played Craps will know this.
AlexCornila 8mo ago
Yeah that is definitely not the relatively easy intuition for this. The relatively easy intuition comes from learning about the Bernoulli trials, binomial distribution and Pascal triangle. Once you understand those you understand why normal distribution is so prevalent. Or just watch this https://youtu.be/AwEaHCjgeXk?si=tV72uauquCHvzkNE https://youtu.be/AwEaHCjgeXk?si=tV72uauquCHvzkNE
- estearum 8mo ago
  They explained the intuition in like two sentences that an 8th grader can understand and test themselves. Sounds simpler than whatever you’re talking about here
trymas 8mo ago
> It's sad that even introductory statistics courses skip this simple intuition. I was probably lucky. We got homework as one of the first lessons in statistics course, for exactly this case. Roll pair of dice, save the result, do it 200 (or some other bigger number) times, plot the histogram, do some maths, maybe provide any conclusions, etc. Such things then definitely stuck with you for a long time.
srean 8mo ago

dwpdwpdwpdwpdwp 8mo ago

Francis Galton, namesake of the Galton Board,(https://en.wikipedia.org/wiki/Galton_board https://en.wikipedia.org/wiki/Galton_board) had this to say of the Central Limit Theorem: "Order in Apparent Chaos.-I know of scarcely any-, thing so apt to impress the imagination as the wonderful form of cosmic order expressed by the " Law of Frequency of Error." The law would have been personified by the Greeks and deified, if they had known of it. It reigns with serenity and • in complete self-effacement amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason." https://galton.org/cgi-bin/searchImages/galton/search/books/natural-inheritance/pages/natural-inheritance_0073.htm https://galton.org/cgi-bin/searchImages/galton/search/books/...

tom-blk 8mo ago

Good read but as others have already mentioned, I would have loved to see some actual math

FabHK 8mo ago

> Place a measuring cup in your backyard every time it rains and note the height of the water when it stops: Your data will conform to a bell curve. That strikes me as unlikely, actually: that the amount of water to fall (per area) across rain showers ("when it stops") is normally distributed. Why would the author think that? Also, not much of "the math that explains" the CLT in the article. The basic conditions are: The samples you add together must be - sufficiently independent - sufficiently well-behaved in the sense of not having huge outliers (finite variance is good enough for this) Not sure either condition holds for rainfall.

deleted 8mo ago
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tsoukase 7mo ago

The Bell curve is the distribution with the maximum information entropy. It surely relates to it being the curve that all others regress to.