But what is the Central Limit Theorem?

A visual introduction to probability's most important theorem
Help fund future projects: / 3blue1brown
Special thanks to these lovely supporters: www.3blue1brown.com/lessons/c...
An equally valuable form of support is to simply share the videos.
Galton board shown in the video: amzn.to/3ZJK8nY
Thanks to these viewers for their contributions to translations
Hebrew: David Bar-On, Omer Tuchfeld
Hindi: Tapender1
Italian: anna-lombardo
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Timestamps
0:00 - Introduction
1:53 - A simplified Galton Board
4:14 - The general idea
6:15 - Dice simulations
8:55 - The true distributions for sums
11:41 - Mean, variance, and standard deviation
15:54 - Unpacking the Gaussian formula
20:47 - The more elegant formulation
25:01 - A concrete example
27:10 - Sample means
28:10 - Underlying assumptions
Correction: 6:37 The narration should say "skewed left"
Correction: 7:15 Again, the narration should say "skews a tiny bit left"
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These animations are largely made using a custom python library, manim. See the FAQ comments here:
www.3blue1brown.com/faq#manim
github.com/3b1b/manim
github.com/ManimCommunity/manim/
You can find code for specific videos and projects here:
github.com/3b1b/videos/
Music by Vincent Rubinetti.
www.vincentrubinetti.com/
Download the music on Bandcamp:
vincerubinetti.bandcamp.com/a...
Stream the music on Spotify:
open.spotify.com/album/1dVyjw...
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3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with KZbin, if you want to stay posted on new videos, subscribe: 3b1b.co/subscribe
Various social media stuffs:
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Twitter: / 3blue1brown
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[@ThisCanNotBTheFuture]

Please consider doing an entire series on probability theory and/or combinatorics.

[@maxdetrickster6524]

I second that!

[@harshsharma03]

Grant replied to a comment in his last video and said that it'd be surprising if he doesn't make it by next year.

[@3blue1brown]

​@@harshsharma03 And I stand by that comment. I made this video in part with the intent of inserting it into that series.

[@harshsharma03]

@@3blue1brown It did seem that way. Thanks for the amazing work Grant, you've helped me more than I can put in words.

[@Pb-rx1lh]

@@3blue1brown I m waiting for u to do a series on theoretical inferential statistics..✨

[@roncho]

I'm an engineering professor far more older than you and I must say whitout a doubt: you are the most skilled professor I have ever seen. The amount of work in this videos is outstandig. They are so flawless that can be considered as art. Congratulations!.

[@friendlyone2706]

Modern young can never understand what it was like in the dark ages of only 60 years ago to be driven to learn more math concepts, and struggling to even find the books in the local & school libraries. This series is GOLD.

[@stevezelaznik5872]

@@friendlyone2706 The best thing for my learning algrebra was the graphing calculator. I could play with the function’s inputs and get a quick sense how it would behave. And this was 20 years ago. I tip my hat to anybody that learned math with a blackboard alone. Luckily have richer teaching tools today

[@friendlyone2706]

@@stevezelaznik5872 When my daughter got her graphing calculator in the 8th grade, as a "blackboard survivor," I was deeply envious.

[@user-mh4gm5my5i]

Real? Are you responsible for what you say?

[@roncho]

@@user-mh4gm5my5i so you are trying to say you are better than mr Grant here?.

[@stratfanstl]

The actual rigorous no-jokes-this-time conclusion from watching 3Blue1Brown videos like this is that Grant deserves some new, yet-to-be invented prize that should be the equivalent of an Oscar for best computer generated imagery, an Emmy for outstanding narration / editing and a Nobel Prize in science for fostering interest in mathematics and science. Amazing, inspired work here.

[@sanssoleilfilm]

yes indeed

[@Doctor_Drew]

Absolutely

[@eyupylmaz9813]

The Grant Prize, sounds catchy enough.

[@kansasllama]

It’s the visualizations for me. Incredible.

[@theproofessayist8441]

Wholeheartedly agree.

[@WhatIsItToBurn]

I have a PhD in applied mathematics, I work in numerical weather prediction as a research scientist. Gaussianity is this hardcore part of the basics of forecasting the weather (even though most atmospheric variables, and their errors, are actually non-Gaussian). This video did a great job at teaching the CLT. I have never seen it explained so well.

[@mlucasl]

As an assumption, it may be because, just as the Galton table, the errors and correlation add up in a "normal distribution" kind of way too, cancelling its effects on the overall distribution? (Haven't seen this topic in years).

[@Edo_Aelio]

@@mlucasl I don't know the weather field, but in geology (mineralogist estimation) the non-gaussian variables could be transformed into gaussians ones to work properly with them

[@friendlyone2706]

High praise -- I watched in its entirety because of your comment. Thank you.

[@Pb-rx1lh]

Yea.. meteorological variables often follow extreme value distributions.. I remember as I took a minor in applied agrometeorology as minor while majorin stats..

[@Chessmapling]

@@mlucasl I thought it was because the weather is more like what is described in chaos theory - small deviations lead to wildly different results. So they don't tend towards something nice like the Gaussian.

[@JaGWiREE]

Of all the years I've supported 3b1b, this video might be the one I was most excited to see pop up.

[@3blue1brown]

Glad to hear it, let me know if there's anything, in particular, you're curious to see in the next part.

[@Pb-rx1lh]

True that!

[@hrishikeshharitas2456]

I simply can't express how much i agree with this! I was introduced to CLT just last week and have been looking forward to the stats series he'd promised.

[@blazerorb]

The next one is my most hyped I think I mean, the next video from 3B1B is always the most hyped, but this time especially. I can’t wait to see if the complex definitions of trig functions come up. Also hope to see a more explicit connection to infinite-dimensionality, or like, infinite independent confounding factors, if that’s a thing. This one gets close when he says that you can generalize to summing different distributions, but I hope we get a good example of when these assumptions are being implied in real-world studies, what can reduce their strength, and how we should temper our understanding of the conclusions.

[@godfreypigott]

@@3blue1brown What happened to your promised second video on convolution?

[@3blue1brown]

Next video, explaining the π and how the function e^(-x^2) arises: kzbin.info/www/bejne/maqbo2qNiNqHZ6s As many helpful commenters have pointed out, at 6:37 and 7:15 the narration should say "skews left" instead of "right". In standard terminology, the skew direction refers to the direction of the longer tail.

[@alikeivani7715]

Great

[@alikeivani7715]

Please make a video on laplace transform

[@ciscoortega9789]

I'm so excited for a visual and intuitive explanation for it! In my stats class, CLT was proven rigorously but I couldn't "see" why. Later in your Discord, I believe someone explained it to me intuitively in terms of moment generating functions, but I'm excited to see if you can leverage visuals for a more elemementary intuitive explanation

[@ericfielding668]

It would be cool to see the Demoivre-Laplace theorem (the original CLT) mentioned.

[@Snowflake_tv]

Cooool! I like solution for WHY

[@GrahamCrannell]

the section unpacking the Gaussian formula is simply a work of art. Giving a graphical intuition about moving from e^(-x) to e^(-x^2), and then to a constant multiplier of the exponent... just absolutely pristine

[@luismonteromunoz4330]

I didn't insertando the step in wich the 'c' desapear, and appeared the 1/2 and the sigma parameter. Can someone explain It? 17:22

[@pavlosurzhenko4048]

@@luismonteromunoz4330 if f(x) is a function, then f(x/sigma) is the same function stretched by sigma along the x axis. He basically plugged in sigma to regulate the standard deviation in the final formula. The 1/2 arises from the fact that e^(-x^2)/sqrt(pi) has the standard deviation of 1/sqrt(2), so you can think about that change as replacing x by x/(sqrt(2) * sigma). The x is squared, so we can square 1/sqrt(2) and get that magical 1/2.

[@dan_pal]

@@luismonteromunoz4330don’t feel bad, he didn’t bother to explain that

[@swanmath8382]

This series of lectures must be incorporated into the math curriculum of all high schools in the world, I was trained in math and as a data scientist, but I have never seen the central limit theorem explained this way. It just made things so easy to understand and intuitive. Well done.

[@domainofscience]

Hooray! A new 3Blue1Brown video!

[@TM_Makeover]

Yoo 🤘

[@domainofscience]

That was really good!

[@giovannironchi5332]

I understand that feeling!!

[@vineetjain5082]

yay

[@Inspirator_AG112]

Another educational KZbin channel discovered. (Your's)

[@kev2582]

Dealing with CLT pretty much every day here. Really impressed with how easily you explain it. By far the most intuitive and easily understood explanation of CLT. Salute!

[@vit3060]

We are all dealing with CLT every day everywhere. That is the Mother Nature law )

[@vikingthedude]

awesome! What field are you in if i may ask?

[@mimid3312]

Thanks for producing such high-quality videos, i'm a maths student who love statistics. I would say this vid gives the clearest and neatest explanation to CLT ever, really inspiring, I sacrificed my sleep time watching it for 3 times!!! Amazed and shocked! Thank you Grant.

[@indiablackwell]

I can't tell you how insanely brilliant you are at taking a universal concept that is vaguely understood and illuminating all the nuance hidden in plain daylight to make this understood on a higher level!!! Genius

[@TonyLeTone]

This video is by far the best introduction to the CLT I've encountered. You are doing the world a great service by putting that much effort into this youtube channel. I adore your work and use your visualizations all the time in my classes (even though I teach in french).

[@feynman_QED]

Am sure you didn't understand anything.

[@0x90meansnop8]

Grant, you are a lifesafer! My exams are in 2 weeks and I have not understood this yet. It's a miracle you are publishing this video online!

[@manaskulkarni4229]

As I was watching this video, I wanted to say that videos from your channel inspire me to learn. Not just mathematics, but anything worth doing. Although I am an engineer and enjoy doing what I do, I have never been a huge fan of pure mathematics. But the way you explain concepts just makes it so easy to understand. Even though I might have to rewatch some videos to fully comprehend the meaning, I really enjoy it and it never feels like a chore. I have watched your videos more than my university lectures. I wish there are more teachers like you in this world. Thank you so much 3Blue1Brown!

[@robinj.p.7187]

Describing the mean of the weights as the center of mass of the distribution was just incredible. And the intuitive matrix multiplication without even mentioning it. You are a great teacher!

[@scrimbingus]

As someone who works with Kalman filtering on a regular basis, this is a very nice video to see. One of the core principles behind the Kalman filter is that all random variables involved must be Gaussian, which seems overly restrictive on the surface. I think this provides an excellent, succinct explanation for why that's actually a reasonable assumption for many systems, since every random process we can directly observe is really just a combination of many smaller processes. I look forward to the next one!

[@Eta_Carinae__]

Yeah, I think it's worth remembering that assuming an RV is normally or lognormally distributed is a pretty minimal assumption, since your basically only saying that your observations are the result of a LC of an unknown number RVs that may or may not be orthogonal to eachother, and that there's _some_ kind of minimal number of RVs, depending on their individual distributions, that your measurements are in excess of. If you find that the distribution isn't normal, that actually gives you some information about the individual distributions themselves.

[@xyzct]

Rudolf E. be all, like, "Dude, the product or convolution of two Gaussian PDFs is Gaussian."

[@EebstertheGreat]

​@@Eta_Carinae__ Some types of data can be assumed to be normally distributed, but not all. Some data is naturally uniformly distributed. Other data is naturally exponentially distributed. For instance, let's say I looked at the distances of home runs in the MLB. That is certainly not normally-distributed, since a lot of home runs are very close to the minimum possible distance. Or let's say I looked at the speeds of atoms of an ideal monatomic gas in thermodynamic equilibrium. These won't be normally-distributed. In fact, they will have a χ distribution with 3 degrees of freedom. Or how about gasoline usage? A lot of the population would be around 0, while the rest would probably look roughly normally-distributed, because a lot of people don't own a car. It's generally not a good idea to just assume data should be normally distributed because it depends on many different factors. Those factors are not necessarily equally important or identically-distributed or independent at all. Typically, you can expect to find normally-distributed data when measurements can span a very large range of values relative to the standard deviation, when no particular special values are preferred, when the distribution should be symmetric with respect to the mean, and when data is clustered around the mean. In other words, it's normal if it's normal.

[@ricardoV94]

Or if you combine the data, e.g, by computing the mean or sum, which can often simplify modelling.

[@Dhruvbala]

Thank you for the stat videos! I've found calculus and linear algebra intuitive, but I've struggled to build an intuition for stat concepts -- even though I'm sure it's fascinating

[@edwardlulofs444]

Yes, and I now see these ideas so embedded in the fabric of the universe that it is responsible for physics entropy, biological diversity, and now the vast science of Complexity.

[@Trenz0]

Same. Statistics has been a lot of memorization and faith for me. Stark contrast with all other mathematical concepts. It must be the numerous layers of abstraction and the fact that seeing these results in a practical manner would simply take too much time

[@edwardlulofs444]

@@Trenz0 Yes, I think much of the memorization comes from people who are not the best teachers. There is an unusual number of autistic people (like me) who do not always have the best communication skills. Statistics is also one of the newer math areas, like 200 years. Calculus goes back 300 to 2000 years. Math is my 1st language, but I am forced to try to speak English.

[@hailmary7283]

That's interesting. For me probability and stats always felt like second nature while calc and linear algebra felt like I had to really bend my mind to understand it at all.

[@edwardlulofs444]

@@hailmary7283 When learning, a lot depends on the 1) teacher, 2) textbook, 3) other students. For me, also, family members encouraged me in science and did not encourage math. In retrospect. perhaps my life would have been more successful if I had concentrated on math instead of physics.

[@paulwilk2854]

I am constantly impressed by how Grant's videos extract the art that is inherent in certain mathematical concepts. What a great video!

[@consistentthoughts826]

If there is a Fields Medal for Math Content creators on KZbin it should be for this channel. Grant Sanderson, you are awesome sir.

[@flummox3d]

You know, I really like math, so I went to a natural science uni to study it. I spent 3 years there, but it was a dry way of learning math and eventually I dropped out. Watching this video (and your videos in general) I understand so much more about probability than what they could have ever thought me.

[@BJ52091]

Probability theorist here. I want to encourage you to not give up on math, and not to let your schooling interfere with your education. Mathematics is not the dry subject presented in classrooms; it is by far the most creative, deeply satisfying, and beautiful activity available to human beings. Dive deep enough into any subject, and you will find math at its core. Reality is how math feels. I specialized in probability because the study of randomness interweaves with truth and beliefs and knowledge and reality in a way that can only be glimpsed through the equations. The study is worth every moment you devote to it. Don’t give up.

[@thewiseturtle]

@@BJ52091 The problem with current academic "math" is that it's more of an unnatural secret code. Real math is what the brain does, and I really hope that some day soon math will be taught in ways that use, and complement, the natural process of mathematical thinking, so that most any intelligent being can use it effectively to both communicate and understand the patterns we find in reality.

[@Snowflake_tv]

Last sentence, Xthought Otaught So did I. I dropped out. But like math and science.

[@SayakKolay]

​@@BJ52091 Hi, I did a Masters in Statistics but found the part on Measure Theoretic Probability unintuitive and poorly taught. Could you please offer some advice regarding this ? Also, can you recommend some good resources ?

[@craftfredo1479]

I would love a series (or just more videos) on Complex Analysis. There is so much geometric intuition behind Complex Analysis that is lost on so many people...

[@3blue1brown]

There are some tentative plans for a video that is at least related to complex analysis in the coming months. I don't have the greatest track record with promises, but stay tuned

[@edwardlulofs444]

It was one of my favorite classes. It is so straight forward and beautiful. It has so many applications. I always recommend it to all my students.

[@jordan6302]

As an actuary, I'd say this is perhaps the best descriptive video/lecture I've ever seen on the CLT. I wish I would have seen this when taking my classes for exam P because the visualizations are so useful in understanding what can be a very dense topic when it's spewed from a chalk board/overhead screen hastily.

[@acobolew1]

Great video, as always. Thank you for doing one on CLT. Also excited to see your convolution video. One thing: @7:16 that's right skew, not left skew (the convention is to define skew by where the tail is, not where the hump is)

[@saintfan7001]

This is correct and needs to be more visible.

[@zh84]

I studied probability and statistics in university and learned about the central limit theorem, then totally forgot it. When I saw this video title I knew I had heard of it, but it took a while to remember it, for the first time in probably twenty-five years. Thank you for explaining it much better than our textbook did!

[@blazerorb]

Seriously, imagine if all our stem subjects had teaching material like this. Then imagine if we made it and exported it to the world.

[@blazerorb]

Ngl, I could see China or India doing it, and then half the world learning from Chinese- or Indian- source lessons plans and curricula. Or like, Finland. Hopefully Canada.

[@austinsimpson8]

One of the most interesting things I've learned in my math undergrad so far is that Brownian motion follows a normal distribution over time (at least, this was shown in the context of diffusion), which you elegantly explained in the first few minutes of the video. We had derived the diffusion equation from a formula modelling simple Brownian motion. I had never seen the connection between abstracted physical science and pure probability theory until then. Great topic!

[@govinddwivedi582]

I really like your style of teaching. The way you help us discover things by slowly unrevealing it, instead of just telling the result, is awesome. Like when you were making the formula for bell curve you just started with e^x and then how slowly slowly, step wise step, by encountering problems and then solving them, you finally reached the formula. That was an awesome mathematical journey. And I enjoyed the ride! Woohoo!

[@stulora3172]

One of those videos of yours, where I know everything (PhD in astrophysics) but your animations and discussion makes it so much clearer than it ever was in my mind! Thank you!

[@danielamurphy8560]

I’m a teaching assistant for introductory econometrics at my university and I’ve recommended your videos to my students! You’re a wonderful resource and to be honest I’ve always wondered why the normal PDF had e and pi in it.

[@feynman_QED]

"I’ve always wondered why the normal PDF had e and pi in it" Please, you could at least omit that.. 😆

[@danielamurphy8560]

@@feynman_QED what do you mean

[@ross302ci]

​@@feynman_QED Being honest about what you don't know is a fantastic personality trait. It makes you a better learner and teacher as well as an all-around more tolerable person.

[@feynman_QED]

​@@ross302ci Man, i agree about honesty, that's true under specific conditions, but not always! She claims to be a teaching assistant at university, GEEZ! She might have omitted that particular and just said that she ignores the meaning of the normal PDF, etc etc. That would sound absolutely normal and understandable. I understand she has used that form of communication just to emphasize how much she values Grant's work, but it eventually turned into a double-sword approach 😄 Besides, I would like to take advantage of this comment to say something about the idea of recommending these videos to her students and suggest to whoever had to read this comment that watching this kind of video is not the way to go to learn a mathematical idea or a subject. Many people mistake watching a video for an effective form of learning: WRONG. You learn by doing and persevering in a solitary, deliberate practice, period. One way to benefit from these videos is this: 1) Study the topic at hand in solitary practice. 2) Spend a huge amount of time engaging with the books and thinking through the main ideas and solving a huge amount of exercises. 3) Share one's ideas with peers to engage in complex conversations and measure how far off is your level. 4) Watch the video and seek to predict, on the basis of what you've learned, what will be next; assess your degree of comprehension, and learn an alternative way of looking at the basic ideas. The beauty of the animations and the original way Grant perceives the mathematical objects will make the rest. They are a supplement to enjoy the time, confirm what you know, see what you've studied differently, and find mathematical inspiration that, one day, might turn into a math degree, for instance. Unfortunately, many people disregard this idea because we live in a historical period in which AI advances and a huge amount of free learning material give the false hope that everyone can become rich by practicing theories that, a few decades ago, seemed possible only for the most gifted people. So, many people, especially those who aim for data science/analysis, AI, and ML, live with the sensation of learning calculus/linear algebra/statistics,/probability by merely watching a couple of videos. Unfortunately, it doesn't work like that. Another recommendation I personally give to those who land on this comment: don't learn programming languages from video tutorials. JUST DO IT!

[@chasewelch1664]

@@feynman_QED 1st comment was not helpful for anyone. 2nd was too long to read given no one has any interest in your opinion after the first comment.

[@Howtheheckarehandleswit]

I would absolutely love to see a series on probability/combinatorics/statistics on this channel. It's the subject I've struggled the most with in math by far. I think your ability to take the time to really think through and understand what the basic building blocks really mean will become a very valuable resource in my and many other people's math journeys.

[@jukmifggugghposer]

I've said this about your videos like a million times before but I love love love the way you always focus on the meaning and intuition behind complicated expressions. The part where you stripped back the equation for the bell curve and slowly built it back up was a great instance of that.

[@goldhealer]

I've finished my studies a long time ago but I love to watch your lessesns as refreshments (If I only could understand English at that level as I can currently 13 years ago😅, your channel would be my top top top). Anyway what I love is like you keep guessing my questions along watching and answering it right away - you truly know your stuff. Keep doing educating us - knowledge is the key. Thank you for it, Mariusz

[@Anonymous-jo2no]

Neither my high school teachers nor my uni lectureres explained this so elegantly. I have just learnt way more about the normal distribution and the CLT from 3B1B's single KZbin video than from a 3mo long statistic course in uni. Our education system is disgusting.

[@shubhmishra66]

Wow I was just looking for resources with in depth knowledge on CLT... Now God drops this!!!

[@erikvanderheide1440]

I have never seen such an intuitive illustration of the Normal distribution. Absolutely amazing!

[@emanuelescarsella3124]

When i started watching this channel the things he was explaining to me where completely new to me and I was watching to learn those new things. Now, after so many years, few exams away from a degree in software engineering, I'm still watching those videos, but not because I don't know the subject, just because I'm sure that he is going to get to conclusions in such a human and reasonable way giving lots of insights and new points of view that I surly never got at a university course... Deam I love this channel❤️

[@maurosobreira8695]

Thank you Grant! You listened to my wishes! I think that is a great video to compose the essence of Statistics - hope you get inspired that way! Thumbs up (and that follows a uniform distribution😅)

[@MARCO-dx4wn]

He Granted you a wish

[@3blue1brown]

My pleasure, I hope to do more probability and stats throughout this year.

[@SayakKolay]

​@@3blue1brown That's great to hear ! Would it be possible for you to do a few on Measure Theoretic Probability ? Found it dry and unintuitive :(

[@omargaber3122]

We would like to congratulate you on reaching 5 million subscribers, this is the largest mathematical channel on KZbin

[@sashazhang6292]

Your linear algebra videos were SO helpful last quarter when I really wanted to understand the intuition/fundamental meanings of the concepts we were learning in class. So many times math classes just become memorizing formulas and theorems, but seeing the concepts and crux of linear algebra visually represented and explained so well by you in just 12 videos was insanely helpful. You are incredible and your videos are such a service for students and education in general!! Next quarter I'm taking Probability theory, I doubt you can put together a series by then, but just putting it out there that I would be eternally grateful for a series on probability theory and statistics down the line! Thank you for everything you do :))

[@nightlord531]

Great vid! For years I kinda smiled and nodded my way through stats classes, understanding the ramifications of CLT but never really intuitively "getting" it. This made things much easier to visualise.

[@DeadlockHolmes]

When you think about it in hindsight, I find the Central Limit Theorem totally mindblowing and maybe the most insane mathematical theorem metaphysically speaking. Thanks for this video

[@drdca8263]

Speaking of its implications for philosophy: John Wentsworth iirc (I hope I’m not getting someone else mixed up) is working on an idea of “natural abstractions” which is sorta based on an idea of, central-limit-theorem-ish things happening (but somewhat more general, so a broader family of distributions) making it so that the number of variables needed to describe enough of a system to be able to describe-well its effects on things which are “far away”, should tend to be much smaller than the number of variables needed to describe the system completely, and also like, what kinds of values those summary variables should be.

[@lucaburghardart]

I love how he finds topics we don't even know exist and makes them interesting.

[@ladyvanda]

The CLT is the basis of stats. It’s a well known theorem.

[@livingroomviewing2987]

Yeah, he's an amazing teacher.

[@albe6923]

i dont believe you never take a statistic class

[@mcbossie1]

@@albe6923 I don’t believe you have ever taken an english class

[@exoticcoder5365]

It’s actually widely used in Machine Learning and Wavelet Transform ( Signal Processing, Image Compression, etc ), it’s a great video for understanding the fundamental

[@acborgia1344]

What I find great about easy to access videos like yours is that they'll make it easier for anyone to understand the intuition behind what they learn at school. Over time, I think the overall level of everyone will increase thanks to that, and we'll have more and more people that can make these fields progress It might be a bit idealistic of a view but I sure hope it's true on the long run

[@spacelemur7955]

Your assumption is probably correct. I slogged through statistics over 40 years ago and never got the intuitive feel, despite some good teachers. After a few years working in statistics, I lacked the confidence to continue, and switched tracks entirely (to translation). While the best minds will grasp this field quickly, the rest would benefit from seeing it from other angles, whereupon understanding might click.

[@sagarpal5746]

One of the most brilliant, creative and coherent explanations of CLT I've ever come across.

[@teegeevee42]

This is so great. Universities should show your videos as "teasers" and only then start the heavy math and formulas.

[@arseniix]

3b1b NEVER disappoints, all videos are absolutely top-notch, absolutely beautiful and absolutely understandable

[@andrewsammons9643]

This is my favorite maths video ever. Another spectacular piece of work, Grant! Hope your 2023 is happy and fulfilling!

[@ivanbooroovooy6775]

This is arguably the best video I watched during my study of statistics, and I spent weeks watching videos to finally find one which explained everything. I would point out the question-answer approach, and these questions are the most intricate ones. This video has enough of everything: it is not a simple reading-changing slides lecture, it explains everything almost to the high-school level, and finally, there are some bold mathematical proof inside, and this video gives you enough information to assemble the CLT proof by yourself. Great job, sir! Would I have enough money for myself - I definitely support you!

[@Banminator7]

You are a crazy good educator my friend, this video was a work of art, masterfully crafted, delightfully beautiful while still highly informative and surprisingly understandable in many levels, thank you very much for it! You're very talented and experienced in highliting the main concepts after building them up perfectly while hinting at a couple very interesting consequences or more complex aspects coming up later, balancing these with immaculate skill, hats off to you!

[@dataandcolours6284]

What he says at 29:52 is in my opinion the single most important thing to point out, despite this video being so great! There are so many people that assume more or less anything follows a normal distribution. But normal distributions aren't really that "normal" and assuming some distribution follows it is often outright crazy. It appears to be an extremely common misconception both in articles, everyday life and many math books when introducing the normal distribution.

[@vigilantcosmicpenguin8721]

Even though it makes sense logically, I was a bit surprised the Galton board isn't technically normal. I guess I just had an intuition that things should be normal.

[@aaronbredon2948]

I got these 2 concepts from this video: You tend to get spiky results when the inputs tend to have spikes. The distance between spikes seems to be related to the input spike distance. You tend get a skewed distribution when the inputs tend towards one side. This is visible on the curve with heavily low value inputs - the left side of the values are a bit above the curve, while the right side is a bit below the curve. It is amazing that the standard deviation concept works in so many cases, but there are situations where it fails. And knowing when it will fail ahead of time and what to do instead is important (and beyond my knowledge)

[@user-hy6cp6xp9f]

Yes! What makes this theorem so beautifully powerful in my mind is that *no matter how spiky* that underlying distribution is, with a high enough sample size N, the distribution will *absolutely* converge onto a standard normal distribution (assuming i.i.d and finite variance). We only saw the skewed and spiky distributions, because there was a low sample size. This is why in real world contexts, a rule of thumb is that a sample size of 30 is good enough to assume that the central limit theorem applies. If you measured the height of 10 people (N=10), it would not be safe to assume that the distribution is normal. However, if you measured at least 30 people (N=30), the assumption gets stronger. If you measured 300 people (N=300), it's almost guaranteed to be normal. It's important to note this is a practical rule of thumb, and that distributions radically different from the normal will take increasingly larger sample sizes until the CLT actually applies. This is why sample sizes in studies are so important, and why it's really easy to lie with statistics. There is quite a bit of math involved to get an intuitive feel for it.

[@aaronbredon2948]

@@user-hy6cp6xp9f and just as the pinball example shows, the individual events don't need to be entirely independent - if combinations of events in the series approximates independence, a normal or normal-like distribution tends to result. When finite variance is not true, you get other curves that look similar but should be approached with quantiles rather than standard deviation. In fact, you can use properly selected quantiles to evaluate what type of curve you have.

[@chasewelch1664]

This couldn't have come at a better time. We just hit on CLT a couple weeks ago in my Engineering probability class. Your video's are always my go to for a deeper understanding of the material and I would say anyone not watching 3B1B is at a disadvantage in STEM. Unmatched visuals and eloquent explanations. Thank you Grant.

[@TheUltamteNerd]

This video was a 30 minute long PTSD trip for my stats class. The entire time I was stressing that I don't remember how to solve these problems and they're definitely gonna be on the final. Thanks for the refresher 3b1b

[@Laogeodritt]

I'm an EE who's done a fair bit of probability and stats (as a non-focal point of my job-I mostly do circuit design) for medical imaging, parameter variation in semiconductor devices and circuit performance variation/sensitivity, and this video's given me a surprising amount of new intuition on the CLT. It also made me realise the connection between sums of random variables and convolution for the first time since I first took prob and stats and signal theory a ≈decade ago. XD Absolutely excellent presentation of the topic as usual, cheers!

[@Suinter]

This is one of my favorite videos so far, the clarity of your explanations is astounding!

[@faisaltasleemft]

One of the best explanation I have ever seen on CLT

[@MikeLeed]

This is amazing. Grant, you are a shepherd of light unlocking the secrets of the universe for us common folks. I can't express how much I appreciate this. Thank you a million times over to infinity.

[@guslackner9270]

Minor correction: At 7:50 the distribution has a RIGHT skew. This is important because skew is meant to give an intuition about how extreme values PULL on the mean more than they pull on other measures of centrality (eg median, mode).

[@ThePotaToh]

The mean is still pulled to the left, not right

[@danrose3233]

@@ThePotaToh Nope.

[@ThePotaToh]

@@danrose3233 What "Nope"? It is even mentioned in the description

[@danrose3233]

@@ThePotaToh For right skew mean>median>mode. The extremes in the tail "pull" the mean in that direction.

[@danrose3233]

@@ThePotaToh Description is wrong.

[@mauricio533]

I am currently doing a course on probability and this video is really helpfull. I also really love the high quality of your videos. All the animations and the little details which make everything crystal clear and allows me to easily visualize the math. Thanks a lot!!

[@anLTproduction]

Those clicks are back! Total mathematical ASMR. Thank you Grant!!

[@sivad1025]

The fact that repeatedly summing any random variable assuredly approaches the normal distribution blows my mind still to this day

[@edwardlulofs444]

Yes, it might be the most important distribution. But there are an infinite number of distributions. I must have seen hundreds in my life.

[@samuels.g.8777]

A series on probability and statistics would be awesome! Everyone in my university hates prob & stat because our teachers are pretty bad, but I'm sure you could explain it really well.

[@TheFireBrozTFB]

I've been waiting for a probability series for a while now! Glad to hear it's being worked on :D. Thanks for the amazing content, and as always, cheers for blowing our minds!

[@rajatjain7894]

Best explanation I have found for central limit theorem so far. It made it easy to deduce that law of large numbers is a direct consequence of central limit theorem. Thanks for this, Grant.

[@PipaQinse233]

Marvellous! I happen to be learning about the normal distribution, and I believe that this brand new video will undoubtedly greatly help me to understand it, as well as other things in statistics. In a nutshell, thank you, 3b1b, for bringing us so many helpful, useful, and interesting videos!

[@blazerorb]

Show it to your prof and get him to at least put it in the “helpful links” page for the course

[@edwardlulofs444]

This video is a great presentation of some of the most important ideas. I could have really used this video before taking one of my hardest classes: college senior level probability and statistics, which I took in 1973. All quarter, I kept asking myself: what were the prerequisites that I was supposed to take but that I must have missed. I have used this information almost every day of my life since then. The world would be so much better off if it was a required class just as freshman algebra class is. Alas, maybe we will be able to teach it better in the future.

[@bgebbq314]

That was the best description and comparison of the difference between variance and standard deviation that I have ever seen. The graphical depiction of variance (as a square shape) versus standard deviation (the square root of variance), producing a line, was a revelation to me.

[@mbraincheck5761]

I've heard of this theorem for quite long time with incomplete knowledges, and this is the video makes me clear, just brilliant and clean, thank you.

[@Dezdichado1000]

CLT is a classic and beautiful. What's more mind boggling is that nowadays you can drop the identically distributed assumption and still get a general CLT (Lindeberg, Lyapunov). Probabilists keep finding a version of CLT in different settings that do not even converge to a Gaussian, but a different distribution like Tracy - Widom, Wigner's semicircles etc.

[@RichardVodden1]

There are very few results as beautiful as the central limit theorem. Thanks so much for the explainer vid!

[@WarzoneMasters]

You don't know how glad I am that you made a video on this topic. I have been trying to understand the subject for a long time and this video helped me in an incredible way

[@Zenoandturtle]

I was looking forward to these series for some time now. Amazing work!

[@oldspicey6001]

I feel like this is one of those videos where I will be pausing more than watching

[@WebmediArt]

You are a true blessing to the realm of mathematical education. Thank you!

[@AmoghA]

Having taken a course on probability and statistics where we analysed many types of distributions, the normal and binomial distributions are by far the most important. I was really excited to watch this 3B1B video and doing so has further improved my understanding. On a side note, I would like to ask Grant to consider doing an entire series, like calculus and linear algebra, on probs and stats. It would be really helpful.

[@Allegrian94]

Thanks Grant, I've wanted to stee probability/statistics videos in your channel for years, and I was just doing a refresher on the topic and found your new videos! I think you should combine them in a statistics playlist just like your "Essence of" series

[@ajred0581]

I just learned about the Central Limit Theorem in my AP Statistics class, but my teacher didn’t explain why it was true. Thank, you as always, for teaching at a deep level but still making it understandable 🙏

[@gabor6259]

Very high quality, comprehensible stuff as always, Grant. Congrats on 5M subs.

[@goldnutter412]

Didn't notice that ! what is the expected range for 10M date challenge !! (not really)

[@GenKoe6917]

What a time to have AP Statistics- I'm loving these probability videos, as the visuals make learning about stats much more intuitive and fun! As of now I've struggled with some of the non-computational aspects of it, such as interpreting different parts of distribution, usually the Normal one. But this video has helped me understand the complexity of Normal distributions and why they matter so much, as well as how they can emerge in multiple views of Statistics as a whole! Again, thank you for making learning mathematics so intuitive and enjoyable! Congratulations on 5 million Subscribers as well 🎉🎉

[@bensapper9064]

Hi I have 250 research papers each revolutionizing the field of mathematics and my great great great great great great grandfather is Euler and I have to say this is the best explanation of interogaussianity and the skewed homomorophic central limit corollary I’ve ever seen

[@johnchessant3012]

Can't wait to see 3b1b's take on the computation of the Gaussian integral, still one of the craziest places for pi to show up (maybe second to the Basel problem which he already covered). Even though the trick is very well-known, I am sure he'll have something new to say. Happy pi day!

[@ES-qe1nh]

Depends on the method I guess

[@Tcrunz]

Thanks for an awesome video! I have some feedback on the visuals that I hope you find useful: sometimes you draw a black rectangle over text that will be revealed later. The colour of these rectangles doesn't match the background exactly. On my TV for whatever reason this difference in colour was very pronounced. Less so on my phone. It was only mildly distracting, but I assume aligning the colours would be a simple fix. Thanks again! I've literally been waiting 5 years for you to explain the central limit theorem ever since my friend tried (and failed) to explain it to me 😅

[@mridulagrawal2370]

brilliant video Grant! I was literally watching videos about the bell curve and the famous (or infamous) calculation of the area underneath it and this video popped up. Great video! Please keep up the great work!

[@jakepimentel]

This has been the most exciting math video I've ever watched! I'm in engineering and I've always hated statistics because it's so unintuitive. You just plig values in ang get values out and you memorize what they mean. For the first time, I actually understand why we do the things we do in stats. I especially like how this video implicitly explains why we need a minimum number of samples. Really great video

[@beginneratstuff]

Probability and statistics are probably my weakest points in math (math that I've specifically learned about in school, anyways) so a full series would be great. Also this is a really good video as usual and I found it to be pretty easy to understand. Of course I would need practice and to re-watch some bits to clear some areas of misunderstanding I have but that's not an issue. Overall, this was very engaging!

[@justarandomdood]

Love this I majored in Statistics but I graduated 2 years ago and haven't been able to use any math skills in any of my projects at work, so I've been missing these. This was an amazing refresher and I even forgot that the 3rd assumption was that varx has to be finite LMAO So glad that you're putting these out, thank you :D

[@martinfisker7438]

Just out of curiosity, what's your job title? I work with engineering, and I could definitely use better statistics skills

[@justarandomdood]

@@martinfisker7438 I got a job as a data analyst, it's more graphing and visualisation than I'd prefer lol. It's a good job but I'm definitely looking for something more mathy (vs PowerBI / Tableau / Qlik programming that I'm doing right now)

[@Karanagi]

Thank you for making these! I studied a small amount of statistics but am faced with the latter half of "use it or lose it". This is a great refresher on the topic. I'm also hopeful that the connections you point out that are new to me, will extend the half-life of my knowledge :)

[@runewalkingshaw6408]

A couple days ago I realized I did not really understand central limit theorem. We had learned about it in my stats class but only briefly. When I tried to read about it online I struggled to get all the ideas presented. This is a very conveniently times video that I appreciate a whole lot!

[@aarjith2580]

Would love a video on complex analysis, mainly on the multivalued functions in it..

[@audr3yaudr3y]

i am someone who cannot comprehend even the basics of what you’re discussing, but that makes these videos so soothing for me! always helps me on bad days like these + i’m so happy u posted today!

[@asilverman1949]

That's the same for me. I appreciate entering a world that holds wonder for me, and even with no deep understanding lets me look under the hood.

[@audr3yaudr3y]

@@asilverman1949 beautiful wording!!!!

[@audr3yaudr3y]

@@asilverman1949 and your page is lovely

[@rubikscubeking8398]

Thank you for your super clear explanation of the standard distribution function. I’ve always wondered about where it come from!

[@maximofernandez196]

14:06 yes please! I really love this channel. Watching all of these videos feels so amazing

[@pendragon7600]

Me when the binomial distribution approximates a normal distribution

[@leonardosaads]

Just a curiosity: almost everything in telecommunications depends on this theorem! it is extremely important!

[@edwardlulofs444]

Absolutely, and everyday I see new applications of it in the world.

[@EngRMP]

Just a brilliant description of basic(?) probability. I thought I understood this already... I didn't. Thanks, so much, for taking the time to put this video together... I can't imagine how long it took... what a labor of love.

[@TKNinja37]

I've been out of school for quite some time, but the only concept I've ever REALLY struggled with understanding has been the standard deviation of a distribution. I'm not going to ever 'use' this info in my life in a meaningful way, but this explanation made sense of every aspect I ever had any questions about. Especially after the last video, before which I'd never heard of a convolution, I saw the dice example and immediately said, "Oh, a rolling average of those fair dice would make the rounded shape of a distribution." I'm always impressed how every 'little' part of your explanations are the right size to wrap your head around, and the total of the 'big' solution becomes so clear when you put those easier 'little' parts together.

[@tommyrosendahl7238]

please consider making a series on Markov Chains and Markov Models to continue the probability theme going :D

[@nylonco7134]

I'm very excited to hear the idea for a video delving into variance. Explaining my trouble with variance: looking at the exponential function, we could absolutely choose to use 2^x or 10^x everywhere and just live with the correction terms ln(2) and ln(10) showing up (sorta like the π vs τ dealio). Out of convenience, e makes a 'better' base. But I can point out a number of 'deeper' reasons to use e as the base for exp than just convenience, and I can point to enough such reasons that using any other number seems 'wrong.' Contrasting this with variance, I'm aware that taking the square of the differences (x-μ) is more convenient, but I can't tell you why it's the 'obvious' or 'correct' choice based on that 'deeper' reasoning. Maybe |x-μ| and |x-μ|^3 use abs() which isn't smooth, but then why not (x-μ)^4, or any other even power? On a desert island, building stats from the start, I don't know how to make that choice for (x-μ)^2 well motivated.

[@Macandcheese1818]

This is one thing I'm still confused about after watching the video as well. Hopefully it becomes clearer in the next video

[@elrichardo1337]

this is probably not the right way to motivate it but perhaps it has something to do with using the “root mean square” (before Bessel’s correction) rather than the arithmetic mean as the “average” deviation? idk im clearly talking out of my ass here and would love to see this properly explained in a future video

[@elrichardo1337]

or, another thought; maybe think of variance geometrically, as the square of the n-dimensional “distance” between the point (mean,mean,mean,…,mean) and the point whose coordinates are your data points?

[@nylonco7134]

@@elrichardo1337 I don't know why I never saw it, it's pretty glaring now that you mention it, but I think you're onto something with the idea of a norm. Almost like I'm asking "if any p-norm works, why do we choose p=2, the Euclidean norm?" For geometry in a flat space, I know (more or less) how to answer that question. If this translates cleanly to variance in stats, I'ma be annoyed that I haven't seen it before.

[@odysseus231]

​@@nylonco7134 It's completely to do with a Euclidean norm. In fact, if you think about it, the standard deviation is exactly the Euclidean norm on the space of centered (mean=0) random variables (up to equality almost everywhere if you know these sorts of things, otherwise don't worry about it).

[@florentinosanchez3969]

Thank you so much for making this. You are the only one in Internet explaining where does formula come from.❤

[@luchoja2004]

the best explanation on the Central Limit Theorem I have ever seen. Amazing work!! Congratulations and thanks so much!!!