But what is the Central Limit Theorem?


0:00 – Introduction
1:53 – A simplified Galton Board
4:14 – The general concept
6:15 – Dice simulations
8:55 – The true distributions for tallies
11:41 – Mean, variance, and standard deviation
15:54 – Unpacking the Gaussian formula
20:47 – The more exquisite formulation
25:01 – A concrete instance
27:10 – Sample means
28:10 – Underlying hypotheses


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You can find code for particular video clips and initiatives here:

Music by Vincent Rubinetti.

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3blue1brown is a channel about animating math, in all senses of the word animate.


  1. Next up, we’ll dive into *why* this theorem is true, i.e. why adding many iid variables tends to produce the e^(-x^2) shape. The hope is for that next video to be out sometime next week.

    • The fourth brother Karamazov

      @kturst s not the same. Here is a definition from a great book on stochastic processes. “A stochastic process is generally defined as any collection of random variables X(t), t is an element in T, defined on a common probability space, where T is a subset of real numbers. (Hoel, 1972)” Random processes and stochastic processes are the same thing. Both refer to collections of random variables. Typically, they are indexed by time and most commonly in a discrete setting. Say I roll a dice. That is a random variable. Now say I roll it once every minute for one hour. Then we can index each this as X_1,…,X_60 where each X_i is the result of rolling the dice at minute i. X_30 would be the result of the rolling of the dice at the 30th minute. This sequence is a collection of random variables X_i and thus is a random process.

    • @kturst s Think about random processes as functions but random. They can be discrete or continuous like normal functions. Random variables on the other hand is as the name tells, just some numbers but random.

    • I was just working on a math puzzle, and in solving that, I wanted to determine what would happen if you keep adding independent and identically distributed numbers in the range of 0 to 1. Specifically, I was looking for the probability distribution only on the range 0 to 1 for any number of random, uniformly distributed numbers, throwing out any points that are higher than 1. This probability distribution will not sum to an area of 1, because it involves adding together the results of infinite sums, where only the distribution of the first random number will have a sum of 1.

      I did this by finding the convolution of 1*1 on the range 0

    • Ive often thought about this without having a better name than “bell curve”. My interest is how it relates to human nature and nature itself. It helps me understand people much better.. Glad to finally see a video on what was before just something in my head.

    • Lorenzo Pla Serrano

      Please, I love your probability videos. ❤

  2. This question has a very simple answer: the central limit theorem is really cool.

  3. Could somebody give a rigorous definition of the functions X_i (at 4:48)? For the sake of specificity, let’s consider throwing a die as our experiment, and let’s say the sample space is {a,b,c,d,e,f}. Then X is a function from {a,b,c,d,e,f} to {1,2,3,4,5,6} that sends a -> 1, b->2, …, f -> 6. Then for a fixed i, what is the definition of the function X_i? (The X_i can’t be equal to X (as functions) each, right? Because if this were the case, then the sum X_1 + … + X_N would be equal to X + … + X (N times).)

  4. Can you explain the difference of the CLT curve from the Poisson distribution? (I thought that they were the same thing.)

  5. Can you spot Messi? 0:52

  6. 17:45, edge of my seat – is the sigma algebra going to catch him and derail the whole video??? But no, the hero deftly dodges and escapes, his honour intact with a brief nod to intervals as generators of the appropriate Borel algebra.

  7. 29:47
    The 3rd red bar just casually shrunk and it kinda bugged me

  8. 09:34 Thanks to Domino’s for sponsoring this video? 😉

  9. Yooooo my favorite theorem has a 3blue1brown video now? Lez goooo

  10. Finally! It has been months since I got my fix in recreational mathematics!

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