The Central Limit Theorem, Clearly Explained!!!

The Central Limit Theorem is a big deal, but it's easy to understand. Here I show you what it is, then I describe why this is useful and fundamental to Statistics!

This StatQuest follows up on the one that describes the normal distribution...

...and the StatQuest on Sampling from a Distribution:

For a complete index of all the StatQuest videos, check out:

If you'd like to support StatQuest, please consider...

Buying The StatQuest Illustrated Guide to Machine Learning!!!

...or...

...a cool StatQuest t-shirt or sweatshirt:

...buying one or two of my songs (or go large and get a whole album!)

...or just donating to StatQuest!

Lastly, if you want to keep up with me as I research and create new StatQuests, follow me on twitter:

#statquest #CLT #statistics

[statquest]

NOTE: Unfortunately I was a little sloppy with my terminology and that the word "samples" can mean different things, so let me try to rephrase it. If we collect 20 measurements and calculate the mean, and then do that a bunch of times (collect 20 measurements and calculate a mean), a histogram of those means will be a normal distribution. This suggests that an individual mean, calculated from 20 measurements, is, in and of itself, normally distributed. For example, if we had a uniform distribution and we collected 20 values from it and calculated the mean, then that mean would be normally distributed. We know this because if we repeated the process (collected another 20 values, calculated the mean, and then did that a bunch of times) the histogram of all the means we calculated would be a normal distribution.

[christophersolomon]

Mr Starmer, I am a professional scientist with many years experience in the academic and commercial worlds and I must say that your videos are truly excellent. They really convey the central ideas so well and run that tightrope between too much detail and not enough perfectly. Keep up the excellent work !

[amitavaroy]

I am a 4th Year UG at IIT Kharagpur and you will be pleased to know that almost everybody on campus loves your lectures on Probability, Statistics and Machine Learning and consider it to be the best resource for cracking company interviews. Absolutely brilliant content!

[chebedi]

If you watch many StatQuest videos, the distribution of BAMs will be approximately normal 😂😂😂😂

[dishantvyas]

I just realized that the entire CLT was encapsulated in the 8s lyrics - "Even if you're not normal, the average is normal!" Hats off to you, man... I never imagined an ukulele being used to teach stats!!

[mugiwara-no-luffy]

The fact that you are still replying to every new comment on a half-decade old video is amazing and commendable! Thanks for this, helping with my stats course for Uni :)

[RebeccaRonDaraf]

I thought I was hopeless with statistics and I was sure I wouldnt pass my college stat exam, but you make it very simple, and you even make me laugh will the songs in the beginning. I cannot thank you enough. I hope god blesses you. Thanks dude.

[shikharkhanna]

Sir, Your way of explaining is beyond Normal in brilliance. Could I request you to please make such enlightening videos on Linear Algebra and other Mathematical concepts in order to interpret the math behind the machine learning algorithms. The academic and text book notation as well as explanations gives me nightmares!

[namedtodream]

Damn this dude is stellar at making statistics engaging!!

[ah]

Great video. I do want to point out that the Central Limit Theorem is why statisticians celebrate the Normal Distribution at all, because let's be honest, the normal density function is supremely ugly to look at and near impossible to fuss with.
The CLT is one of those "too good to be true" laws of the universe, and it is actually more miraculous than this video presents itself.


The most generalized form claims that the sum (not just the mean, which is just the sum divided by a constant) of any random variables will be roughly normally distributed. These random variables don't even need to come from the same distribution. You can sample from a uniform, a beta, a lognormal, an inverse gaussian, and the sum of those 4 values will be normally distributed. (fine print, the variances and means need to be in comparable range otherwise one sample will dominate).


It's also the reason why waiting time starts to become normally distributed, because it is the sum of exponential (which is a gamma distribution, which converges to normal very fast).
It is also the reason why most variables in life are normally distributed, because you can usually break them down into sums of smaller categories of unknown distributions.

[monikgupta]

Cauchy has some practical implications, like decay of radio active material in nuclear fall out, or chemical decomposition of material, where process tends to slow down at the end.

[petemurphy]

Hi, I just wanted to thank you for the videos,
I am doing a degree in statistics at the moment, my general method for learning is to work through what the professor give me (which I find very confusing), then come to your videos to get an easy to understand explanation.
You are really helping me out with my degree and I want to say thanks!!!

[abalter]

Josh--you are an inspiring teacher. Tidbit about distributions that don't follow the CLT. I believe the condition for the CLT to hold is that at least the first and second moments of the distribution are finite. There are many phenomena in nature that are, more or less, modeled by power law distributions (Pareto, Zipf, etc.) or ones with power law tails (Levy). Any distribution with a tail that decays slower than x^(-3) (i.e. x^-a where a<3) will have an infinite variance. If a<2, then the mean is infinite as well. These distributions can't truly model anything physical, but sometimes apply for a meaningful range of values. Therefore one would want to apply the CLT carefully for these cases.

[JCA]

Right now I’m studying to take the first actuarial exam in probability, and I just discovered your channel. You just earned a new subscriber!

[haifa]

GOD BLESS YOU, HONESTLY I WAS LOST. TILL I FOUND THESE VIDEOS. ITS REALLY VALUABLE TO ME. THANK YOU

[konstantinlevin]

Thanks a lot! I've tried the examples you gave with python. I sampled from uniform and exponential distributions, computed means and draw histograms and bam! This actually feels like magic. I'm looking forward to understand the theorem more. I read the wikipedia page and it actually seems like there are lot to learn!

[skewer]

spend 10 mins on your videos and cleared my 10 years doubt, paypal donate just sent, thank you so much, will watch all of your videos

[sidalimounib]

I have not found a single video that explains this better than you do. Great work + 1 sub

[irwinlxrry]

i just graduated from pharmacy and started a job that requires knowledge about statistics and your channel helps a lot! thank you!

[surajthapa]

Thanks, thanks and lots of thanks... I love your way of explanation BAM!!!. Can you please make videos on the following topics-
1. Bayes for ML, I mean how Bayes helps us to find the best parameter of a model and probability of a prediction.
2. MCMC sampling methods.