The Central Limit Theorem, Clearly Explained!!!

[@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. ALSO: If you want to play with the central limit theorem, and see it in action, check out this page: cltapp.fly.dev/ Support StatQuest by buying my books The StatQuest Illustrated Guide to Machine Learning, The StatQuest Illustrated Guide to Neural Networks and AI, or a Study Guide or Merch!!! statquest.org/statquest-store/

[@andreaxue376]

I wonder since there is a rule of thumb for the sample size at each draw(at least 30), is there any rule of thumb for the number of times you have to repeat the process to get a normal distribution?

[@statquest]

@@andreaxue376 Are you asking how many collections of 30 samples we would need in order to get a histogram of the means to look like a normal distribution? I don't know. I guess the answer is somewhat subjective. However, you could make an objective criteria, like how many collections of 30 samples would you need until a K-S test gives a p-value > 0.05. (A K-S test compares distributions). Hmm... An interesting question.

[@aditya4974]

BAM! Thanks again! "Even if I'm not normal, the average is normal" is indeed the best way for me to remember the Central Limit Theorem :D

[@statquest]

@@aditya4974 Awesome! :)

[@alonsom.donayre1992]

I got same doubt when i see the video because im from latam and we make a diference between samples and random measurements.

[@amitavaroy5723]

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!

[@statquest]

Wow!!! That is great! Thank you very much. Maybe one day soon I can visit. :)

[@amitavaroy5723]

@@statquest IIT would be very happy to host you, do visit :)

[@sathwikshettyiitb285]

@@amitavaroy5723 Yup

[@burstingsanta2710]

@@statquest Same at IIT BHU, you are pretty popular among engineering students! Everyone just refers you for anyone starting ML

[@statquest]

@@burstingsanta2710 That's so cool. Thank you!

[@christophersolomon633]

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 !

[@statquest]

Wow, thanks!!

[@legendrams548]

@@statquest your explanations with slides are truly awesome! 👍👍👍

[@MikeKay1978]

I only watch them for intro songs 😊

[@dishantvyas977]

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!!

[@statquest]

bam!

[@shudu4683]

This channel is a treasure.

[@statquest]

Thank you! :)

[@r.s.10]

that was indeed very clearly explained hah you've won yourself another subscriber!

[@chebedi]

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

[@statquest]

BAM! :)

[@avazB]

@@statquest you are a great man!!!

[@simongross3122]

Do I have to watch at least 30?

[@christopherody5806]

@@simongross3122 Only in the wild!

[@fredhasopinions]

Great theory, but that implies that the BAMs are uniformly distributed. Which, considering he can’t just start a video with “BAM”, might wreck our wee theory haha

[@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 :)

[@statquest]

bam! :)

[@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.

[@statquest]

Hooray!!! I'm so glad my videos are helpful! :)

[@Pixxi3]

Same! I have wasted dayyss trying to understand these theories! This channel was a life saver!!!!

[@namedtodream9895]

Damn this dude is stellar at making statistics engaging!!

[@statquest]

Thanks! :)

[@anthonychow6732]

Triple BAM!

[@shikharkhanna5404]

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!

[@statquest]

Thank you!!! One day I'll do it. In the mean time, check out 3Blue1Brown - he's got a series on Linear Algebra. It's good. When I make my own series I'm going to focus more on how the math is applied in practice (to statistics and machine learning), but his videos will give you a great start.

[@elsavelaz]

@@statquest looking forward to your explanations of lin algebra and yes 3Blue1Brown is great and I would love to see how you explain the application in ML

[@whatyouwantyouare]

@@elsavelaz I heard the book "Hacking the matrix" does a great job of explaining Linear algebra with a view towards CS/ML ... maybe it would help

[@ah2522]

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.

[@leanvo3880]

I got your idea. I am thinking about the convolution of LTI system which is kind of sums, those sums would be a normal distribution as well, no matter what distrbuted input is. thank for the comment.

[@Zenoandturtle]

My math lecturer told me exactly that, she was amazing. She told me that the significance of Normal distribution was related to CLT, in that plotting sample size (30, 30 +)of any distribution function yielded to our beloved bell curve.

[@DM-py7pj]

waiting time of? Any waiting time? E.g. waiting for a medical treatment

[@juliagschwend]

TRIPLE BAM!

[@JCA51698]

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!

[@statquest]

Thanks and good luck!

[@petemurphy7164]

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!!!

[@yungzed]

did u get ur degree yet

[@retsyalapiza2622]

hi, I'm also studying undergraduate statistics. may I connect with you?

[@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

[@statquest]

Awesome! Thanks for filling in all the details! :)

[@cantkeepitin]

The Cauchy has a strong physical and mathematical background. E.g. the conf interval for the mean of a normal distribution with unknown sigma has a Cauchy distrbution if we have one sample. Also dividing normal samples gives a Cauchy. And firing in a uniform random angle, the projection to a line would be a Cauchy distribution. That can explain why archers sometimes make really bad shots.

[@merryjoy48]

Recently been working on modelling the effects of shocks in production in large firms in an economy to the shocks in the production of whole economy. The proposition is that the share in value added by the firms to the total GDP of the economy is log-normally distributed with a power law tail (Pareto). Hence we couldn't apply CLT as previous studies had done so.

[@brenorb]

There are plenty of things which can be modeled as a Pareto distribution. That's why the 80/20 principle (also called Pareto principle) is so famous, which gives a Pareto distribution with a=1.16. Also, if a distribution gets close to a Pareto, it still converges to normal, but can take an unreasonable amount of time. Taleb writes about it beautifully in his book Statistical Consequences of Fat Tails under the name of sub-asymptotic analysis.

[@haifa6004]

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

[@monikgupta6687]

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.

[@sidalimounib589]

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

[@statquest]

Thank you so much! BAM! :)

[@88skewer]

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

[@statquest]

Awesome, thank you!

[@zjj-VIM]

wow, first to see your video, i think you video is very good, because i can understand what you say. My first language is not English and i don't have much confidence about my English. But your English can make me understand without translate. Thank you my friend.

[@statquest]

Thank you! 😃

[@uniquekatnoria5380]

I am new to statistics and have trouble understanding the formal terms stated in books. The content from this channel really makes it easy to get intuition and understand the underlying principles. Great work!!

[@statquest]

Thank you!

[@blackpearl2386]

The first line of this video explained everything.

[@irwinlxrry]

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

[@statquest]

BAM! :)

[@konstantinlevin8651]

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!

[@statquest]

You're off to a great start!

[@sankalpvk18]

Hands down the best channel on YT to learn statistics. Thanks for sharing your knowledge.

[@statquest]

Wow, thanks!

[@charlyslgado]

Why can't all teachers be like you? Thanks for the amazing content!

[@statquest]

Thanks! :)

[@johnmolokach_staff-southga3529]

Because teaching talent is not uniformly distributed =]

[@ciensalud]

@@johnmolokach_staff-southga3529 TRIPLE BAM!!!

[@Rohan-ce1sy]

Thanks for the crystal clear explanation Josh. BAM !!!

[@statquest]

Thank you!

[@andrewbetz535]

This channel is an absolute gem 💎

[@statquest]

Thanks!

[@somashekarreddy2650]

This one deserves an award

[@statquest]

Thank you!

[@kevalprajapati5365]

How can you have dislikes on your videos? I think it is also because of CLT. BAM!!!! I became a great fan because of the way you teach the concept. I will never forget the CLT in my life. BAM !!!

[@statquest]

BAM! :)

[@ravitan85]

"Even if you're not normal, don't worry the average is normal". That's so deep.

[@statquest]

bam! :)

[@JoyceSalvadorthewanderer]

Your "Triple Bam!" encouraged me more to review Stat subject for my FE exam, thank you wizard! :D

[@statquest]

Good luck! :)

[@danspeed93]

I've met folks hoping that we could understand this concept only looking at formulas. I wish your video existed earlier, thank you, never too late to understand!

[@statquest]

Thanks!

[@GibranMakyanie]

YOOOOO, YOU ARE MY EXAM SAVIOUR!!!! PLEASE KEEP THIS CHANNEL UP AND GOING. The way you say 'clearly explained' really reflects. Keep up the good work please!!!!!

[@marisa4942]

Thank you for this informative and fun video! Just to confirm, to justify using CLT, we need to know 1) Xi are i.i.d 2) the mean and the variance is finite (can be calculated) 3) num_observations >= 20 Love the little tune! "The average is Normal ~ "

[@statquest]

Yes! That is correct. However, the number of observations doesn't always need to be >= 20. Smaller sample sizes can work.

[@marisa4942]

@@statquest Got it! Thank you so much!

[@davidecoldebella8270]

Wish had discovered you sooner

[@michellestacybudu-kumi9178]

Me too

[@HarpreetSingh-ke2zk]

I have seen many animated ways to describe mathematical/probabilistic concepts. But your one is short and simple that can stay in mind.

[@statquest]

Thank you very much! :)

[@moli1218]

Thank you! I love the way you explain the statistics. Much easier to understand with examples. I really hope I can find these videos earlier. Thank you for all the help.

[@statquest]

I'm so happy to hear that you like my videos! :)

[@mfp123]

I literally laughed so hard at the “Who cares?” I wasn’t expecting to laugh while trying to understand statistics. You’re good..!!👍🏻

[@statquest]

bam! :)

[@huseyincelikel7527]

When i see your videos two words coming in my mind : "Bam", "Hooray" 😂

[@statquest]

Hooray!!!! :)

[@atrichatterjee5068]

@@statquest bummer

[@luminesc]

It's such a simple and obvious concept but it didn't click in my head until you showed it. Thanks!

[@statquest]

Bam! :)

[@tommcnally3231]

My new favourite pastime is listening to Sal Khan say "Sampling distribution of the sample means" over and over. Ps. learning maths from Khan Academy, followed by watching these videos, is a really effective way of learning statistics.

[@statquest]

Cool! :)

[@kusumkumari6894]

I am doing the same 🥰.

[@abbasjivani7166]

The guy made the concept easy peasy lemon squeezy!!😎 Absolutely loved the way the things were elabrated.😍

[@statquest]

Thanks!

[@denniswixon3592]

Enjoyed your video very much. I have been teaching statistics and programming statistical on and off for 50 years and this is one of the best explanations I have seen. I particularly appreciate your pointing out that a sample size of 30 is not a magic number. I wish you added that consistency of the data affects the needed sample size for generalization, but it's probably in another lecture. It's good to see you are reaching so many students. Keep up the good work.

[@statquest]

Thank you very much! :)

[@Christopheralb]

"Triple Bam" lol. I like how you fluctuated the tone of your voice too. So many teachers could learn from you on the delivery of information. Anyways, thanks for helping me brush up on stats stuff for possible interview questions. Love your vids man!

[@statquest]

Good luck!

[@SOUVIK_RAY_]

Just came across your channel. You explain every concepts with so much simplicity. The examples are spot on and helps to relate the concept with the problem at hand. Great work StatQuest!

[@statquest]

Thank you very much! :)

[@chetlund4465]

The best and clearest explanation of the central limit theorem I have ever seen & heard.

[@statquest]

Hooray!

[@colinhall7481]

This an amazing lesson Josh. Every student in statistics could benefit from this video alone.

[@statquest]

Thank you!

[@douglasnadysgoncalves7432]

WHAT THE HELL!! I AM IMPRESSED! Well done mate, thank you very much.... In the beginning I was like, what the hek is this song?? and at the end I was like BAM! now I get it... I will probably take this for the rest of my life.

[@statquest]

bam!

[@Cass_i]

I get so enthusuatic when he goes "BAM" 🤣🤣🤣

[@statquest]

Hooray! :) BAM!!!!

[@3someFootball]

Its incredibly clear explanation. I just got lucky to find your channel while I was starting to find statistic boring...Thank you so much for your sense of humor and your great ability to explain something in a very simple way, i know it takes a lot of experience and knowledge.

[@statquest]

Thank you! :)

[@surajthapa4160]

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.

[@mycotina6438]

Woah! This is a gem. Central limit theorem intuitively explained!

[@statquest]

Thanks!

[@juliecongress6278]

The video and source is extremely helpful in understanding concepts. The visual examples are great and the humor helps demystify difficult topics. Thanks Josh!! I wouldn't be able to make it through my classes without it!

[@statquest]

Glad it was helpful!

[@asianslayer555]

I finally understand this after so many years! Thanks and Double BAM!

[@statquest]

Happy to help!

[@carolinejo]

I SPENT HOURSSSSS NOT UNDERSTANDING and then BAM suddenly i UNDERSTOOD

[@statquest]

bam! :)

[@muskygaming69]

It's the fact that you calculate the mean of 20 samples to get one mean at 1:27 and afterwards at 1:51 getting one mean per sample in your explanation that gets me wondering if I really understand or not. The sampling frequency seems to be the most important notion to grasp this concept as 1000 samples with a mean calculated every 20 samples shows a mean distribution that is normal whatever the random variable initial distribution. edit : I just saw the note in the comments so I understand better now thanks !

[@statquest]

bam!

[@GravityGrid]

Your 7 min KZbin video was more useful and clearly explained than my 2 hour lecture. Thank you!

[@statquest]

Wow! :)

[@greeshmajith2752]

Iam happy that i perfectly understand the concept for the first time after learning it so many times.. Please put more videos

[@statquest]

Thank you very much! You can find all of my videos organized here: statquest.org/video-index/

[@amardeepsingh9001]

You are awesome Josh. I already knew the concept but felt just now ;)

[@statquest]

Thanks!

[@akshaypatel5468]

You have made life too easy man. Thanks a lot.

[@statquest]

Happy to help!

[@farsky22]

Regards from Brazil, one of my favorites channels! Really didatic

[@statquest]

Muito obrigado!

[@adityajaiswal7911]

The intro song summarizes it all. BAM!!

[@statquest]

Yes!

[@YoulooseMu]

i luv your classes thank you from brazil!!!

[@statquest]

Thanks! :)

[@anshulzade6355]

great way of teaching. Keep it up. The world needs it. Thanks

[@statquest]

Thank you!

[@averyjones2079]

"Saturday" a vivacious tune Josh keep up the music

[@statquest]

Thank you very much! :)

[@willyoctavianus8691]

oof.. this video is quite underrated... well narrated, interesting, and simple

[@statquest]

Thank you very much!

[@kushaltm6325]

Thanks again Josh. Today my prof taught CLT in the class and as usual am here to understand what his words actually mean !! :)

[@statquest]

Hooray! I'm glad the video helps! :)

[@Learn_SAS-du8lr]

You've made me visualize statistics. When I now look at a model output at work or in a presentation, I can relate that to mice height, mice weight, gene expression and actually explain it, suggest another method and why it might provide better results. Although I'll have a graduate degree in the data science soon, it's the day I finish working through your videos I will confidently say that I am a data scientist. Thank you for teaching me to love statistics!

[@statquest]

BAM! :)

[@JimmyCheng]

reviewing stats for my ml course, found these videos super useful, thanks!

[@statquest]

Awesome! Good luck with your course. :)

[@takeiteasy3525]

Holy shit, just discovered your channel and just in time.... thank you so much for doing these little lessons in a way that I can understand them. Plus, I crack up everytime you say 'BAM.'

[@statquest]

bam! :)

[@Cass_i]

Wow. I can adopt some of your teaching techniques for future classes I may have. You're very good

[@statquest]

Thank you! :)

[@JemRochelle]

Thank you for this video! The Central Limit Theorem was making my head spin but your video made it finally click! You have gained a subscriber :)

[@statquest]

Hooray! Thank you.

[@robhuntington8504]

Sorry 2 Qs 1. Just to be 100% clear - When you say at 1:30 "20 random samples" you mean a random sample of 20? 2. The labels on Y axis are throwing me off. For example, on the uniform distribution how can all values have a probability of 1.0? My first thought was "1 means 100% probability of that value occurring" But they can't all have a 100% probability of occurring. I'm starting to suspect that 1 is referring to relative probability (even though that's not something I 'm super familiar with).

[@statquest]

These are good questions!1) I mean that we collected 20 data points. Unfortunately, as you observed, "sample" is a somewhat vague term. I'll try to be more careful in the future. 2) Probability isn't the y-axis value for a specific position along the x-axis (that's actually called "likelihood" - see my video Probability vs Likelihood for more details: kzbin.info/www/bejne/porbf4aLebh5fpY ). Probability is the area under the line (or curve or whatever the shape you continuous distribution has) between two points on the x-axis. So, to calculate the probability of observing something between 0 and 0.5, you integrate the function between 0 and 0.5 to solve for the area under the line. In this case, with the uniform distribution, the line is set to y=1. The integral of this line between 0 and 0.5 = 0.5. So the probability of observing something between 0 and 0.5 is 0.5. The probability of observing something between 0 and 1 is the integral of the line (y=1) from 0 to 1. This integral = 1. NOTE: With the uniform distribution, the area under the line is always a rectangle, so you can, more easily, solve for the probability by just multiplying the width of the rectangle by the height of the rectangle. Does this make sense?

[@robhuntington8504]

@@statquest Thank you that is helpful. I think I "knew" that at one point about area under the curve but forgot somewhere along the way. I'm also going to watch your other video on Probability vs Likelihood

[@statquest]

I think the mistake you made is very common - and with the uniform distribution, it's super common. So no shame there. If you have time, you should also check out one of my videos on Maximum Likelihood - it will help you understand why people would even care about calculating likelihoods. kzbin.info/www/bejne/jpbTiaeibr5-rcU

[@phoenixnair]

The BAM! earned my subscription. This is really entertaining.

[@statquest]

Hooray!!! :)

[@nividinsights8190]

These videos make my day. I'm a Quant Tutor and it really comes in Handy!

[@statquest]

Awesome! :)

[@Putteponken17]

Thank you so much for your videos, I really need to visualize this with some simple examples and you do this excellently! Keep it up dude!

[@statquest]

Thank you!

[@chiragpalan9780]

"Even if you are not normal averagre is normal" CLT

[@statquest]

:)

[@angelfrancisco8128]

Dude! Your videos are a joy to watch! Thanks for this gift to the world!

[@statquest]

Wow, thank you!

[@sb-hf7tw]

Sir, my question is that, why there doesn't exist the mean of Cauchy distribution even if it is continuous.

[@statquest]

I think the simplest explanation is that the tails for the Cauchy distribution are too "fat". If you compare a normal distribution to a Cauchy distribution, the tails in the normal distribution get smaller much faster than the tails in the Cauchy distribution. For the normal distribution, when we collect a large number of measurements, most of them will be from the middle (near the mean) and only a few will come from the tails. This allows the estimated average to converge on the center of the distribution as the sample size is increased. In contrast, a large sample from a Cauchy distribution will have a lot of measurements from the tails, making the average value unstable - it could be a value near the middle, but it could also be a value near the edge. Increasing the sample size simply increases the chance you'll get more measurements from the edges that prevent the average from converging on the center of distribution. Does that make sense? If you want to see the math, there are plenty of webpages that will walk you through it.

[@sb-hf7tw]

@@statquest very very thanks sir for this

[@profealexandrasierra]

I love the music of the intro! So cool! Thanks for this videos ❤

[@statquest]

Glad you enjoy it!

[@赵宛冰]

You have worked in biostatistics for twenty years!Awosome!

[@statquest]

Thanks! :)

[@tinglingwei1056]

Thank you for this fun and easy to understand explanation. I’m wondering why CLT is true, do you happen to have a video on this? Thanks again! 😊

[@statquest]

Unfortunately I don't have a video on that yet. :(

[@tinglingwei1056]

@@statquest Thank you so much for replying! 😃

[@muralikrishna9499]

The central limit theorem does not apply to Pareto distributions since the mean and variance are infinite! Bammm!

[@YourGirlPratiksha]

😂😂

[@vahegizhlaryan5052]

Even if you are not normal...the average...is normal!!! The most inspiring thing I have seen😂

[@statquest]

bam!

[@venicetimones4853]

the BAM!!! gets me every time.

[@statquest]

:)

[@nathanx.675]

I graduated from college in May and thought it was time to say goodbye to this wonderful channel. I even got a little emotional thinking about the time I've spent here and how much this channel has helped me. I now realized how premature that was [facepalm] and how naive and clueless I was back in May. As a grad student, I'm back here again for a data science class. I guess life does always find a way to mess with you lmao. Just thought this is pretty funny and wanna share. Anyways, Quest on.

[@statquest]

Double BAM! Glad StatQuest is still helpful! Quest on!!!

[@chyldstudios]

next video: quadruple bam!!!!

[@statquest]

Dang!!! :)

[@mushfiqurrahmanshishir8055]

you deserve WAY more subscribers..

[@statquest]

Thanks!

[@kunalshukla1236]

Quadruple Bam !! The distribution of 'the number of times you say "Bam" in your videos', in not Normal!

[@statquest]

That's awesome! You made me laugh out loud. :)

[@JuanuHaedo]

Quintuple BAM!! The distribution of the mean of 'the number of times you say "Bam" in your videos', IS Normal!

[@statquest]

@@JuanuHaedo I love it! This thread of comments is probably my all time favorite. :)

[@naveencena7004]

Bam! apply central limit theorem to make it normal

[@Lphanova]

THANK YOU SO MUCH! I have been looking for some videos for a while to finally understand statistics and I would never believe that learning this subject in English (and not in my mother tongue) will help me!

[@statquest]

Glad it helped!

[@shkmamun]

"After we collect 10 samples.." should be "10 times of 20 (or n) samples..." Am I correct?

[@statquest]

I'm a little loose with my use of the word "sample", and for that I apologize. Sometimes I use "sample" to refer to an individual, but technically a sample is a collection of individuals that represent a population. Google "Random Sample" for more details.

[@chujingxl]

Thank you! You are a wonderful teacher! The theory has been explained so clearly. It is easy to understand.

[@statquest]

Glad it was helpful!

[@TheKnrumsey]

While I appreciate parts of this video for being clear and easy to understand, it is very wrong in terms of the fine print. Although the *population mean* of a Cauchy distribution is undefined, you can ALWAYS calculate a sample mean. The CLT does rely on having a finite *population mean*, but that's not the important part of the fine print anyways! The part about the sample size is far more important. There are many distributions in real life (such as income for certain groups) which may require far more than 30 samples for the CLT to provide an accurate approximation.

[@prrr7308]

And for any distributions which have not finite expected value (population mean), you can calculate the finite sample mean, and you MAY NOT realize that you estimate infinity with your sample mean calculations. Anyway, one of CLT (yes, there are many!) is for the standardized random variables, i.e., subtract the sample mean and divide this by the (corrected) sample standard deviation. The approximate distribution will be the standard normal one, if the expected value and the variance of the original distribution exist. And the histogram is wrong for equidistant based columns!

[@hakandemir101]

Thank you very much to provide us the more understandable way of teaching. It is just simple and pure.

[@statquest]

Thanks! :)

[@swapnilchavan7076]

Amazing explanation.... Lots of love from India😍

[@statquest]

Thanks! :)

[@Jupiter1423]

The central limit theorem is perhaps one of the most amazing mathmatical discoveries in history. Most amazingly the mean of the normal is the same.

[@statquest]

yep!

[@ariacube07]

i am binge watching your videos for my statistics exam. wish me luck.

[@kvjqxzz5905]

good luck matey

[@statquest]

Good luck and let me know how it goes. :)

[@GregThatcher]

Thanks!

[@statquest]

TRIPLE BAM!!! Thank you so much for supporting StatQuest! :)

[@ShroukAbdulshafy]

I like how you explain things in a funny and simple way. Thank you so much!

[@statquest]

Thank you!

[@obelix2545]

very helpful for a level stats, thank you

[@statquest]

Glad it was helpful!