Introduction to the Central Limit Theorem

I discuss the central limit theorem, a very important concept in the world of statistics. I illustrate the concept by sampling from two different distributions, and for both distributions plot the sampling distribution of the sample mean for various sample sizes. I also discuss why the central limit theorem is important in statistics, and work through a probability calculation. (For the most part this is a non-technical treatment, and simply illustrates the important implications of the central limit theorem.)
For those using R, here is the R code to find the probability for the example in this video:
Finding the (approximate) probability that the mean salary of 100 randomly selected employees exceeds $66,000:
1-pnorm(66000,62000,32000/sqrt(100))
[1] 0.1056498
Or, standardizing:
1-pnorm((66000-62000)/(32000/sqrt(100)))
[1] 0.1056498
1-pnorm(1.25)
[1] 0.1056498

[@jbstatistics]

I'm a statistics professor in the Department of Mathematics and Statistics at the University of Guelph.

[@5RiverFlow2020]

I wish you were my prof. Also can you do videos on Moment generating functions?

[@JustMoseyinAround]

Many thanks from York University

[@gardenmenuuu]

@@sanchitakanta1018 LOL

[@RSB_777]

Informative Video. Keep moving making new videos

[@emmanuelekele6169]

Hello sir

[@yagayagaBabaYaga]

Watching in 2020 for my stats diploma. Just realized this is an 8 year old video. Jeremy Balka, your channel is a gold mine. You are amazing! I will always remember you. Thanks!

[@jbstatistics]

Thanks for the feedback. I'm a little overly restrained in this one, and possibly a touch boring, but I felt that the original was a little over the top and irritating in some spots. I'm glad you liked the normal distribution video! Stats is definitely something to get excited about!

[@ucheumolu4345]

I never really comment on videos but this was so helpful It would be an insult to not thank you. So, THANK YOU! You have saved me

[@jbstatistics]

There are different formats of standard normal table. I have videos outlining how to use a standard normal table for two main types of standard normal table (one that gives the area to the left of the value of z, and the other that gives the area between 0 and a positive value of z).

[@damiankonieczek5792]

My teacher has spent hours trying to teach us this. You did this in 13 minutes and 13 seconds.
Great job and thank you:)

[@newbie8051]

Hey just curious what are you pursuing now ? (as you were studying stats 5yrs ago)

[@codeblueocean3520]

The world need to know😂

[@thandolwethu3430]

😂here for this

[@musclesmalone]

Extremely, extremely helpful. I'm going through a data science masters and I'm finding myself increasingly turning to youtube and getting a primer/intuition of a concept before listening to my actual lectures. This week is CLT and law of large numbers and after this video I'm in a lot better shape to assimilate the material. Thank you!

[@jbstatistics]

Thanks for the compliment! I'm glad you liked it, and I'm very glad to be of help!

[@senorfootball2460]

Very well explained, and good examples! I find examples are extremely important to learn stats, so this helped.

[@jbstatistics]

Thanks! I'm glad you found it helpful.

[@sammendimasa784]

Thank you so much for these videos. I am taking stat for engineers and I am literally teaching myself everything by watching your videos.

[@icathianrain2298]

tbh literally the best video on CLT I've ever watched, thank you so much, thank those statisticians so much

[@themathguy3149]

Best video series about statistics in this whole youtube wildlife, thank you so much for existing and making everything better

[@sams5922]

Thank you so much for these videos. Between the textbook and my professor, I could NOT figure this out till I watched your video. They have been so helpful, especially with everything being online/ remote now.

[@manutdsparta]

I have been watching many of your videos recently. Thank you for your (fast) videos as well as you explain them very clearly with your voice. Enjoyment to watch and learn!

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Bravo. His teaching is beyond perfection. Amazing.

[@karltorento3358]

I love you so much man! I'm studying for the CFAs and your video explained CLT perfectly :D

[@jbstatistics]

Thanks! I'm glad I could be of help!

[@Tiffany_3x]

God Bless You! I am a little more confident about the final exam after watching your series of videos! Thank You!

[@whutismyname]

Definitely the best video on explaining CLT! Thank you!

[@gialinhpham6303]

Thank you very much for your admirable kindness. Your explanation is so comprehensive that I can save much time.

[@RahulBhasin]

One of the best video for understanding CLT.. thanks a lot...!!!

[@jbstatistics]

You're welcome, and thanks for the compliment!

[@ratikeshsharma1624]

This is magic how you taught us this difficult concept easily.

[@rickgeng463]

I can’t tell you how thankful I am of this video!!!

[@jbstatistics]

I'm glad you found it helpful!

[@IIAndersII]

I use your videos as inspiration when I prepare for teaching my class - thank you for the perfect explanation

[@jbstatistics]

I'm glad to hear that! Thanks so much for the compliment!

[@QuadDrums]

I really appreciate these videos, I hope to be a teacher who can help my students understand as well as you do.

[@jbstatistics]

When we draw a single sample, the sample mean will take on a single value. But if we were to draw a different sample, the sample mean would take on a different value. Before we draw our sample, we can think of the sample mean as a random variable with a probability distribution. The CLT tells us something about that probability distribution. You might want to watch my video "Sampling Distributions: Introduction to the Concept", which discusses this notion in greater detail. Cheers.

[@diencai1812]

I have learnt so much watching your statistics videos. Thank you for sharing your insight on the subject

[@meraj786ful]

Best Video explanation on CLT on the whole youtube. Thanks a lot

[@raviteja5125]

I have been confused for years but not anymore. Excellent explanation! Thank you very very much.

[@jbstatistics]

I'm glad to be of help!

[@armanozcan7983]

This channel never disappoints.

[@jbstatistics]

You are very welcome, and I'm glad to be of help!

[@charitharatnayake9147]

WOAAAHH NICE BOY!!! This will exactly helps me to pass tomorrow's exam...

[@malugaoaprilrose3946]

Thank you so much for this video, especially the word problem that you gave. It helped me pinpoint the main idea of this topic. You are such a blessing for learners during this quarantine. Thank you very much.

[@jbstatistics]

That's great Vinayak! I'm glad to hear it!

[@parthbhatia2478]

Amazing way of explaining CLT. Thank you so much!!

[@deepeshnair4375]

Clean sweep!! Clarity is wonderful!

[@jaekl3337]

You sir deserve a medal for explaining this stuff in a 13 minute video!! I was so confused.. thanks !!!!!!

[@jbstatistics]

You are very welcome!

[@jbstatistics]

Thanks! I'm glad to hear it helped.

[@ajklein40]

This was super helpful, thank you! I like how clearly into statistics you are. Really helps me to pay attention.

[@jbstatistics]

Thanks! I'm glad to be of help!

[@hounamao7140]

you're a fucking god of explanation!

[@jbstatistics]

Thanks!

[@oprabin]

Houna Mao
True that!

[@roadkil899]

I have an stat exam tommorow.. You saved me... Thank you so much Sir :)

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For what it’s worth, I would just like to let you know that your hard work does not go unappreciated!

[@betsegawlemmaamersho1638]

All your videos I watched are concise and simple. I do not think any of the concepts can be explained more simpler. You are amazing teacher

[@jbstatistics]

Thanks!

[@abhinavsharma1976]

Best explanation so far!

[@jbstatistics]

You are very welcome Tobias! I hope your studies are going well!

[@mathhack8647]

Woke up, checked this Vidéo before even have my coffee, I knew C.L.T longtime ago but now I go it much better. now I can explain it to my daughter in a btter bay . Thanks you .

[@adhoc3018]

Hey man, I've been watching some of your videos and they have really helped me to understand better statistics. In the past it seemed so difficult to me, but thanks to you I'm making good progress. I hope you are doing fine :)

[@xiwang4918]

Really clear explanation. Thank you a lot! I have understand this more!

[@fisslimen]

Great example! Helped me understand why CLT is used

[@sofianb86]

This was an amaizing explanation. It was very helpful, thank you!

[@BinethTharupama]

Thank you very much,
Understood every single thing..!! (Y)

[@samad.chouihat4222]

the number of views in this channel does not match the number of subscriptions . This guy should have more than a million subscritptions . i come here whenever i get confused about something , thanks dude and greetings from Algerian Sahara

[@jbstatistics]

That's good to hear!

[@Emmzillla]

Great video! I mised the lecuture on CLT in math class due to jury duty. This video helped so much!

[@atikahauliaputri6677]

Thank you so much for the beautiful explanation! It helps me a lot, I'm not kidding.

[@anuragsahu2893]

This is amazingly beautiful. How am I going to tell my mentor "please watch this video" :) Thanks for crystal clear explanation with robust example.

[@hashmarker4994]

Thank You!Its been Years since the Video has been Uploaded,But still Thanks!!

[@jbstatistics]

Hi Karthik. It is the number of observations used to calculate the mean that is important. In practice we typically draw only a single sample. If that sample has 5000 observations, say, and our sample mean is thus the mean of 5000 observations, then the sampling distribution of the sample mean will be approximately normal in that situation.

[@PassengerT_]

Really explicit explanation! good job!

[@jbstatistics]

+Weiji Hong Thanks!

[@John-lf3xf]

Weiji Hong I don't think you know what explicitly means

[@JimbobFaz]

This is all finally making sense! 😀 After many years of sort of getting this I understand it now so much better. So basically Xbar is a random variable all of it's own, with it's own mean and s.d ect, and varies depending on which sample we randomly pick from the population right? When I think about it like this it makes a lot more sense. Thanks for these brilliant videos.

[@FHO72]

i love how this is just straight to the point. I hate when videos and BOOKS always start with an example. just give me the god damn definition already! so thanks.

[@aaronforester82]

Best video on Central Limit Theorem. Do you have a virtual tip jar I can throw some virtual dollars in?

[@jbstatistics]

Thanks for the compliment. I'm just glad I can be of help. Cheers.

[@Dennaton]

@@jbstatistics what a legend

[@regularviewer1682]

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[@sanchitakanta1018]

@@jbstatistics In the last example while we are calculating the probability of the average being greater than 1.25 sigma.
The average is always in the middle of the normal distribution right?
Z value =0.
Then how can it be greater than 1.25 Sigma?
Can you please explain.

[@jbstatistics]

@@sanchitakanta1018 You're mixing up the true (theoretical) mean, and the sample mean. The normal distribution is centred at the true mean. The question asks for a probability involving the sample mean.

[@grantx3026]

Crystal Clear now. GJ!

[@brunoassumpcao]

My former statistics professor (great dude) used to say that without central limit theorem, we wouldn't be here. I laughted then, I cried over my tests, then I eventually learned... and everything makes sense once we realize the awesomeness of this mathematical theorem. Now I do the same for my colleagues :)

[@sarthakraheja13]

Awesome explanation. Thanks.

[@arjetaallamani5930]

You are the greatest!Thanks a lot!

[@jamesfullwood7788]

This video was excellent. Thank you for this!!!!

[@annasarahkramerfazendeiro8030]

Thank you so much! This was very well explained!

[@gautamhathiwala7267]

So beautifully explained....
Thank you so much, Sir....

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The best and clearest explanation I have ever found!!!!!!!!
Keep the good job #######

[@jbstatistics]

Thanks so much for the kind words!

[@arkadipbasu2348]

Wonderful Explanation, thanks a lot

[@Zerpentile93]

Your videos are as good as Khan Academy. Thanks for helping us with the maths!

[@garthenar]

clear, concise and professional. perfect lecture.

[@jbstatistics]

+garthenar Thanks!

[@seb6302]

Loved this!

[@saptarc]

Thanks. Awesome tutorial and example.

[@jbstatistics]

You are very welcome. Thanks for the compliment!

[@naved591]

Thank you so much professor. Finally I grasp it

[@biswajitnandi4304]

WOW ! nice explanation ! easy, understandable and well described !

[@jbstatistics]

Thanks!

[@Maya_s1999]

I just keep coming here despite all the textbooks I keep buying! Thanks so much again for being on Yotube.

[@jbstatistics]

You're very welcome!

[@jbstatistics]

Thanks and you're welcome!

[@armandpirgu3709]

Very well explained, i would recommend this to everyone that is banging their head on the wall, trying to figure out. Thank you

[@jbstatistics]

Thanks for the kind words!

[@elvinjafarli6257]

Great explanation, which means you know very well what you are teaching. Thanks!

[@jbstatistics]

Thanks for the compliment!

[@LinLin009]

great exploration with nice illustration!
Thanks,

[@gilgil5602]

Very good video! It tells us why CLT is such important. I was wondering whether you could make another video explaining the CLT intuitively? Why the limiting distribution is normal instead of exponential, gamma, or any other distributions? What is the essence of the CLT?

[@uclalse]

Best video this far on the CLT! I have watched around 10. This one did it.

[@jbstatistics]

I'm glad to be of help. Thanks for the compliment!

[@jbstatistics]

You are very welcome Dineo!

[@asbinanceasbinance1473]

Thanx a lot. This helps very much to understand.

[@stefanofedele4820]

Thank you so much for clarifying me such an important concept of statistics!

[@jbstatistics]

You are very welcome!

[@ethenhumbert9905]

I learn more in 13:13 with your explanations than three hours in class each week plus tutoring.

[@jbstatistics]

I'm glad I could be of help!

[@jahanvikhedwal1107]

Best explanation ever!

[@ravimishra339]

Amazing, great explanation!!

[@afmmarques]

Another great explanation, thanks! Greetings from Portugal

[@jbstatistics]

Hi Vinayak. In its simplest form, the CLT applies to the mean of independent and identically distributed random variables. If we are sampling from a finite population, then if the sampling is done without replacement the observations are not independent. So to perfectly satisfy the conditions of the CLT, we'd need to be sampling with replacement. But if we are sampling only a small fraction of a large finite population, then there isn't much of a difference between with and without replacement.

[@prabirdas8430]

Many many thanks for this nice explanation

[@shuaibx360]

good explanation, appreciate the work!!

[@jbstatistics]

Hi Vinayak. The very last example involves the average salary of 100 employees. The distribution of individual salaries is probably not normal, but the central limit theorem tells us that the distribution of the mean salary of 100 employees will be approximately normal. That's what allows us to calculate an approximate probability based on the normal distribution. We're drawing a single sample, as we typically do, but it's a single sample of 100 employees.

[@jbstatistics]

You are very welcome.

[@wenguanshen7275]

Man, you are just so genius !

[@pomegranate8593]

this is so sick, cheers!

[@AnneAnimanga]

love this !
im like binge watching all your vids .

[@jbstatistics]

Thanks! I'm glad to be of help!

[@CanDoSo_org]

Great class, thank you.