The Central Limit Theorem, Clearly Explained!!! 

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?

@@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.

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

@@aditya4974 Awesome! :)

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

BAM! :)

@@statquest you are a great man!!!

*a little correction the sample means of BAMS will be normally distributed

Do I have to watch at least 30?

@@simongross3122 Only in the wild!

Wow, thanks!!

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

I only watch them for intro songs 😊

Thank you! :)

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

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

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

@@amitavaroy5723 Yup

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

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

bam! :)

bam!

Thanks! :)

Triple BAM!

did u get ur degree yet

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

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.

@@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

@@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

Thank you very much! :)

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.

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.

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

TRIPLE BAM!

Thanks and good luck!

Thank you! :)

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

Thank you so much! BAM! :)

Glad it was helpful!

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

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

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

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.

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.

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.

Wow, thanks!

Thank you!

You're off to a great start!

Thanks! :)

Good luck! :)

Thank you! :)

Hooray!

Thanks!

Bam! :)

Hooray! Thank you.

Thank you very much! :)

Thanks! :)

Because teaching talent is not uniformly distributed =]

@@johnmolokach_staff-southga3529 TRIPLE BAM!!!

Awesome, thank you!

Thank you very much! :)

Awesome! Good luck with your course. :)

Muito obrigado!

Glad it helped!

Wow, thank you!

Thanks!

Thanks! :)

bam! :)

Thanks!

BAM! :)

Hooray!!!! :)

@@statquest bummer

Happy to help!

BAM! :)

Cool! :)

I am doing the same 🥰.

Awesome! :)

Thank you!

Thank you very much! :)

Hooray! :) BAM!!!!

Thank you!

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

😂😂

Hooray!!! :)

Happy to help!

Wow! :)

:)

Thank you very much! :)

Thanks! :)

:)

bam!

Dang!!! :)

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.

@@statquest very very thanks sir for this

Thank you!

Glad it was helpful!

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

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

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

Bam! apply central limit theorem to make it normal

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: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-pYxNSUDSFH4.html ). 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?

@@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

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. ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-XepXtl9YKwc.html

Glad I can help! :)

Thanks!

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.

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!

Thanks!

Have you seen the linear models StatQuests? Factorial design is a type of linear model. If you have time, watch those - they'll get you 80% of the way there - there are few extra details (like how to check for interactions and what not) that I don't cover - but the main ideas are all there. Here are the links: Linear Regression: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-nk2CQITm_eo.html Multiple Regression: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-zITIFTsivN8.html t-tests and ANOVA: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-NF5_btOaCig.html Design Matrices: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-2UYx-qjJGSs.html That last video (which builds on all the previous ones, is the most important thing. If you understand design matrices, you're just a step away from factorial design.

@@statquest ok sir.

Glad you enjoyed it!

Glad it helped!

Glad you enjoy it!

Thank you! :)

Thanks! :)

You're very welcome!

BAM! :)

Glad you like them!

I'll put that on the to-do list. The more people that ask for it, the more I'll push it to the top of the list.

Thank you very much! :)

Thank you! :)

Thank you! :)

Nice!

You're welcome! :)

BAM! :)

Thank you!