Central limit theorem | Inferential statistics | Probability and Statistics | Khan Academy

[@gopikarajanikanth4482]

Honestly, this almost 10 minute video helped me understand something we were learning in class for like 2 weeks! Thank you so much!

[@johnanderson8765]

that two weeks of study is the reason you can enjoy this clip so much.

[@aniruddha4672]

@crni195 Like a true engineer you had to point out that you are one lol

[@natureloving785]

Superb Khan sir, am very pleased to study statistics as I watched your 10 minutes videos

[@shakuntalakamade]

are you alive ?

[@matthewwroblewski8752]

Very, very cool stuff. Also, you're obviously a smart guy, Sal. But at the same time, you're incredibly accommodating to us students. Thank you for your sincerity and empathy, sir.

[@esmeralda4884]

You explain this better than a textbook. You are a great!

[@adamromero]

Why can't teachers explain things this clearly, why do they have to act all scholarly?

[@deepakrp]

because they don't understand it either!

[@alenjose3903]

did u watch the next few lessons?

[@yuningliu6300]

because they are scholars, not pragmatist, like you !

[@glennhenshaw7474]

Well I think what's mostly going on is that there are different levels of understanding for a topic. Sometimes, if the explanation is not clear it's because the instructor is making room for some of the subtleties down the line. For example, imagine how confusing Newtonian mechanics would be if instructors always fixed the tiny error due to relativity. Or, like someone said below--maybe they don't have a clear understanding yet.

[@blakeelzinga1168]

a true mastery means that one can teach it simply and clearly

[@SuperYtc1]

Fantastic explanation. A shame that most teachers are not educated enough to be able to understand and explain things like this to their students.

[@sophiefeng2740]

so true

[@mrknarf4438]

...and when they are educated, they start believing it's super obvious so they just tell you what it is without examples and in depth explanations and jump straight to the following topic, expecting you to have not only understood but also interiorized the concept.

[@Saiphel]

@@mrknarf4438 This so much. I love when they have 200 students in front of them and when the teacher asks for an answer to a question and nobody answer it's everyone else's fault. They never think maybe it's their fault they suck at teaching. Sal is amazing, I only wish topics were more in depth.

[@jasminespence6452]

Dear Sal. I always skip all my calculus and statistics lectures and come straight to your videos. This has been the secret to my success in university. Thank you!

[@i6mi6]

Drinking game: drink whenever you hear the word "sample"

[@TeeNanners]

i6mi6 I might need to play that game to get over my probability score... D;

[@rogersyversen3633]

no

[@nellyaviles9342]

lets just kill braincells before the exam..

[@samdavepollard]

Are you intoxicating that I'm insinuated?

[@skyepaul261]

I'm trying to study here shhh

[@meganmaloney192]

These videos have been tremendously helpful! Thank you SO MUCH for making them! The concepts make so much more sense when I can see them being worked out.

[@RachelLovelace]

So cool. I took Stats 101 about nine years ago, and these videos were there for me. I'm back in grad school now, and you're videos are helping me with Applied Stats once again. You rock!

[@soumyaranjandas7394]

Can u explain me in somewhere.... actually I didn't get what is related to central limit theorem. Is it Sample size or no. Of samples from which we calculate mean.

[@mimireyes04]

cramming for my stats exam tomorrow

[@EmberArcher]

Cramming for my stats final today

[@carterwest9504]

same except in 20 minutes

[@swatibiswas6796]

Me rn

[@CrunchyDark]

Not cramming. :P trying to learn it as much as possible.

[@thesevenkg]

@@CrunchyDark big flex

[@stelun56]

I have been a mathematician all my life. I dropped out after middle school but started to get bored, so I bought some mathematics books with the answers at the back and used them to self-study for university entrance in the UK. That was 50 years ago before I graduated with scholarships for Oxbridge after gaining a double first-class in pure mathematics and theoretical computing. With hindsight, I feel your videos would have been really useful for statistics which I dropped for pure mathematics, applied mathematics, and physics. You are always highly recommended to all my tutees struggling with their education during this pandemic. Excellent material!

[@BboyFadi]

you are amazing , thank you , not only for this video , but for all your videos that i have been using for 3 years :)

[@Bella_Noches_]

This video summed up in almost 10 minutes what I have been trying to understand in my textbook for the past week. Good stuff...thank you!

[@hxh2025]

Thank you! Thank you so much! Thank you very much! I have been in Intro to Econometric class for 2 months already. I feel I understand more from your video for 10 minutes than in class for 2 months

[@mostinho7]

Done thanks 4:30 looking at the SAMPLE MEANS (taking a sample of n measurements, then averaging those n measurements is the sample mean), doing this for x samples of n measurements we have x sample means The distribution of these sample means tend towards a normal distribution as we take more samples. Also as the sample size the number of measurements in each sample increases, the sample means distribution approximates normal even more

[@Wooktent]

I was given 3 20 minute videos on this subject and I didn't understand a thing they were trying to tell me, but I watch one 10 minute video from you here, and I completely understand this now. Thank you so much, KA.

[@bluesky-mi2sx]

This video was posted 10 Years ago and still so useful! Such a crazy thing

[@8dannygirl]

You just saved my life brv.....you explained in 9 minutes and 49 seconds ,what ive been trying to understand for the last 2 hours.

[@kwsatl6066]

went to lecture today and read the chapter and was clueless. I watched the first 7 minutes of this and the concept is crystal clear!!

[@thuuyenphamnguyen8899]

my teacher just mentioned the central limit theorem and did not explain it (in 1 week!) :D and I just spent 10 mins to watch this clip to understand what he tried to explain in 1 week (and no one understand) :D thank you so much!

[@hou950]

You are helping me get through my graduate level quantitative analysis classes. Thank you so much ! =)

[@SuperTurkeyKing]

OMG!!!u r much better then my lecturer!!! He talks like a computer n I can just keep copying the solution of the examples during the class!!!! I understand much better becoz of u!!!!!!thx a lot!!!!!!!!!!!

[@nazifataha8868]

It's sad how many people commented here saying that their teacher could not explain it. It seems most teachers are not good at their jobs. Where in the world would we be if we didn't have contents like Khan Academy?? Thanks to the internet. Thanks to people like Sal

[@kristennwang]

I never took a stats class in high school or college and the bootcamp class I am currently taking does not do a good job at explaining this theorem. So who do I turn to? Sal! I grew up with you and you are still helping me learn even in my near thirties. Thank you!!!!

[@pratibhas2468]

It's been 13 years since upload and people like me are still using these videos... Great explanation!

[@18gshock]

Nothing much to say how good you are, the video tells it all, keep up the good job!!!

[@MayankMehta-pr9eg]

Once again Khan Academy saved me from the state of I am not able to understand to how easy is this stuff. Thanks

[@hennageorge3259]

If x = 1,3,4 or 6 and the sample size is 4, there would be 4*4*4*4 possibilities i.e. 4^4 possibilities =a maximum of 256 possible outcomes so by taking 10,000 samples you will be repeating each 1 about 40 times.

[@noahschuler6388]

Henna George yes, and if you take an infinite amount of samples, the distribution of the sample means will show the probability of getting each sample.

[@aprilhicks3474]

You just made life so much more interesting. Love you Sal! Will donate soon.

[@That1KoreanGuy]

This channel is a life saver!

[@TheSevenofMine]

I like it that it's called Khan Academy. KHAAAAAAN!!

[@Abubakar-ht5ee]

you guys are shaping history. thank you.

[@LYLxd]

YOU ARE SO AMAZING. PLEASE KEEP DOING WHAT YOU'RE DOING. I need to pass my exams...

[@ahmethamdicelik1277]

have you passed?

[@obinnadaniel2001]

I mean this is brilliant! Got me thinking and understanding deeply.

[@rodrigomorgado5524]

A good way to explain CLT. From an unknown discrete distribution to converge to a normal distribution.

[@richaunfacey5447]

You have a great channel. I am in a master's degree program, and I still use your site.

[@usharagunath23448]

may god bless you Sal!! You are my guru you are the voice in my head as i solve math

[@ChitimachanPanda]

"1" is one element of the sample, so is another "1", as well as "3" and "6", therefore there are 4 total elements that comprise the whole sample, thus the sample size, n, is equal to 4. It's 4 in this case because that is the sample size this person decided to use for his test. Higher sample sizes usually lead to more accurate tests. If I say "What is the sample size of all possible outcomes on rolling a die?", there would be 1, 2, 3, 4, 5 and 6, meaning n=6.

[@lllBOLTlll]

i've figured him out... He went to a good school and learned this beginner subjects and mastered them because of good teachers, and then he words it into a 10 min video and impresses all of us...

[@yitianxiao6650]

thanks. This is way more straight forward than the aihl textbook

[@mracuraon18s]

The U-looking symbol you're talking about is the greek letter μ or Mu. It represents the mean. In this video, he used an x-bar (just an x with a bar above) to represent the mean because it is was specifically for a SAMPLE. In other words, you use μ for a population mean and use x-bar for a sample population.

[@joed9229]

khan is awesome ! Im in this course that could not explain this well. I need to know the principle and Khan blew it out of the water ! I know the principle and the APPLICATION ! sweet

[@serachrysanthemum9687]

Heck yeah, this is a great motivating video... gives an outline of the idea and why it's so cool and important!

[@pritivalecha3147]

thanksss, i loved your video, i was looking to prove CLT and it clarified niceee

[@jordantan5023]

Very Good example ... I was having problems with figuring out how the individual mean element was obtained...

[@sinchumithun1]

I think there is a flaw, the normal distribution is more and more approximated when we increase the number of trials. As we increase the sample size we get the variance goes down and in the limit we get a delta function around the mean, which is a consequence of law of large numbers.

[@ytkv]

Best explanation out there. Thanks, Sal!

[@samadhashmi]

I love statistics!

[@khanabdulumarkhan6713]

Clear explanation & better

[@jamesrobsonza7752]

Much better explanation than my textbook, thank you so much

[@SamahGomaa]

I am a big fan of khan Academy ❤❤

[@Nizchul]

You are changing the world!..... seriously!....

[@solticpanzer]

This made understanding this theorem a lot easier. Also, that is a really bad 4.

[@purps45]

Just remember, this only applies to finding the Mean or Sum. I've heard people try to claim the CLT means you can treat any PDF like a Normal Distribution if you take enough samples.

[@jamesbrown7885]

what is pdf

[@ylast3756]

your explanation is more elegant than a 45 min lecture from a top40 US college.

[@vishnuvardhan2608]

Sal.... Your my rockstar!

[@versus13]

Your welcome

[@ayonrab]

God bless you ! I got this after 10 years...

[@diencai1812]

Thank you for your clear explanation. You are a world class educator!

[@winterfell14]

Oh man, that is SO logical!

[@tango2olo]

Mr. Khan, you are a Prophet in the world of education. Please start making vedios on training the so called Teachers, on "How to Teach" a complicated stuff in a lucid manner! They all need salvation too. :-p

[@lorrainepinon]

helped me to pass my statistic class. thank you!

[@matevarga1401]

This was a very useful video. Thank you so much. Clear and interesting explanation. Although the "peak" of the normal distribution should be around 3.5 in your example, not 2.75. Since that's the mean. Right?

[@WilliamJSSequeira]

Jeez, thanks for driving it home! You need to get with a publisher and go wide, you explain in the most basic, and common fundamental way for easy learning.

[@LeahInTheRye]

Thanks! Got a test that includes this section next week and it had me stumped

[@GirlGirlicious]

You, sir, are The Real MVP!

[@aj-tg]

Thanks Sal !

[@NikitaSharma-bs4gg]

Thank you 💚💚💚

[@ShroukAbdulshafy]

Very helpful! thank you so much, Sir!

[@Lunaizle]

this man khan do anything

[@choppertalk8291]

Very clear. Thank you

[@임상일-h7f]

Your video is easy to understand. Thank you^^

[@rideshareescapades4281]

I love you man it's a lot of fun learning this stuff. Thanks a lot

[@Zurh1994]

Central limit theorem = mind blown

[@cherrychapstickgurl]

this helped me with psychology! thank u sal

[@uvwxyz91]

+cherrychapstickgurl wut?lol

[@cherrychapstickgurl]

***** that's right! you need math in psychology too

[@eugenedeschamps164]

yha there is is how crazy do you think you patient is and if he going to kill anybody there is a general equation is to start guessing the medication that you think that fits the med book for that diagnoses then again guess the doses you gonna cram down his esophagus the more the better because the more patients you overdose is better for the bank book at the end

[@goktugguvercin8069]

You can use the word "observations" for the elements in a sample.

[@davidmbeckmann]

Very instructive video!

[@pianoforte17xx48]

You can say "4 observations in Sample ..." . Eases understanding

[@檜皮猫]

So correct me if I'm wrong: The central limit theorem demonstrates how larger sample sizes and a larger number of samples will lead to a spread more similar to a normal distribution.

[@chopper84a]

I heard about the elegance of math: think I just got it!

[@shareef3743]

damn this is actually really cool

[@andthatsallll]

I believe the symbol for mean is a U looking symbol... The one you did was for standard deviation ... either way you helped me out thanks

[@lorassa727]

THANK YOU!! Your videos are amazing, such an amazing and intelligent man!

[@EclipZeMuzik]

awesome video!!

[@SandroAndrade]

I'd like to really thank you for your videos. All of them with great didatics and only now made some aspects of statistics very clear for me :)

[@richenjoshi7054]

Thank god you mentioned "frequency distribution".

[@floh223]

The video is simply beyond cool. Thank you so much sir.

[@chaitanyatuckley4666]

Beautiful. Made my day.

[@69erthx1138]

The tails of the bell curve should represent low frequency mean samples, like those lacking sufficient permutation, e.g. [1,1,1,1]. Is this a correct assertion?

[@kthwkr]

A resistor. Contains lots of electrons. Thermal energy is causing the electrons to bounce around. Some bounce big. Some bounce small. The amount of bounce creates a voltage. At any one instance there is a total voltage from all the bouncing electrons that produces some peak total voltage as all those voltages add together. You could divide that total voltage by the number of electrons for an average but why bother since that divisor is a constant because the number of the electrons doesn't change. In the example in the video he divides by four. But it's not necessary to divide since every average in his example has the same divisor, four. So the divisor just becomes a scaling factor and the scale does not matter in the CLT. It's the total that matters. So the total voltage from all those electrons over and over is going to change around and be slightly different. AND!! that total voltage will be within the bounds of a gaussian distribution because of the CTL. And that is why thermal noise from a resistor has a Gaussian distribution. The probability of the distribution voltage from individual electrons can be most any kind of probability function. But the many TOTALs of those voltages will be gaussian, even though they may have individually been created within some probability distribution based in quantum physics probably beyond our comprehension. So it is not necessary to know the probability distribution of the voltages from a single electron because what matters is that the total from many electrons will be gaussian. And finally, if you take any probability distribution and convolute it with itself over and over, the result will approach gaussian. Or if you take any waveform and convolute it with itself over and over the result will be gaussian.

[@quinnculver]

Very nice explanation. Hats off.

[@AbirChoudhury101]

Thanks for the explanation!

[@guptaaniket274ag]

Thankyu so so much!!!!!

[@danweasly]

Thank you very much for this very clear explanation!

[@rachitadehury2829]

Thank u so very much u helped me a lot

[@karafofubuntu]

Thank you so much for a clear explanation.

[@quazinuzhat1515]

you nailed it. thanx a lot

[@BurkeyAcademy]

If you don't take samples from the distribution (of the random variable) and average, then you are not talking about the central limit theorem. He could have done this more simply, but you can't avoid samples. I demonstrate it by sampling phone numbers from the phone book-- the averages (sample means) have a bell-shaped distribution. The random variables are digits, averaging samples of 7 digits make the distribution. Actually, summing does it, not averaging. But you add before you average.

[@999tktktktk]

it's really helpful

[@shahronak47]

Great explanation!!

[@flyingspur930]

you are much better than my professor