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

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Introduction to the central limit theorem and the sampling distribution of the mean
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Probability and statistics on Khan Academy: We dare you to go through a day in which you never consider or use probability. Did you check the weather forecast? Busted! Did you decide to go through the drive through lane vs walk in? Busted again! We are constantly creating hypotheses, making predictions, testing, and analyzing. Our lives are full of probabilities! Statistics is related to probability because much of the data we use when determining probable outcomes comes from our understanding of statistics. In these tutorials, we will cover a range of topics, some which include: independent events, dependent probability, combinatorics, hypothesis testing, descriptive statistics, random variables, probability distributions, regression, and inferential statistics. So buckle up and hop on for a wild ride. We bet you're going to be challenged AND love it!
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[@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.

[@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 ?

[@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

[@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.

[@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

[@esmeralda4884]

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

[@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

[@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!

[@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!

[@hexiaohuang857]

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

[@hou950]

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

[@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.

[@bluesky-mi2sx]

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

[@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.

[@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!

[@serachrysanthemum9687]

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

[@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!!

[@obinnadaniel2001]

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

[@aprilhicks3474]

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

[@iraklismoutidis204]

Thank you for this explanation. I would like to ask you a question. If we got a distribution (probably heavy tailed) and take many samples with different size (for example the first sample will have 30 samples, the second 40, the third 34, the fourth 51 and so on) will then the distribution of those samples be normal?

[@diencai1812]

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

[@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!!!!

[@usharagunath23448]

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

[@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

[@TheSevenofMine]

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

[@ytkv]

Best explanation out there. Thanks, Sal!

[@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?

[@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.

[@pratibhas2468]

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

[@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!

[@Abubakar-ht5ee]

you guys are shaping history. thank you.

[@rodrigomorgado5524]

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

[@18gshock]

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

[@LeahInTheRye]

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

[@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

[@jamesrobsonza7752]

Much better explanation than my textbook, thank you so much

[@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 :)

[@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.

[@J0EH3AD]

Thank you for very nice and instructive Probability series. I would like to see mathematical analysis of random walks - will you create it?

[@jordantan5023]

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

[@lorrainepinon]

helped me to pass my statistic class. thank you!

[@floh223]

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

[@lorassa727]

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

[@That1KoreanGuy]

This channel is a life saver!

[@rideshareescapades4281]

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

[@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.

[@davidmbeckmann]

Very instructive video!

[@user-mn1od5il7w]

Your video is easy to understand. Thank you^^

[@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.

[@winterfell14]

Oh man, that is SO logical!

[@misterwessels]

Thank you very much for this very clear explanation!

[@goktugguvercin8069]

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

[@nasratullah5596]

This is extremely well explained ! Thank you teacher.

[@LYLxd]

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

[@ahmethamdicelik1277]

have you passed?

[@AbirChoudhury101]

Thanks for the explanation!

[@kushagra452]

As the sample size (i.e "n") increases the sample distribution starts looking more and more like the normal distribution. Does it have to do with also the number of samples taken? If I were to take 10000 or 100 sample of the same sample size (n) then would both have the normal distribution? Thanks!

[@ShroukAbdulshafy]

Very helpful! thank you so much, Sir!

[@richaunfacey5447]

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

[@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!!!!!!!!!!!

[@karafofubuntu]

Thank you so much for a clear explanation.

[@yitianxiao6650]

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

[@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

[@ayonrab]

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

[@firaskais]

Did you mean to right the symbol of the mean instead of variance in the center of graph toward the beginning?

[@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...

[@rachitadehury2829]

Thank u so very much u helped me a lot

[@chopper84a]

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

[@rjeption]

Khan is getting me through college

[@vishnuvardhan2608]

Sal.... Your my rockstar!

[@versus13]

Your welcome

[@LFSPharaoh]

I just felt like this central limit theorem thing is super important to learn along with standard deviation... and the 80-20 rule :) I was thinking, is this whole "normal distribution" thing really as simple as this? - The data keeps coming from the same "odds" or the same source, aka the original sample you drew. Because the odds haven't changed, and you're originating the data from the same set that hasn't been modified half way through the experiment, isn't it sort of expected that you should end up with a normal distribution? Like when you play the same arcade game, or you're at the circus, no matter who plays and for how many times, the odds are the same? Just seems like a complicated way of saying "the odds haven't changed", so "on average" you'll end up with "X" at a certain rate or whatever. Unless I'm missing something lol

[@lerneninverschiedenenforme7513]

First explanation, that I saw, that gets the dimensions of the variables right. Interesting though is, that the CLT is differently defined in Japan.... In Japan every Sample S_i must have a mean of u.

[@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.

[@quazinuzhat1515]

you nailed it. thanx a lot

[@pritivalecha3147]

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

[@shahronak47]

Great explanation!!

[@deepkushagra]

thanks for your existence i have a question if the sample size is six, but instead of numbers we have categorical variables. E.g, S¹=[pc, laptop, pc, mobile, mobile, mobile]. now we can't take the mean. how can central limit theorem help us now?

[@shareef3743]

damn this is actually really cool

[@choppertalk8291]

Very clear. Thank you

[@chaitanyatuckley4666]

Beautiful. Made my day.

[@SaikatChakrabarti]

great xplanations sir...thanks so much...

[@madorkas5400]

would there ever be an instance where there is skew on the "normal distribution curve" that CLT produces?

[@ylast3756]

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

[@user-uj8up5jd6j]

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.

[@GirlGirlicious]

You, sir, are The Real MVP!

[@samadhashmi]

I love statistics!

[@lovelivelife8950]

THANK YOU!!!

[@tilakchanddhital137]

U r helping me a lot :D

[@quinnculver]

Very nice explanation. Hats off.

[@ramachandrajr9565]

Does that mean normally distributed data is a characteristic of not the population distribution but the act of drawing a simple random sample itself?

[@Mlledalia91]

thank u, so helpfull!

[@brendapham1325]

how do you denote the number of times you have samples? example: so you say sample size n=4, but 100 of those samples, how do we denote that?

[@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

[@999tktktktk]

it's really helpful

[@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.

[@amolbuch8713]

You are just awesome..!

[@Zurh1994]

Central limit theorem = mind blown

[@ZbladeVX]

great vid!