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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 Watch the next lesson: https://www.khanacademy.org/math/probability/statistics-inferential/sampling_distribution/v/sampling-distribution-of-the-sample-mean?utm_source=YT&utm_medium=Desc&utm_campaign=ProbabilityandStatistics Missed the previous lesson? https://www.khanacademy.org/math/probability/statistics-inferential/normal_distribution/v/ck12-org-more-empirical-rule-and-z-score-practice?utm_source=YT&utm_medium=Desc&utm_campaign=ProbabilityandStatistics 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! About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content. For free. For everyone. Forever. #YouCanLearnAnything Subscribe to KhanAcademy’s Probability and Statistics channel: https://www.youtube.com/channel/UCRXuOXLW3LcQLWvxbZiIZ0w?sub_confirmation=1 Subscribe to KhanAcademy: https://www.youtube.com/subscription_center?add_user=khanacademy
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Text Comments (210)
Nikki B. (10 days ago)
why can't we take the 2 and 5 as samples?
Göktuğ Güvercin (29 days ago)
You can use the word "observations" for the elements in a sample.
Carmen Rae (1 month ago)
Stats sucks
Rish Kum (1 month ago)
What just happened at 0:37
chemics bear (1 month ago)
You are wrong that you put the mean at 2.75 the mean should be at 3.5
Saheen jon (1 month ago)
Niveda Sivaprakasam (1 month ago)
If the sample size is infinity, how can we compute mean for that set?
donuts5 (2 months ago)
but the sample mean should be the same as the population mean which is 3.5. so the peak of the frequency distributions should be at 3.5 not 2.75, which is close to where Sal drew it
Kushagra Kumar (2 months ago)
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!
you guys are shaping history. thank you.
임상일 (5 months ago)
Your video is easy to understand. Thank you^^
Atharva #breakthrough (6 months ago)
but i have heard that once you get a 2.75 it is impossible to get exactly the same value again. then how can we get a bell curve?
John Landon Miller (6 months ago)
Too much droning goddamit
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.
kinsa khan (7 months ago)
Where can I get the proof to this theorem ???
Roger Syversen (9 months ago)
he is sampling by Benford's law
Hassan adamu (10 months ago)
An insurance company has 25000 customers if the annual claim of the company is a random variable with mean 320 and standard deviation 540. approximate the probability that the total annual claim of the company exceeds 8.3million naira
Jin Tao (10 months ago)
I don't think that if the sample goes infinity,then the mean will be very close to 2.75 as he assumed in the video. I think it will be 3.3333…for 1*2/6+3*1/6+4*1/6+6*2/6=10/3.Anyone agrees with me?
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?
HeyImRod (10 months ago)
Firas Kais (11 months ago)
Did you mean to right the symbol of the mean instead of variance in the center of graph toward the beginning?
La Ri (11 months ago)
I nearly cried because there's no single thing that I understand about central limit theorem.But you sir made it so easy for me.Thank you 😭
kkrose (1 year ago)
that's super cool
Hazem Maadawi (1 year ago)
John Mare (1 year ago)
what is p0hat? my professor keeps using it and i havent seen it in the other videos either, please, anyone explain this
Suzanna Reinhardt (1 year ago)
It's the empirical probability. So basically it's how many successes you have in your experiment over your total number of attempts
Just Mimi (1 year ago)
cramming for my stats exam tomorrow
Joe Knight (1 year ago)
what a good looking 4
UNH Psychology (1 year ago)
This is painful
hovenmoz (1 year ago)
what if your crazy original distribution was not symmetric. Would your repeated sampling averages still result in a normal distribution?
Laubotje (1 year ago)
Centril li.imit theory is big If tru.......
William JSS (1 year ago)
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.
Kristian R. Brasel (1 year ago)
if your sample size was infinite wouldn't you get the same mean every time?
Pharaoh on LFS (1 year ago)
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
Gihan Panditha (1 year ago)
TheImaxify (1 year ago)
You never mentioned that n (sample size) should go to infinity before this happens! please correct it.
kushagra deep (1 year ago)
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?
Uranus (1 year ago)
Fantastic explanation. A shame that most teachers are not educated enough to be able to understand and explain things like this to their students.
Victor Serra (1 year ago)
He gets 5 samples and then averages them. What is the size of the original samples? 5 as well?
MD Abir Choudhury (1 year ago)
Thanks for the explanation!
Jonathan Cavazos (1 year ago)
Show me the Data!
AbsorbIt (1 year ago)
Is that a Greek 4? lol
Adam Madam (1 year ago)
centreal limit theorem for dummies huh? I sure am a dummy so no problem with that for sure.
Sarnen (1 year ago)
This is a boring, rambling, over-explanation that never seemed to get to the point until the last minute. I was better off google'ing the definition than listening to his longwinded anecdotes. GET. TO. THE. POINT. I'm not here for a comedian or to listen to a verbal flood. The reason people are here is because they're having a hard time learning with their peers. Meaning they need the bottom line, faster, with less distractions from concepts, anecdotes, and words that aren't directly and immediately related to the topic.
Mrs. N J (1 year ago)
Hi do you do tutoring offline?
Errin Bass (1 year ago)
Great video!
Megan Taylor (1 year ago)
How would you calculate the mean and standard of the sample distribution that would result from take n number of samples from a distribution?
GirlGirlicious (1 year ago)
You, sir, are The Real MVP!
Justine Marie (1 year ago)
I swear to god he keeps saying meme and not mean. The internet is ruining me XD
Ramachandra jr (1 year ago)
Does that mean normally distributed data is a characteristic of not the population distribution but the act of drawing a simple random sample itself?
Junisha Kaul (1 year ago)
The man is so not right about the concept. For the central limit theory to apply n >30. He constructed a normally distributed curve using a sample size of 4, i.e n=4. Are the other viewers delusional ?? I think the guy might have been high on acid.
Junisha Kaul (1 year ago)
Man there's an error. You cant have more than 256 samples. Do the math.
Ramachandra jr (1 year ago)
@Junisha Sampling is different from possible combinations. We draw a sample here, that means we are going to roll a die and note down the result. The same way we note down the result of 10000 such samples. Sal here is assuming that he'd get 4 with some blah blah probability and 5 with some... probability(just assumptions).
Junisha Kaul (1 year ago)
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.
Ramachandra jr (1 year ago)
And why is that?
Roberto Martini (1 year ago)
i6mi6 (2 years ago)
Drinking game: drink whenever you hear the word "sample"
Skye Paul (6 months ago)
I'm trying to study here shhh
Sam Dave Pollard (6 months ago)
Are you intoxicating that I'm insinuated?
nelly aviles (8 months ago)
lets just kill braincells before the exam..
Roger Syversen (9 months ago)
Jonathan Lunenfeld (1 year ago)
i6mi6 I might need to play that game to get over my probability score... D;
Mark Rudis (2 years ago)
in one word, gazilion!
Sera Chrysanthemum (2 years ago)
Heck yeah, this is a great motivating video... gives an outline of the idea and why it's so cool and important!
Obinna Daniel (2 years ago)
I mean this is brilliant! Got me thinking and understanding deeply.
Henna George (2 years ago)
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.
brenda pham (2 years ago)
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?
notthatbl (2 years ago)
He says that with an infinite sample size, it approaches a normal distribution, but shouldn't it just be a straight line at the mean? (albeit this is a normal distribution with a standard deviation of 0)
Steve Buscemi (11 months ago)
notthatbl what he means is not the size of each sample (if you had one sample of an infinite size, yeah, it'd be one precise point), but if you had an infinite number of samples (infinite samples with only 4 in each sample)
Kristian R. Brasel (1 year ago)
I had the same question, I think what he is trying to say is that as the sample size grows it gets closer to a normal distribution. However there must be a point of diminishing return, because once the sample size reaches the number of events you will not get an approximation but the actual mean, which would be the same every time. Another interpretation could be if each sample was of increasing sample size until it reached the number of events. That would likely give you a normal distribution.
Junisha Kaul (1 year ago)
If the SD is 0 there wouldn't be a normal distribution.
张梓径 (2 years ago)
Simple is beautiful
Rachita Dehury (2 years ago)
Thank u so very much u helped me a lot
Koen Vos (2 years ago)
Best explanation out there. Thanks, Sal!
craft with NIKKI (9 months ago)
It's only explanation.... I need it's proof, but explanation is well defined.
Love Live Life (2 years ago)
Quazi Nuzhat (2 years ago)
you nailed it. thanx a lot
esmeralda4884 (2 years ago)
You explain this better than a textbook. You are a great!
Vishnu Vardhan (2 years ago)
Sal.... Your my rockstar!
Salvador Bueno (1 year ago)
Your welcome
Matthew wroblewski (2 years ago)
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.
baldy hardnut (6 months ago)
smart guy is an understatement lol
Samad Hashmi (2 years ago)
I love statistics!
Shareef (2 years ago)
damn this is actually really cool
James (2 years ago)
Yeah fuck this I'm going to bed
Q Q (1 year ago)
If its a sign wouldn't that mean it is the comment you needed to see? Would imply you should work your ass off so you're not cramming last day...
Renee Knezovich (1 year ago)
This is not the comment I needed to see at 11:14pm with my exam in two days (not tomorrow, but the day after) hahahaha XD Perhaps its a sign lol
Don The Best (3 years ago)
What happens to the standart deviation if we work on means of sample instead of means of one large sample ? Someooone? :)
Dont you want to make a video about Dunnett test, Bonnferroni-Holm es correction etc.?
Asther Phoenix (2 years ago)
check out their channel . they have it
Fadi Hasi (3 years ago)
you are amazing , thank you , not only for this video , but for all your videos that i have been using for 3 years :)
Tilak Chand Dhital (3 years ago)
U r helping me a lot :D
Megan Maloney (3 years ago)
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.
Khamis Buol (3 years ago)
Kieran Sibley (3 years ago)
cherrychapstickgurl (3 years ago)
this helped me with psychology! thank u sal
Eugene deschamps (1 year ago)
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
cherrychapstickgurl (3 years ago)
+General Dash that's right! you need math in psychology too
General Dash (3 years ago)
+cherrychapstickgurl wut?lol
Joe Dunbar (3 years ago)
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
Oisin Hughes (3 years ago)
thanks very much for putting this together  - your a role model in helping folks understand the world around them.
Sera Chrysanthemum (2 years ago)
There are many people like Sal on YouTube, though he's one of the OG's, which is something that's friggin awesome about this world!
atiger92 (3 years ago)
At 7:24 the narrator says that a sample size of 4 and a sample size of 20 would have the same mean.  That is not correct!  They might be close, but it highly unlikely that the mean of any sample sizes would have exactly the same mean.  Khan Academy...you posted something misleading.  I'd suggest vetting the narrators more closely and make sure they don't tarnish your reputation.
Bobby Bullock (3 years ago)
+atiger92 I think you misunderstood the comment - he meant that the means of the sampling distributions are not a function of the number of observations in a sample with the same probability distribution (i.e., equal chances of rolling a 1, 3, 4, 6).  In statistics it is very common to refer to the idea of an "infinite sample" to make points and in that case, taking an infinite number of observations from a sample size of 4 and a sample size of 20 would indeed produce the exact same mean (again given the same chances as before).  Of course he understands that if he rolled his dice 4 times and took the mean of that, then rolled the dice 20 times and took the mean of those rolls, that they would not likely be equal.  However, if he took the mean of an infinite number of 4-rolls, it would be equal to the mean of an infinite number of 20-rolls.
Amol Buch (3 years ago)
You are just awesome..!
TheSnow720 (3 years ago)
I watched this video and didn't understand a thing at all. Then I went to class and read my textbook and it all became crystal clear!
ronak shah (3 years ago)
Great explanation!!
ZbladeVX (3 years ago)
great vid!
latesq1 (3 years ago)
What does the central limit theorem tell us about samples from a bi-modal distribution ?
tiivc (3 years ago)
The whole point of the central limit theorem is that it applies to all distributions at all times as long as they have a well-defined mean and variance.
TheSevenofMine (3 years ago)
I like it that it's called Khan Academy. KHAAAAAAN!!
KWS ATL (3 years ago)
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!!
Hannibal EnemyofRome (3 years ago)
Very instructive video!
Gopika Rajanikanth (4 years ago)
Honestly, this almost 10 minute video helped me understand something we were learning in class for like 2 weeks! Thank you so much!
crni195 (1 year ago)
lol, thats not bad.. we did this in 1 lesson and they expect you to know this on oral exam and im not even studying mathematics, im in civil engeneering
Nils Berg (4 years ago)
You're saving my ass! Thank you!
Nithin Reddy (4 years ago)
follow jbstatistics
felixthainsomniac (4 years ago)
thank you
Eric Antolovic (4 years ago)
Khan Academy you are the bomb dig.
danielle4823 (4 years ago)
Its unclear if we need to increase the sample size of Si or if we need to increase the number of samples S that we take from the population
Dustin Watson (4 years ago)
You need to increase the sample size of Si. The rule of thumb is if you took a sample size of >=40  and plotted their means then you would get a normal dist curve.  (30 can also work if the population variance is known, but its usually not, hence why you are using the CLT)
Musliston (4 years ago)
Great! That was very clear ^^
rafaelaprende (4 years ago)
Thank you! :-)
YUNOS RIDUWAN (4 years ago)
cunt academy LOL
eric (4 years ago)
Glorified Truth (4 years ago)
WOW!!!!  That's hilarious!  Please, do you have any other gems to share???
Trainwhrek (4 years ago)
SNORE FEST loljk good lesson -.-
Richard Guzman (4 years ago)
Fronte999 (5 years ago)
I don't understand. That sampling to determine the mean tends to cluster in a pattern that looks like a bell is not a surprise. It doesn't tell us that a '2' or a '5' are impossible or not or what the first chart looked like, except as to it's mathematical mean. So what was all of the stuff at the beginning of the video about?  All of the elaboration that it is not a regular pattern or that '2' and '5' are possible or not possible. Every set of data has a mean, and taking potshots at it would of course look like a bell curve. No matter what the data looks like. So what was the point of telling us what it looked liked?
Dustin Watson (4 years ago)
Regression Analysis (and statisitics) is fundamental to any scientific experiment as well. 
Marcos Ponce (4 years ago)
When you have a bell curve, a normal distribution, you are then capable of meeting assumptions in another statistical concept known as regression analysis. Regression analysis, to me, is absolutely amazing. It is widely used in economics but also in a plethora of other fields as well.
MrHav1k (5 years ago)
Well explained, but the arithmetic for the actual numbers is a bitch. 

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