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.
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.
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.
...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.
@@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.
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!
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!
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.
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!
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
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!!!!!!!!!!!
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
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!
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!!!!
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.
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
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...
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.
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.
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.
"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.
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
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.
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?
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.
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?
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.
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
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.
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.
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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.
Honestly, this almost 10 minute video helped me understand something we were learning in class for like 2 weeks! Thank you so much!
that two weeks of study is the reason you can enjoy this clip so much.
@crni195 Like a true engineer you had to point out that you are one lol
Superb Khan sir, am very pleased to study statistics as I watched your 10 minutes videos
are you alive ?
Why can't teachers explain things this clearly, why do they have to act all scholarly?
because they don't understand it either!
did u watch the next few lessons?
because they are scholars, not pragmatist, like you !
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.
a true mastery means that one can teach it simply and clearly
You explain this better than a textbook. You are a great!
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.
cramming for my stats exam tomorrow
Cramming for my stats final today
same except in 20 minutes
Me rn
Not cramming. :P trying to learn it as much as possible.
@@CrunchyDark big flex
Fantastic explanation. A shame that most teachers are not educated enough to be able to understand and explain things like this to their students.
so true
...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.
@@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.
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!
Drinking game: drink whenever you hear the word "sample"
i6mi6 I might need to play that game to get over my probability score... D;
no
lets just kill braincells before the exam..
Are you intoxicating that I'm insinuated?
I'm trying to study here shhh
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!
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.
you are amazing , thank you , not only for this video , but for all your videos that i have been using for 3 years :)
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!
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!
This video was posted 10 Years ago and still so useful!
Such a crazy thing
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
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.
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!!!!!!!!!!!
You are helping me get through my graduate level quantitative analysis classes. Thank you so much ! =)
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
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!
It's been 13 years since upload and people like me are still using these videos... Great explanation!
Clear explanation & better
Nothing much to say how good you are, the video tells it all, keep up the good job!!!
Once again Khan Academy saved me from the state of I am not able to understand to how easy is this stuff. Thanks
You just made life so much more interesting. Love you Sal! Will donate soon.
thanks. This is way more straight forward than the aihl textbook
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!!
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!!!!
you guys are shaping history. thank you.
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.
You have a great channel. I am in a master's degree program, and I still use your site.
This channel is a life saver!
I like it that it's called Khan Academy. KHAAAAAAN!!
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
Thanks Sal !
I mean this is brilliant! Got me thinking and understanding deeply.
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...
A good way to explain CLT. From an unknown discrete distribution to converge to a normal distribution.
YOU ARE SO AMAZING. PLEASE KEEP DOING WHAT YOU'RE DOING.
I need to pass my exams...
have you passed?
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.
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.
thanksss, i loved your video, i was looking to prove CLT and it clarified niceee
I love statistics!
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.
You are changing the world!..... seriously!....
This made understanding this theorem a lot easier.
Also, that is a really bad 4.
may god bless you Sal!!
You are my guru
you are the voice in my head as i solve math
Very Good example ... I was having problems with figuring out how the individual mean element was obtained...
"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.
this man khan do anything
Very helpful! thank you so much, Sir!
I am a big fan of khan Academy ❤❤
You can say "4 observations in Sample ..." . Eases understanding
Heck yeah, this is a great motivating video... gives an outline of the idea and why it's so cool and important!
your explanation is more elegant than a 45 min lecture from a top40 US college.
you are much better than my professor
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
Thankyu so so much!!!!!
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.
God bless you ! I got this after 10 years...
Thank god you mentioned "frequency distribution".
Central limit theorem = mind blown
Best explanation out there. Thanks, Sal!
Much better explanation than my textbook, thank you so much
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?
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.
what is pdf
This was beautiful
Thank you 💚💚💚
You, sir, are The Real MVP!
great work..
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
Very clear. Thank you
helped me to pass my statistic class. thank you!
Your video is easy to understand. Thank you^^
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?
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.
Thanks! Got a test that includes this section next week and it had me stumped
damn this is actually really cool
You should become a professor
Beautiful. Made my day.
Oh man, that is SO logical!
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
this is gonna save my statistics grade
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.
awesome video!!
Thank you for your clear explanation. You are a world class educator!
you nailed it. thanx a lot
it's really helpful
You can use the word "observations" for the elements in a sample.
Thank u so very much u helped me a lot
Thanks for the explanation!
Very instructive video!
Very nice explanation. Hats off.
ty💙
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.
very nice, kudos to you
Thanks
I love you man
it's a lot of fun learning this stuff.
Thanks a lot
I heard about the elegance of math: think I just got it!
Very good video. Minus some points for not being Indian
thank you!