Great job, jbstatistics! On-line education is the best. Those expensive brick & mortar universities are a rip-off with ego-centered professors. This has been repeatedly attested to by so many frustrated students in their online responses to these UA-cam videos. It is time for a real paradigm shit in educational institutions, but I am certain the long ago established institutions are doing everything possible to slow this process. Keep the videos coming jbstatistics!!
I think it's biased since you before class and you when you were watching this videos is you in " different state of mind" Maybe after you being confused in the class and it's about to be a quiz or an exam before you hit up on the video, the ability of your concentration is heighten therefore your answer isn't enough to reject the null !!!!
I don't know about ur professor but usually in universities they do derivation for all kind of distributions (for example this chi-squared distribution function is derived from gamma distribution) but here in this video the youtuber just stated what is what and just showed the distribution function he did not derive it or prove this distribution. soo.. if indeed ur professor did derive this distribution ,then it is ur mistake of not being able too understand
It is nearly critical to pace the video in such a way that the viewer has time to grasp what's being presented. Especially when studying scientific subjects with way to much information, without focusing on the students intuition. I found your video to be excellent. Thank you for clearing up my understanding of this distribution.
Fantastic video! Loved the graphic where you compared graphs of different Chi-square distributions with different degrees of freedom. Very insightful and so helpful!! Thank you!
This is a quite good video with one thing missing, the problem. Which problem does this distrubution help us to solve? Why do we need it? How and by whom was discovered in historical perspective?
This is a video introduction to the chi-square distribution. It's not a discussion of the extremely large number of practical scenarios in which this distribution comes into play; it is a brief introductory discussion of the distribution. I could discuss the chi-square distribution, its applications, its historical relevance, full derivations of how it arises mathematically, why the name, points of confusion, great moments in statistical history involving the chi-square distribution, etc., but that would be a *very* different sort of video and would be many hours long. This is a 5 minute introductory video on the basics of the chi-square probability distribution.
@@jbstatistics Understood. However, given your talent for making videos please maybe consider doing some expository content of that type, as it is noticeably missing from UA-cam. It seems to be a problem in statistics teaching generally. Every teacher just gives the cookbook "here's what the formula is, here is how to do X test." No explanation of how these mathematical objects were discovered and in what context, no derivation from first principles, no "why" things work the way they do. Even the textbooks I have don't explain where most of this stuff comes from.
what if the df = 1, can we interpret it as the distribution is skewed to the right? secondly, for mode you mentioned df atleast 2 may be to deduct it with - 2. so what if df is 1?
Thanks from a curious beginner... just wondering which software uses tables and which ones use algorithms to do the definite integrals of pdf (ie get the area under the curve)... also wondering about non-standard normal distributions... is there a pdf equation for chi-square that includes non-standard normal distributions, ie N(0, 2)... or N(3,4) or N(-3, -2)?... how about any other distribution? I am under the impression that chi-square is only for standard normal distributions.... is this true?
The difference of two independent normal variables itself has a normal distribution. Is it true that the difference between two independent chi-squared variables has a chi-squared distribution? Explain
Yes, in the sense that as the DF increase the chi-square distribution becomes closer to normal. Since the sum of k independent squared standard normal random variables has a chi-square distribution with k degrees of freedom, the CLT tells us that as the DF increase the chi-square distribution will become more normal.
Chi-squared has k degrees of freedom. What you're talking about is the distribution of the sample variance, which is a chi-squared distribution with n-1 degrees of freedom, where n is the sample size. Your confusion is somehow similar to the confusion between standard normal distribution and normal distribution. The former is a normal distribution where mean=0 and variance=1 (or std=1), and the latter is the general concept. Same here: chi-squared distribution is the general concept, and the distribution of the sample variance is one example. Hope that helps.
Just chi-square. The mode of the t distribution, for example, is 0 for any DF. The mode of the F distribution changes, depending on the DF, but is ~1 for larger DF.
That's the pdf of the chi-square distribution. This video doesn't involve a mathematical derivation of the distribution. The derivation is often covered in a first course in mathematical statistics, and while the derivation is not super complicated, it is far beyond the scope of this video.
The pdf does not have "infinite probability". The integral of the pdf from 0 to infinity is 1 (in other words, the area under the entire curve is 1). That doesn't stop the *height* of the curve tending toward infinity as x tends to 0. The height of the curve can't be negative, but there is no upper bound.
i meant 1 not 0 on the cdf sorry. Isn't the height of the curve the probability? If the height of the curve tends towards infinity as x tends to 0, and 0 is the minimum value... how is that integration including 0 not inf?
The height of the curve at any given point is *not* a probability. Areas under the curve are probabilities. We need the entire area to equal 1, but there's no upper bound on the height. The height can tend to infinity if, as is the case here, the area is still 1. If you're asking how it's possible for an area to be finite if the height tends to infinity, then I'm not going to get into that explanation right now. For a simple example, find the area under the curve f(x) = 1/2sqrt(x) between 0 and 1.
Other than the usual interpretation of a pdf? Values in a small interval near 0 are more likely to occur than values in an interval elsewhere that is of the same width. Are you looking for more than that?
Yes you're right of course. I meant more out of curiosity: is there an intuitive explanation for those shapes; is there a fundamental difference between df 2 and 3?
I don't have a great intuitive explanation for why there is the change in shape at DF = 2, but I'll give you a little motivation for it. First, it's not hard to show mathematically (by differentiating the pdf f(x)) that the pdf is strictly decreasing in x for DF 2. But without actually carrying that out differentiation, we could guess something like that would happen. A squared standard normal random variable has a chi-square distribution with 1 DF. The pdf of the SND is symmetric about 0 (with a peak at 0), so it stands to reason that the squared rv will have the peak of its pdf at 0. And we also know that the sum of k squared independent standard normal random variables has a chi-square distribution with k degrees of freedom. The central limit theorem tells us that the distribution of that sum will get closer to the normal distribution as the number of summed terms increases. So, the chi-square distributions becomes "more normal" as the DF increases. So, armed with our knowledge of the SND and CLT, we know going in that with 1 DF the max value of the chi-square pdf will occur at 0, then, for larger DF the pdf will be increasing to a maximum then decreasing. The fact that the change occurs at exactly k = 2 we can easily show mathematically, but I don't have an intuitive explanation for that being the precise point.
Hello, thank you for the video. Just yesterday I didn't understand or couldn't apply a khi2 test. Now it's getting better but i still don't understand the mechanics of it and especially that chi2 distribution. I can't explain it to myself and can't plot it either. I tried to plot the chi-squared distribution with one degree of freedom with the following Octave code as a base : ---------------------------------------------------------------------------------------------------------------------- x = [-3:0.1:3]; sigma = 1; mu = 0; fx = (1/sqrt(2*pi*sigma^2)*exp(-(x-mu).^2/(2*sigma^2))); plot(x, fx); ---------------------------------------------------------------------------------------------------------------------- this plots a gaussian distribution with mean=0 and variance=1. Now to "sqare" it, i tried the following : fx = (1/sqrt(2*pi*sigma^2)*exp(-(x-mu).^2/(2*sigma^2))).^2; % just squaring the formula of the gaussian and as i pretty much expetected, i just got the same curve/bell plot but with lower values (obviously for x^2 for 0
@@fishermanwithfishes2286 There's nothing stolen here. Degrees of freedom actually tells u the number of independent variables on which a state is dependent. So here the chi distribution is the sum of squares of k random variables that follow normal distribution. So it obviously depends on those k random variables which means the degree of freedom is k.
From what I understand, Greek speakers of the modern Greek language have very different ways of pronouncing some Greek letter names, when compared to the way we pronounce them in North America. In my neck of the woods, people would wonder what I was getting at if I were to pronounce pi as "pee", or mu as "me". My pronunciations might possibly offend the ear of a speaker of modern Greek, but they are standard in my circles. Cheers.
"i" pronounced "ay" is specific to the anglophone world though, in ancient greek it should probably be closer to khy or hhy but who cares, your explanation is so perfectly made anyway
the saddest thing is that I understood in 5 minutes what my professor has been trying to explain for forty five minutes. Thanks a lot!
You are very welcome!
Great job, jbstatistics! On-line education is the best. Those expensive brick & mortar universities are a rip-off with ego-centered professors. This has been repeatedly attested to by so many frustrated students in their online responses to these UA-cam videos. It is time for a real paradigm shit in educational institutions, but I am certain the long ago established institutions are doing everything possible to slow this process. Keep the videos coming jbstatistics!!
where did you take your college?
I think it's biased since you before class and you when you were watching this videos is you in " different state of mind"
Maybe after you being confused in the class and it's about to be a quiz or an exam before you hit up on the video, the ability of your concentration is heighten therefore your answer isn't enough to reject the null !!!!
I don't know about ur professor but usually in universities they do derivation for all kind of distributions (for example this chi-squared distribution function is derived from gamma distribution) but here in this video the youtuber just stated what is what and just showed the distribution function he did not derive it or prove this distribution. soo.. if indeed ur professor did derive this distribution ,then it is ur mistake of not being able too understand
It is nearly critical to pace the video in such a way that the viewer has time to grasp what's being presented. Especially when studying scientific subjects with way to much information, without focusing on the students intuition. I found your video to be excellent. Thank you for clearing up my understanding of this distribution.
I have watched lots of vedios to get the base of this topic but none of them were helpful ,thank you so much sir u have taught it just
Fantastic video! Loved the graphic where you compared graphs of different Chi-square distributions with different degrees of freedom. Very insightful and so helpful!! Thank you!
This is a quite good video with one thing missing, the problem. Which problem does this distrubution help us to solve? Why do we need it? How and by whom was discovered in historical perspective?
This is a video introduction to the chi-square distribution. It's not a discussion of the extremely large number of practical scenarios in which this distribution comes into play; it is a brief introductory discussion of the distribution. I could discuss the chi-square distribution, its applications, its historical relevance, full derivations of how it arises mathematically, why the name, points of confusion, great moments in statistical history involving the chi-square distribution, etc., but that would be a *very* different sort of video and would be many hours long. This is a 5 minute introductory video on the basics of the chi-square probability distribution.
@@jbstatistics Understood. However, given your talent for making videos please maybe consider doing some expository content of that type, as it is noticeably missing from UA-cam. It seems to be a problem in statistics teaching generally. Every teacher just gives the cookbook "here's what the formula is, here is how to do X test." No explanation of how these mathematical objects were discovered and in what context, no derivation from first principles, no "why" things work the way they do. Even the textbooks I have don't explain where most of this stuff comes from.
what do you mean by 'degrees of freedom',. you gotta explain that first I believe
Is it due to the Central Limit Theorem that the chi square dist appears to look like a normal distribution with larger degrees of freedom?
what if the df = 1, can we interpret it as the distribution is skewed to the right? secondly, for mode you mentioned df atleast 2 may be to deduct it with - 2. so what if df is 1?
Then the mode is 0
Thanks from a curious beginner... just wondering which software uses tables and which ones use algorithms to do the definite integrals of pdf (ie get the area under the curve)... also wondering about non-standard normal distributions... is there a pdf equation for chi-square that includes non-standard normal distributions, ie N(0, 2)... or N(3,4) or N(-3, -2)?... how about any other distribution? I am under the impression that chi-square is only for standard normal distributions.... is this true?
The difference of two independent normal variables itself has a normal distribution. Is it true that the difference between two independent chi-squared variables has a chi-squared distribution? Explain
As the degrees of freedom go up isn't that kind of showing you "the central limit theorem"?
Yes, in the sense that as the DF increase the chi-square distribution becomes closer to normal. Since the sum of k independent squared standard normal random variables has a chi-square distribution with k degrees of freedom, the CLT tells us that as the DF increase the chi-square distribution will become more normal.
For k=12,why the max pt located at 12-2=10? Is there any equation?
The mode of occurs at DF - 2, as long as DF >=2. This can be shown by taking the derivative of the pdf and setting it equal to 0.
Can you do one on the gamma distribution?
Then what will be the mode of the chi square distribution if mean is given
Doesn't the chi squared distribution have k-1 degrees of freedom?
no. When you are finding confidence interval of variance, it has k-1 degree of freedom.
Chi squared distribution has k degree of freedom.
Chi-squared has k degrees of freedom. What you're talking about is the distribution of the sample variance, which is a chi-squared distribution with n-1 degrees of freedom, where n is the sample size.
Your confusion is somehow similar to the confusion between standard normal distribution and normal distribution. The former is a normal distribution where mean=0 and variance=1 (or std=1), and the latter is the general concept. Same here: chi-squared distribution is the general concept, and the distribution of the sample variance is one example.
Hope that helps.
The mode of the distribution equals dof minus 2
Is this statement valid for all distributions or just chi-sq?
Just chi-square. The mode of the t distribution, for example, is 0 for any DF. The mode of the F distribution changes, depending on the DF, but is ~1 for larger DF.
Please do a video on Gamma distribution!
This tutorial video is so good
Thanks!
How does the equation in 1:32 suddenly come about?
That's the pdf of the chi-square distribution. This video doesn't involve a mathematical derivation of the distribution. The derivation is often covered in a first course in mathematical statistics, and while the derivation is not super complicated, it is far beyond the scope of this video.
Ha!! I agree with Houna Mao! This is an awesome video!! Thank you!
Does density function of Chi-square belongs to a well-known family of distribution ?
Yes, its a special case of the gamma function
how does a pdf have an infinite probability... isn't a CDF (-inf to inf) supposed to be 0?
The pdf does not have "infinite probability". The integral of the pdf from 0 to infinity is 1 (in other words, the area under the entire curve is 1). That doesn't stop the *height* of the curve tending toward infinity as x tends to 0. The height of the curve can't be negative, but there is no upper bound.
i meant 1 not 0 on the cdf sorry. Isn't the height of the curve the probability? If the height of the curve tends towards infinity as x tends to 0, and 0 is the minimum value... how is that integration including 0 not inf?
The height of the curve at any given point is *not* a probability. Areas under the curve are probabilities. We need the entire area to equal 1, but there's no upper bound on the height. The height can tend to infinity if, as is the case here, the area is still 1. If you're asking how it's possible for an area to be finite if the height tends to infinity, then I'm not going to get into that explanation right now. For a simple example, find the area under the curve f(x) = 1/2sqrt(x) between 0 and 1.
Thanks!
Thank you so much for the amazing explanation🤍
what is degree of freedom
sample size minus 1
Great video, thanks a lot.
Great job. It helped me a lot. Thank you!
Thanks for the video, very helpful.
Humanity thanks you for your contribution
Please make a video on Gamma distribution.
2:58 ... What's the interpretation of the pdf being greatest at zero for df of 1 and 2?
Other than the usual interpretation of a pdf? Values in a small interval near 0 are more likely to occur than values in an interval elsewhere that is of the same width. Are you looking for more than that?
Yes you're right of course. I meant more out of curiosity: is there an intuitive explanation for those shapes; is there a fundamental difference between df 2 and 3?
I don't have a great intuitive explanation for why there is the change in shape at DF = 2, but I'll give you a little motivation for it.
First, it's not hard to show mathematically (by differentiating the pdf f(x)) that the pdf is strictly decreasing in x for DF 2.
But without actually carrying that out differentiation, we could guess something like that would happen. A squared standard normal random variable has a chi-square distribution with 1 DF. The pdf of the SND is symmetric about 0 (with a peak at 0), so it stands to reason that the squared rv will have the peak of its pdf at 0. And we also know that the sum of k squared independent standard normal random variables has a chi-square distribution with k degrees of freedom. The central limit theorem tells us that the distribution of that sum will get closer to the normal distribution as the number of summed terms increases. So, the chi-square distributions becomes "more normal" as the DF increases. So, armed with our knowledge of the SND and CLT, we know going in that with 1 DF the max value of the chi-square pdf will occur at 0, then, for larger DF the pdf will be increasing to a maximum then decreasing. The fact that the change occurs at exactly k = 2 we can easily show mathematically, but I don't have an intuitive explanation for that being the precise point.
I think you've helped me with these degrees of freedom, thanks! I have no idea why everybody calls you all those nasty, nasty names.
Are you Brian Will ?
Thank you very much
Hello, thank you for the video. Just yesterday I didn't understand or couldn't apply a khi2 test.
Now it's getting better but i still don't understand the mechanics of it and especially that chi2 distribution. I can't explain it to myself and can't plot it either. I tried to plot the chi-squared distribution with one degree of freedom with the following Octave code as a base :
----------------------------------------------------------------------------------------------------------------------
x = [-3:0.1:3];
sigma = 1;
mu = 0;
fx = (1/sqrt(2*pi*sigma^2)*exp(-(x-mu).^2/(2*sigma^2)));
plot(x, fx);
----------------------------------------------------------------------------------------------------------------------
this plots a gaussian distribution with mean=0 and variance=1.
Now to "sqare" it, i tried the following :
fx = (1/sqrt(2*pi*sigma^2)*exp(-(x-mu).^2/(2*sigma^2))).^2; % just squaring the formula of the gaussian
and as i pretty much expetected, i just got the same curve/bell plot but with lower values (obviously for x^2 for 0
why they named it degree of freedom...whats so free in it?
i guess they steal this idea from kinematics
@@fishermanwithfishes2286 There's nothing stolen here.
Degrees of freedom actually tells u the number of independent variables on which a state is dependent. So here the chi distribution is the sum of squares of k random variables that follow normal distribution. So it obviously depends on those k random variables which means the degree of freedom is k.
Hey you're doing a great job thank you so much!
Excellent explanation Thanks a lot
You are welcome Akash! I'm glad you found it helpful.
very nice. very clear..
Dude you are amazing! keep the great work up :)
Bruh This is dope , Thank you!
Fantastic
Thank you
Superb😇
Very straightforward, thanks!
+Natalie Shen You're welcome Natalie!
could you tell me how to prove that Prob((X^2, with d.f. n)>1) is increasing in 'n', using the definition of chi-square?
brilliant!
Thank you so much!!
wow thank you for this : )
Helped a lot
🎉🎉🎉🎉🎉 fantastic
Nice one
Brilliant!
THANK YOU
Thanks
0:33 it is not jade , its pronounced "ZEE"
nice one
kobra-kaiiiii
YOU SHOULD MAKE A FAST VERSION :)
I do have a fast version for this one! This slower version might not be quite as exciting, but it is a little better :) Cheers.
play it in 2X speed, you get a fast version
어렵군요 하하
어쩌면 조금
Gamma,beta,lognormal, weibull
X = Hee not chi ,,, got ear cancer from that
From what I understand, Greek speakers of the modern Greek language have very different ways of pronouncing some Greek letter names, when compared to the way we pronounce them in North America. In my neck of the woods, people would wonder what I was getting at if I were to pronounce pi as "pee", or mu as "me". My pronunciations might possibly offend the ear of a speaker of modern Greek, but they are standard in my
circles. Cheers.
+jbstatistics you're right ... my comment was kind of mean sorry ... cheers for the good work
No worries. I understand that it might sound strange to your ear. All the best.
"i" pronounced "ay" is specific to the anglophone world though, in ancient greek it should probably be closer to khy or hhy but who cares, your explanation is so perfectly made anyway
MrPsilokomenos
Better be sorry. He's helping us
Thank you so much