Proof that the Sample Variance is an Unbiased Estimator of the Population Variance
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- เผยแพร่เมื่อ 5 ก.ย. 2024
- A proof that the sample variance (with n-1 in the denominator) is an unbiased estimator of the population variance.
In this proof I use the fact that the sampling distribution of the sample mean has a mean of mu and a variance of sigma^2/n. If you need that to be shown as well, I show that in this video: • Deriving the Mean and ... .
In 2016, after graduating 2 years ago, I still watch your video to review some fundamental ideas in statistics. It does help me a lot. Appreciate your dedication in making these clips.
Thanks for your kind words. It's nice to be reminded every now and then that my videos still make a difference. It's been a long day for me, and your comment came at just the right time. Thanks again.
me too man! graduated few years back but gotta relearn these theory since I'm going back to school
I am taking my first econometrics course, and my professor just did a review of prob & stats. He did not take the time to go through the steps, so I was beginning to feel anxious about moving foward. This video helped me better understand the concepts he just went over. Thank you!
Great video! - the only place I have found online that intuitively explains why we divide by (n-1) instead of n. Lots of articles just leave it at either "...because there is (n-1) degrees of freedom." or "...because there are (n-1) independent variables". This video goes deeper but still keeps it intuitive. Thank you!
You are very welcome. Thanks for the compliment!
I guess it was more mathematical than intuitive. I didn't get the intuition yet.
Agreed, this video provides no intuition.
Jup, this makes it clear.
This is the best (read: most useful) proof I've come across. Thank you!
indeed
I covered this proof when I was back at school, and I was looking for a reference to refresh my memory. I spent over an hour googling trying to remember the details of the proof, but all pages I ran into are ambiguous with no clear/consistent notations
this 6 minutes videos are concise and clear, and saved me spending more time!
Thank you!
Great video for a difficult concept... There really should be more likes on this.
Cleanest, simplest and most importantly, rigorous proof why we divide by (n-1) and not n. Thank you for this video!
You sir rock!! I used this to prove something related, that the MLE of the sample variance is an unbiased estimator of the population variance when the population mean is known
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Did you make a video on it?
thank you for this video. i'm reading the casella and berger book right now, and they do a proof similar to this, but they take very large leaps between each step of their proofs.
having it shown in this way was very helpful.
+clancym1 You are very welcome. I'm glad you found it helpful!
Sorry, but can you say the book's name and its edition?
Congratulations on your videos. I've downloaded all your videos and I'm sawing them to learn about statistics. I'm doing my PhD and I found out your videos the most didactic ones. You make formulas and "conceptual" statistics easy to understand. For example, regarding "degrees of freedom", I was watching several videos to understand why we divided by the degrees of freedom, and after watching other tutorials I found out that you had a video for that, and when I've seen your video I've completely understood this concept, something that I could not with other videos. That's awesome. Your way of explaining is clear and pure. If you do not mind, just one recommendation: You could organize all your videos in playlists so that we can simply start to understand conceptually how the statistics are organized. For instance, I've organized your videos in folders like "Basics of probability", "Probability distributions", "Inference statistics", etc. But if you directly organize your videos as you think they should be, I think it will help us with only a first look to understand how statistics work.
This is the only TH-cam video on explaining why n-1 is needed and makes sense. Congrats and thank you
Exactly the example I was looking for. I am reviewing statistics after years away from university. Thanks a lot, mister.
You are very welcome!
Looked for this proof a couple times, but this is by far the best resource, thanks!
You're welcome. I like this one too!
One of the best explanations for why n-1 is used for sample variance. Thank you so much!
what a relief. I have been looking for proof that does not skip steps. This is a straight forward proof! thanks!
You are very welcome!
Really nice work Prof Balka. I appreciate the care and effort that has gone into this and all your videos. Thank you for opening the door to understanding in this way.
This video just clarified what I had been confused about for a long time. Thank you very much sir.
thank u for ur efforts, great video, great tutorial!!. what a sad thing that in my country schools aren't doing their job, instead of cultivating interest, they only make math seem tedious, and they get nicely paid for doing this. I found math actually interesting many years after graduation and ur videos explain things crystal clear, and u do it for free! thank god my Englisch is good and thank God for having TH-camrs like u . god bless u! hope u produce more great stuff!!
🤣
I really appreciate the clarity of this video! Well done!!
same feeling! better than videos ive watched previously!
this video was extremely helpful to me. i have no idea how i would have figured it out without it. It is the best video on youtube teaching this topic.
I have been searching for a video explaining this and clicked on it because I saw my initials lol. This was an awesome vid and it really gave me what I was looking for and i am not being biased here.
This video was great. In fact all of your videos that i have watched are brilliant.
All of your videos are amazing. They are very helpful with my mathematics class. I am grateful for your help!
my lecturer did this in 5 steps in the lecture notes, thanks for actually teaching me it
Well, a very underrated statistics youtube channel !
Indeed :)
This video is superb. It clears my long standing doubt. Thank you very much.
u just become my favorite youtuber. thank you!
I'm glad to be of help!
That is a proof that I was looking for a long time! Thanks a lot!
A clear and straightforward explanation!
Thanks!
THIS IS MAGIC
thanks for such a good and clear explanation
I don't think you could have explained this any better. Nice job!
You are very welcome! Thanks for the kind words!
Thank you so much for this explanation. The formula is a rule of thumb but it is hard to find the explanation of it. Your video is just perfect.
This is gold. ACTUAL explanations of this are like rocking-horse poop. Thank you.
You are welcome.
Sending you lots of hugs, this saved me ❤❤❤❤😭😭
The expectation for me to understand this video is the sum of the each time I re-watching it with the expectation to be able to understand it, minus the time that with expectation I had that I need to watch it again, plus the new expectation that hoping I will finally understand it after already watching it one more time, divided by my negative expectation of giving up the fact that I cannot understand it but need to re-watch it one more time again... :/
For anyone else like me who was confused as to why, at 3:50 or so, Sum(Xbar) becomes nXbar, whereas in the combined term, Sum(2XiXbar) becomes 2XbarSum(Xi) instead of 2nXbarSum(Xi), I think I figured it out:
In the combined term Sum(2XiXbar), the sum function is applying to Xi, so the constants can be factored out like any other addition -- i.e. 2x+2x+2x=2(x+x+x) -- whereas in the term Sum(Xbar), there is no Xi for the sum function to apply to, so the Xbar is itself being summed n times -- i.e. x+x+x=3x. I'm not 100% sure this is right, so if you know better, confirm or correct me as needed :), but I think this is what is going on.
I do have a naaaagging statistics question; expectations and assumptions. I know we take a lot of things as a given in statistics and one tutor said it nicely when I asked why "Because a lot of mathematicians worked it out a long time ago so we don't have to."
But I do wonder...how do we trust those base assumptions? How can we as plebians do the mathematics on those base assumptions? Are we even intelligent enough to do so?
I find myself asking "Why" a lot. That's WHY I took statistics...and so of course every time an assumption is made...I want to know why.
You remind me that there are teachers out there that I can fully understand the first time through..... thank you
+northcarolinaname You are very welcome!
this is a must have video in statistics. This video gets all ideas in statistics together.
Thanks for this video. Such a complicated topic is explained in such an easy manner. Hats off to you🙇♂️
I'm glad to be of help!
Finally!!! A proof versus explaining, “Obviously then, you divide by the Degrees of Freedom.”
This explanation was so helpful! Thank you so much!!
You are very welcome!
Excellent explanation and walk-through. Great content!
Great video. Helped me understand the concept.
Thank you!
Thank you so much.. I could not understand this in class but you made it so clear !
Great! I'm glad you found this helpful! All the best.
Clear as daylight. Thank you sir.
Only you made me understood. Thank you very much!!!!
sometimes hand-wavy explanations don't really convince me. Thanks for this
how did u derive eqns for E( x^2) and E(xbar^2) ??
You do a lot of reference to previous videos, please mention the name or provide a link in the description.
It is too haphazard to go to all videos and then try to find what was being referred to!
The explanation and structure of videos is great and well thought of, kudos!
Thank you so much for your videos :)
You are very welcome!
Best Explanation Yet!! Thanks!
Much Thanks from Ethiopia. It was helpful.
You might, quite simply be awesome!
I do my best. I'll let others decide if that results in awesomeness :)
It is one of those videos where you wish that TH-cam had donate button. Crisp and to the point
That was magic! An unbiased estimator🤯
So natural and elegant.
+杨博文 Thanks!
such a good video....... thanks for helping me clean an important concept!!!!
Your videos are amazing.
Insanely well explained
Truly useful video. Let me share with you some of my thoughts. I am wondering about the invention of the sample variance formula. who invented it first? Is he Bessel or not? How he got the concept of (n-1)? and although there are many different approaches to prove it, How that man proved it? which method he used? You may say: "what are these silly questions"? asking about history not about statistics!!! Anyway, these were some flies in my mind I liked to share them with you.
One more thing, (n-1) is always defined and explained as (the degrees of freedom) and most of people when they explain it they try to explain why it is called degrees of freedom and why we subtract only one not more ... because we loose one degree of freedom when we calculated the mean ..... My question is what the relation between their philosophical talk and the mathematical proof ???
Sorry for all this headache. Thank you for reading my comment.
Great presentation of the proof, many thanks.
BLESS YOU you beautiful human thank you so much
beautifully done! thanks!
Dear jb, thanks for your awesome videos! I would be lost without your videos in my classes :)
I have a quick question about the relationship established in minute 3:56. Why can we take 2x̄ in front of the summation (i.e. have 2x̄ * ∑xi) but in the next term have n * x̄^2 from ∑x̄^2. Why is the first not 2*n*x̄ * ∑xi? The sum of a constant (here 2x̄) isn't the constant if we sum over more than one term but rather the constant * n. Where am I missing something?
Thanks for clarifying & keep up the great work!
- Anka
Looking forward to this clarification aswell. Thank you very much for your vids JB
please do you have demo por var(s^2)=2sigma^4/(n-1)
THANKS A LOT!!
This was extremely useful and clear :)
+mai ahmed You are very welcome!
simply brilliant
I’m glad I found this video
You sir, are the man!! Great explanation!
Thanks Matthew!
Thank you so much maaaan!
You are amazing! Thank you!!
This video was a life saver.
This is just great
Thanks Daniel!
This was so helpful, thanks a lot!
You are very welcome!
THANK YOU
You are very welcome!
bravo!
but i have doubt in other exercise can you help me...i really appreciate it...
0:50 I wonder if the first of the two relationships actually defines the expectation operator E. It is the thing that arithmetically characterises a collection by an homogenous substitute. And the second can be derived can't it?
At 2:15 you say [E(Xbar)]² equals μ² because you expect the sample mean to equal the population mean. So we're talking formally not about the mean of the sample but the mean of the means of all possible samples weighted by their respective probabilities?
Awesome video! Thank you!
I was able to understand it. Thanks.
This is very clear and helpful. Thanks alot!
You saved my midtrem :D!, thank you
Very useful! Thanks!
You are very welcome!
Great video! I have a question though-- Why are we taking the expected value of the equation?
I'm showing that, on average, the sample variance equals the population variance. In other words, S^2 is an unbiased estimator of sigma^2. Unbiasedness is a good property for an estimator to have.
@@jbstatistics Thanks for responding! I really appreciated the depth of your video and your response!
great explanation sir!
Bravo. Beautiful video
Hello, I have a little confusion about X bar. As you presented in the video, X bar is a constant that can be factored out from the expectation. Doesn't this lead to E(X bar) = X bar which is obviously false cause E(X bar) = miu?
X bar is a constant *with respect to the summation*, not the expectation. X bar came outside of the summation, not the expectation.
We are summing over the n sample values, and at each of these n values, the sample mean is the same. But here we are viewing X bar, like S^2, as a random variable that only takes on a value once we get the sample.
@@jbstatistics Thank you for the reply! I think that clears me up. I have also done some research myself and I came across someone claiming that the sample statistics have the so called 'duality' of being a RV and a constant. Thus, one can do algebras depending on specific contexts. Is this equivalent to your explanation? I'm a bit lack of confidence on this because the one who claimed that doesn't put any reference. ; )
Clearest explanation I''ve found yet
+MarcoGorelli Thanks!
This is awesome. Thank you so much!
+Hankun Luo (Eric) You are very welcome Eric!
Thanks a lot Sir. You explained it in a very simple way.:):):)
Why do we use n-1 for a sample variance that does NOT act as an estimator for a "corresponding" population variance?
hi there, you are awsome. One problem 1:49 have Ex^2 at the top, while 5:14 have E sub i X ^ 2... good to be consistent as you always try to be... glad to clarify
Every step of the logic is crystal clear, BUT I CAN'T INTUITIVELY SEE WHY?!
This is beautiful my friend
Wow wow wow....
Now I got rid of this thorny term (n-1) in the denominator 😃
at 5:05
E(X1 ^2) = sigma^2+miu^2... why we can equate the random variable x1 to the population variance (sigma ^2) and population mean (miu^2)?
thank you, very elegant!
***** You are very welcome!
Thanks so much !!
cleared my doubts
You are very welcome Ashish. I'm glad to be of help!
can u explain me how to prove ..the expected value of the sample mean equals the true mean (μ) of the
population?
I have a video deriving the mean and variance of the sample mean at th-cam.com/video/7mYDHbrLEQo/w-d-xo.html. Cheers.
Great explaination
Thanks!