Proof that the Sample Variance is an Unbiased Estimator of the Population Variance
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- Опубликовано: 22 авг 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!
This is the only RUclips video on explaining why n-1 is needed and makes sense. Congrats and thank you
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.
Looked for this proof a couple times, but this is by far the best resource, thanks!
You're welcome. I like this one too!
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?
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?
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!
One of the best explanations for why n-1 is used for sample variance. Thank you so much!
Exactly the example I was looking for. I am reviewing statistics after years away from university. Thanks a lot, mister.
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.
This video was great. In fact all of your videos that i have watched are brilliant.
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.
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 RUclipsrs like u . god bless u! hope u produce more great stuff!!
🤣
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 really appreciate the clarity of this video! Well done!!
same feeling! better than videos ive watched previously!
Well, a very underrated statistics youtube channel !
Indeed :)
my lecturer did this in 5 steps in the lecture notes, thanks for actually teaching me it
That is a proof that I was looking for a long time! Thanks a lot!
All of your videos are amazing. They are very helpful with my mathematics class. I am grateful for your help!
u just become my favorite youtuber. thank you!
I'm glad to be of help!
This video is superb. It clears my long standing doubt. Thank you very much.
This is gold. ACTUAL explanations of this are like rocking-horse poop. Thank you.
You are welcome.
thanks for such a good and clear explanation
A clear and straightforward explanation!
Thanks!
Sending you lots of hugs, this saved me ❤❤❤❤😭😭
THIS IS MAGIC
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.
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!
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.
Best Explanation Yet!! Thanks!
I don't think you could have explained this any better. Nice job!
You are very welcome! Thanks for the kind words!
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!
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... :/
Clear as daylight. Thank you sir.
this is a must have video in statistics. This video gets all ideas in statistics together.
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.
Thank you so much for your videos :)
You are very welcome!
Only you made me understood. Thank you very much!!!!
That was magic! An unbiased estimator🤯
Finally!!! A proof versus explaining, “Obviously then, you divide by the Degrees of Freedom.”
Excellent explanation and walk-through. Great content!
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.
Great video. Helped me understand the concept.
Thank you!
This explanation was so helpful! Thank you so much!!
You are very welcome!
such a good video....... thanks for helping me clean an important concept!!!!
Much Thanks from Ethiopia. It was helpful.
This is very clear and helpful. Thanks alot!
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.
Your videos are amazing.
So natural and elegant.
+杨博文 Thanks!
You saved my midtrem :D!, thank you
You might, quite simply be awesome!
I do my best. I'll let others decide if that results in awesomeness :)
sometimes hand-wavy explanations don't really convince me. Thanks for this
Insanely well explained
This video was a life saver.
It is one of those videos where you wish that RUclips had donate button. Crisp and to the point
Great presentation of the proof, many thanks.
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!
beautifully done! thanks!
You sir, are the man!! Great explanation!
Thanks Matthew!
Thank you so much maaaan!
This was so helpful, thanks a lot!
You are very welcome!
simply brilliant
I’m glad I found this video
Awesome video! Thank you!
great explanation sir!
THANKS A LOT!!
This was extremely useful and clear :)
+mai ahmed You are very welcome!
This is just great
Thanks Daniel!
I was able to understand it. Thanks.
Very useful! Thanks!
You are very welcome!
how did u derive eqns for E( x^2) and E(xbar^2) ??
THANK YOU
You are very welcome!
Thanks a lot Sir. You explained it in a very simple way.:):):)
This is awesome. Thank you so much!
+Hankun Luo (Eric) You are very welcome Eric!
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
This is beautiful my friend
Clearest explanation I''ve found yet
+MarcoGorelli Thanks!
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
thank you, very elegant!
***** You are very welcome!
Wow wow wow....
Now I got rid of this thorny term (n-1) in the denominator 😃
god i wish my prof teaches this much clear
BLESS YOU you beautiful human thank you so much
Great explaination
Thanks!
Every step of the logic is crystal clear, BUT I CAN'T INTUITIVELY SEE WHY?!
thank youu so much for this video
Actual goat
Elegant proof you got right here. It's sad to see those shitty udacity videos on top of the search bar instead of this.
Thanks! My videos come up near the top for a number of search keyword combinations, but yes, some are buried.
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 ruclips.net/video/7mYDHbrLEQo/видео.html. Cheers.
thanks a lot! very useful
Awsome video!
Thank you so much
Great video, easy to understand. Thank you, my friend. :D
-- Great Video
You are very welcome, and thanks for the kind words!
Thank you Sir , Thank youuu!