Proof that the Sample Variance is an Unbiased Estimator of the Population Variance

A proof that the sample variance (with n-1 in the denominator) is an unbiased estimator of the population variance.

[saaqibz]

Great video for a difficult concept... There really should be more likes on this.

[annabethhild]

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!

[poopoo]

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.

[excelskillstopaythebillspa]

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!

[willhancock]

Finally!!! A proof versus explaining, “Obviously then, you divide by the Degrees of Freedom.”

[Justjoshingyou]

This is the best (read: most useful) proof I've come across. Thank you!

[ktursts]

how did u derive eqns for E( x^2) and E(xbar^2) ??

[MorriganSlayde]

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.

[MrGamer]

Cleanest, simplest and most importantly, rigorous proof why we divide by (n-1) and not n. Thank you for this video!

[gjsnuggle]

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
.

[muhannedalsaif]

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!

[ordiv]

Why do we not use |Xi - x̄ | instead of (Xi - x̄ )² ?

[Dekike]

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.

[leecherlarry]

so in this video you use capital S, capital X, and in the degrees of freedom video you use lower S and lower x. why ?

[ricardotalavera]

please do you have demo por var(s^2)=2sigma^4/(n-1)

[الحافظالصغير-هر]

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.

[luisschmidt]

Exactly the example I was looking for. I am reviewing statistics after years away from university. Thanks a lot, mister.

[clancym]

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.

[梦醒红楼]

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 YouTubers like u . god bless u! hope u produce more great stuff!!

[claudiamesaaparicio]

bravo!
but i have doubt in other exercise can you help me...i really appreciate it...