What is an unbiased estimator? Proof sample mean is unbiased and why we divide by n-1 for sample var
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- เผยแพร่เมื่อ 20 ก.ค. 2024
- In this video I discuss the basic idea behind unbiased estimators and provide the proof that the sample mean is an unbiased estimator. Also, I show a proof for a sample standard variance estimator that uses n in the denominator, and show that it is a biased estimator, therefore we use n-1 in the denominator to obtain an unbiased estimator for the population variance.
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This stuff was giving me nightmares 😫 but you've simplified it in the best way possible. Thank you 🇰🇪
Same here
I spent last 2 days trying to wrap around my head estimators and what it means to be unbiased. You explained me in minutes what I could not understand for days. I dont know how to thank you. You are the best. Thanks for the beautiful video
👍🏻🔥
It is a very good video that simply describes some jargon which usually is ignored in the literature.
Thank you.
Wow. Incredible. The best proof of sample variance on TH-cam. Thank you!
All of your videos are amazing!! As an Msc student I am checking out your videos for catch up and brushing up my informations. I am very happy to watch all of your videos they are clear and answering needs. Thanks!!
Am so happy I understood the concept. I found the finer details of the concept I was looking for.Thank you
Amazing! I was chasing to understand the meaning of biased and unbiased, but this video explains in a very simple way and with great explanation too. Thank you so much for the details.
Yay! Happy to hear you found this video to be helpful :-)
I'm in a statistic / probability class this semester, which makes you, my new best friend 😁.
Thank you for the great explanation 👍👏
You are the Best. You definitely deserve a ton more views and subscribers.
How amazingly you have explained this complicated thing is just beyond articulation ! Thank you so much
Let's not get carried away
@@hrob6381 No, this was where I got stuck but this video cleared my doubts. So I am not getting carried away 😊
@@SSCthanos beyond articulation? Really?
Yes ofcourse, for days i was finding the explanation for the concept but just one day before my exam I encountered this video. Thus beyond articulation.
@@SSCthanos fair enough. Although you seem to be articulating it fairly well.
You are my personal hero for the month and probably the following months too cos I'm gonna start studying everything from your videos now
Happy to hear you found my videos to be helpful :-)
Your channel is criminally underrated! Most videos on this topic will simply "proof" this empirically or talk about degrees of freedom without connecting it to anything. This is the first in dozens of videos I found that actually provides mathematical proof! Your explanation was excellent! I got to say at this point it's not super intuitive for me why it's -1 (and not any other number to make the Variance larger), but I can appreciate how the math supports it.
I just saw that you have tons of other videos on statistics and, if they are anything like this one, I know I will probably end up watching them and learning so much (=
Thank you for putting in so much time and energy! And for sharing your amazing Knowledge!
WOW! now we're taking! this is the best, literally the best! academic, clear, perfect! thank you so so much! maybe I put too many exclamation marks, but I mean it! THANKYOU THANKYOU THANKYOU
Such a good video, clear all my confusion about this topic, wish my professor can be half good as you.
This was very helpful and easy to understand. Thankyou so much
Thanks a lot for the video! Very clear and precise!
Nice one, thanks for making such nice video on statistics
This is meaningful material. A book I read on the same topic was a eureka moment for me. "Game Theory and the Pursuit of Algorithmic Fairness" by Jack Frostwell
just subscribed your channel and recommends everyone reading this...
Amazing explanation! Thank you so much!
Hii Michelle ,thankyou for your wonderful and complete explanation
BEST VIDEO EVER! THANK U SO MUCH!
Beautiful. Amazing. I was waiting to see this kind of an explanation. Thanks
Glad it was helpful!
Very clear explanation. Thank you!!
Clear explanation, good work.
Thanks for breaking it down. and i mean the simple things like the meaning of an estimator. you the best ma'am.
Awesome! I'm happy to hear that you found this video to be helpful :-)
Thank you.Excellent teaching.
Finally, someone made it! Thanks!
So well explained, thanks a ton
This was so helpful! Thank you so much.
Nice explanation ❤
Hi Michelle, thanks for your work! But I still have some qustions. At 13:44, you substituted E(xi2) with sigma2 and miu2. I don't think you can do that. Because the xi in var(xi) = E(xi2)-(E(xbar))2 is the value from the whole population, but xi in equation (∑E(xi2)-nE(xbar2)/n) is the value taken from the sample. So, the sigma in equation E(xi2)=miu+sigma2 means the sigma of our sample, rather than the whole population.
Mostly its given that E(xi) = myu
That means for any Xi regardless of where it is from its E(Xi) is myu
i think, many people explain this by interchanging X for both. It will be better if they use different variable for xi for population and xi for the sample.
Perfect video! Thanks
Wow,, Thank you for the wonderful video.
Great, clear explanation! One small thing: On the computation done in color green and then color blue (around 12:44 and 13:44) I think you failed to carry down the square of mu. Meaning your final derivation was sigma^2 + mu where it should have been sigma^2 + mu^2
I think she addressed it at 13:56
Yeah, I noticed it about 30 seconds later and corrected it in the video. Sorry for any confusion for that mistake!
Thank you you are so helpful!
Great explanation! Thank you.
Very clear, thank you so much !
Great tutorial , thank you
Wow thank you so much for your explanation
Im really so glad that you use different colors for deriving something out of the main problem ❤
It helps us to understand better
💓Again Thank you so much😄
Thanks for clarification! One question here. Why var(X bar) equals to sigma^2 / n?
I have a video discussing this question here: th-cam.com/video/XymFs3eLDpQ/w-d-xo.htmlsi=uWfZpTGzePAd22ju
At around 5:11, after pulling the 1/n and the Sigma out, you said that E(xi) = Miu (the true mean of the population). But xi as you said in the beginning was an observation that we chose randomly, ie. it's a specific value (like a number), and so shoudn't the expected value of a number be itself (E(xi) = xi)? How could it be the mean of the population? Could someone help me to understand that part?
Good question. Thanks for this post. The mean of the random variable xi is always mu, regardless of i. This is an assumption for the proof. If I were to observe several random values of x (obtain a sample), those values would be coming from the same population where the mean of x values is mu.
Michelle is correct. It's true that E(xi)=xi for all numbers xi, but your mistake (and it's a common one) is that xi is not a number. It is a random variable that will result in some number after a chance process. The mean of the random variable xi is the population mean mu.
@@MattSmith-il4tc Do you mean that if we treat the xi in E(xi) is a random variable, that means that single xi varies and the expected value of this single sample is the population mean mu?
Thankyou so much...this cleared all my doubts.
Great..very helpful explain
how did you express Var(x bar) in terms of expected value of (x bar square) and (expected value of x bar) square .
Where can I read more theory about it.
You are amazing....thanks for this video
this is so easy to understand now. ty
Very Wonderful 😢🎉❤
God bless you soo much.
Great explanation!
Clear really. Thanks!
You're welcome! :-)
This was so COOOOOL !!!
Thank you so much...
I am speaking from Bangladesh
Thanks a lot. Love from Bangladesh. You have a great voice and accent too.
Amazing explanation❤
amazing ma'am loved it
You demonstrate that Xbar is an unbiased estimator of mu without assuming that Xbar follows a normal distribution centered around mu with variance equal to sigma_square/n. However, to show that S_square is a biased estimator of the variance sigma_square, you do make this assumption since you substitute var(Xbar) with sigma_square/n (at 13:02). Would it be possible to do the demonstration without this assumption/substitution?
Careful. Notice that I do not assume that the data is normally distributed in this video. I do not need the normality assumption for either proof in this video. Rather I use the definition of variance to find the variance of X-bar near minute 13.
@@Stats4Everyone TYVM. From your answer and en.wikipedia.org/wiki/Standard_deviation#Standard_deviation_of_the_mean, I finally got it.
@@bertrandduguesclin826 Thank you so much for that link! I was confused in the same place...
if we are given a pdf of 4 values of x with their probabilities in terms of theta, then we find an estimator for the mean theta-hat and then we find the mean square error in terms of theta (should it be in terms of theta?), how can we find if it it mean square consistent. I am unsure because n=4 for my questions so I can't see how it makes sense to consider the limit as n goes to infinity. Please could someone shed some light. Thank you
thank you so much!
This is very powerful 👏 🙌 👌💪
Thank you!!
hey I have a question when u showed us how the expected value of sigma squared is biased estimator is called mathematical prove in econometrics right ?
Thank you so much😊
what the heck, this is diamond.
Thanks from Taiwan.
Thank you very much
More than excellent,
Finally got it ❤️
At 13:00, you equate var ( x bar) with square of sigma divided by n. I cannot get my head around this step. How is variance of sample means same as population mean divided by sample size?
Great question! Thank you Kurien Abraham for this post. Here is a video I made to try to address this question: th-cam.com/video/XymFs3eLDpQ/w-d-xo.html
Please let me know if you have any follow-up questions :-)
7:06 you said the expression is divided by n-1 to get the unbiased estimator. Will that work for any other number?
Thank you
Please more videos on Statistical inferences
Thanks for the video this helped me a lot. But in my course ists the other way around when you have 1/n its an unibiased estimator and when you have 1/n-1 its biased so now im lost again😂
In your course, if the estimator for sample variance? For example, if you are estimating a mean, the unbiased estimator would have n in the denomiator... Though for sample variance, the proof that I provide in this video is correct. Here is another source that might be helpful: en.wikipedia.org/wiki/Bias_of_an_estimator#:~:text=Sample%20variance,-Main%20article%3A%20Sample&text=Dividing%20instead%20by%20n%20%E2%88%92%201,results%20in%20a%20biased%20estimator.
good work
perfect!
Awesome...
any proof for SDOM? I don't get it why doe have root(N) as denominator in the normal distribution SDOM
This video may be helpful: th-cam.com/video/XymFs3eLDpQ/w-d-xo.html
Thank you❤
Good
I dont understand why E(xi) = u at the first place? I mean Capital Xi denotes the units of population lets says it has N units. And small xi denotes units of sample , it has n units. So E(Xi) should be equal to u (population mean) but how we can say E(xi)=u ? Since xi is a just a small subset of population units that is Xi , by defination of sample.
Help me.
Thanks for this comment! A sample is a subset of the population. Sorry for any confusing regarding notation... in this video, I do not use Capital Xi and lowercase xi, because I am referring to the same objects. For example, let us think about a small population. Suppose my population is the following set:
{3, 5, 6, 2, 1, 7, 8, 10}
the population average, mu, is 5.25. Also, the expected value for any member of this set is 5.25.
mu = E(Xi) = 5.25
Now, suppose I were to take a random sample of 3 objects from this population:
{5, 1, 8}
Here, the sample mean, Xbar, is 4.67. This sample mean is an estimate of the population mean. Though, the population mean is not changed by us taking this sample. It still holds true that mu = E(Xi) = 5.25.
very helpful and clear
Awesome! Happy to hear that this video was helpful!
❤❤❤❤
can you talk about various other estimators !! the best thing is that you are fluent in the subject .. clap .. clap ..
Legendary
Mam, I have a doubt at 12:18 , why we are taking sigma² for var(xi) instead of S²?
I think is because sigma square itself is the symbol of variance and in the video, she was just explaining the definition of variance in order to do continue the calculations in the previous steps.
That sigma square is just a symbol for the concept of “variance”.
wow this video is very understanderble
Just thx😊
at 12:44 mark should it not be sigma squared + mu squared = E(x sub i squared)
I noticed this mistake about 30 seconds later and corrected it in the video. Sorry for any confusion!!
I should have finished the video but I just did now. Thanks for clearing that up for me
@5:21 why is the mean of all the Xi is equal to the same Mu?
Good question. Thanks for the comment. All the Xi come from the same population, therefore they all have the same population mean, mu.
Where you from? I love your accent!
In general when we use n-1 is biasdness or
Sorry...i meant n and not n-1
@@perischerono987Do you have an example in mind? We use n in the denominator for x-bar so that it is an unbiased estimator for the population mean, and use n-1 in the denominator of s^2 so that it is an unbiased estimator for the population variance. Every estimator needs it own proof for unbiasedness... In other words, in general, we need to show that
E(estimator) = population parameter
godammit perfect !
Like from India
why doesn't 2x bar not have an n? 9:27
2xbar is a constant since there is no "i" subscript, therefore:
sum (2xbar*xi) = 2xbar*sum(xi)
For example, here are some numbers we can plug in to show that the above statement is true:
suppose n is 3, and x1 = 2, x2 = 4, and x3 = 9, therefore xbar = 5
sum (2xbar*xi) = 2*5*2 + 2*5*4 + 2*5*9 = 10*2 + 10*4 + 10*9 = 10*(2+4+9) = 10*15 = 150
2xbar*sum(xi) = 2*5*(2+4+9) = 10*15 = 150
I hope this makes sense and is helpful
Ok
Mam, I am from India 🇮🇳, thanks mam for this video, really help in my exam ......😊
Hence proved
ok, I got something. you don't get this reading Ronald e Walpole's book of the shelf.
amazing
Great explanation!