Sampling Distributions: Deriving the Mean and Variance of the Sample Mean

I derive the mean and variance of the sampling distribution of the sample mean.

[jbstatistics]

Thanks for the input. I'm well aware that this one is far from exciting, not something people have a great deal of trouble with, and unlikely to be a hot search topic. I put it up for completeness, as I'm trying to get video support together for all sections of my intro stats notes. Getting lots of hits is way down on my priority list. That said, I appreciate the input as far as hot topics go, as I will take this kind of thing into account in the future. Cheers.

[emptyxnes]

Besides being great at explaining, thank you for sounding excited in teaching. Not sure if you have coffee prior to recording, but your voice really helps. Again thank you for uploading.

[jbstatistics]

Yes, X1 is a random variable representing the first draw from the distribution. X1 through Xn are all independent random variables with the same probability distribution.

[jingyiwang]

Thank you so much for this detailed explanation about the derivation of the mean and variance of the sample mean. I finally understand it!

[bebla]

been looking for that last variance formula explanation for a while, found it thank you very much sir

[MrVasilist]

THANK YOU!!! I was looking for over an hour on web with no analytical proof. All i found it "you do that, then some maths that i dont tell, intervene and then it is proven" Thank you.

[jbstatistics]

X1 is a random variable representing a randomly selected value from the population. This population has a mean of mu and variance of sigma^2, so those are the mean and variance of X1.

[anthonyng]

Thank you for the video, great refresher to someone who is picking up statistics after awhile

[PandyV]

Thank you! I always keep forgetting this. There´s also another formula that has an (n-1) in the denominator. I think it´s also a sampling variance.

[afmmarques]

Thanks, I got this idea from watching the other video of poutine fast food which made sense for me, but when I think of people I still get somewhat confused at the idea that a randomly selected person has expected value and variance equal to the population. Thanks!

And a simple suggestion, you should put this video in the sampling distribution playlist! ;)

[kabascoolr]

Wdf? I have always been confused about the 1/sqrt(n) for the sample standard deviation. It's the first time it's ever made sense. Thanks

[akku]

Its like being taught statistics by Seth Rogen

[Leonardo-jvls]

I never understood one simple thing. Is the "n" the number of elements inside a single sample, or is it the total number of samples?

[valeriereid]

Thank you so very much for clearing up my confusion. Your video certainly helps

[patrikbak]

I'm wondering, if X1, ..., Xn are independent, does it mean that a sample may contain repeated values, i.e. anytime we talk about a random sample of size n, it might actually be one value n-times?

[sevenkulqwerty]

It is ok that E[X] = µ but why should E[X1] be equal to µ? Is X1 a random variable?

[erazn]

Wonderful wonderful video!! Really explained things well

[timetravelerqc]

Why the expectation of a single sample is not the sample value of that single sample? If we have only one sample, we can only know that the average value is its own value.

[jshin]

always thanks. have a nice sunday or saturday or monday or (My country is on sunday now. )

[afmmarques]

Hi, thanks for these videos, they're really great! If you could just explain something to me, why is the expected value and the variance of a sample like X1 equal to the expected value and variance of the population? I'm not getting this concept very clearly