Deriving the Mean and Variance of the Sample Mean

[suzyolds]

I'm confused about the definition of the Xi's. If the Xi's are independently drawn observations, how can each Xi have a variance? Isn't each Xi a single observation?

[thekittulegend]

You are a really good educator, Sir you helped me understand very clearly thank you 🙏

[tuankietnguyenthanh]

it's great, I've been kinda confused about this for few days and now finally understand it. Your effort is much appreciated!

[GoodLuckForever-wikb]

Thanks for explaining this concept in such a simple way.

[suleymansahal]

Thank you for these great videos. I understand the logic now but in the past something always confused me, maybe others too. I think the relation between Xbar r.v. and X1, X2, ..., Xn r.v.s is not clear. Once you form this relation, the rest is relatively easy. At first I always thought we should incorporate the number of trials in these equations. Later, I figured that out by myself, hopefully correctly. What i understood: Xbar is a new variable which has its own probability distribution and we do not know it. X1, X2 ... Xn (with capital Xs) also have their distributions and their distributions are identical. In each trial we get one observation for Xbar r.v. which is (x1, x2, ..., xn)/n. The number of trials makes the distribution more normal. Maybe that point needs a little deeper explanation.

[sazik]

Can you please help me understand why at 2:39 you say "When a random variable gets multiplied by a constant, its variance gets multiplied by the square of that constant". Can you show a quick proof of that?

[KrishanSingh-gzop]

At 00:27, do you mean that expected mean of X1 = Expected mean of X2 = expected mean of X3 mean of Xn= μ?
And variance of X1=Variance of X2....=variance of Xn= (σ)^2 ?

[গোলামমোস্তফা-শ৮থ]

All are good...But I want to know, what does varience of sample mean actually mean? Is there any real example? Like E(x bar) is= mu(pop. Mean)..we can prove it by example..

[jeffreylin]

@jbstatistics Can the samples of the sampling distribution overlap?

[manupandit]

What is the difference between sample variance which is S2 and the variance of the sample mean which is sigma squared / n

[kevinmorfol]

omg seeing the proofs makes me feel like i finally get it! thats crazy!

[গোলামমোস্তফা-শ৮থ]

If x1 is just a observation then how E(x1) is equal to population mean mu? Can you explain?

[yaweli]

What if we have a huge population and we have sampling distributions of means - say we have about 50 sample means. I guess the mean of the sample means will be the population mean. How will the variance of the sample means look like ? Also, I suspect it’s pop variance if the sample means are large enough.

[bmgri]

At t=2:40 WHY does the 1/n get squared. That's the part I can't understand. Can you please try to show me how to prove that part slowly using the definition of variance??

[MrKorkak]

these are the best videos about statistics and probability... please exponential and gamma dist. to continuos dist.

[gags-villsounds]

where can I find the proof for the third step 02:59 ?

[gatsu]

What would be the variance of the square of the sample mean?

[ivoriankoua]

So the Mean have a Mean ! what's the Meaning of this Mean ?

[syedahmedali]

no doubt you have such a great quality of teaching

[medicalstudentscorner]

Respected Sir,
At 3:18 I didn't get how the variance of the independent observations is equal to the variance of the population. I got mean is the same in both the cases but the variance i didn't get...
Could you please help me on that or am I missing something???
Thank you..