About sample mean and the mean of various samples

Question

What I would like to know it is the precise use of both statistics -- the sample mean and the mean of various samples. I'm getting a bit confused with the notation in my textbook and sometimes I cannot make a clear distinction between both statistics. For instance, when we talk about "Xi" (i=1,2...) aleatory variables from one population that approaches a normal distribution, I can assume we are talking about various samples from one population that approaches a normal distribution? I would be glad if someone could give me a clear difference in the use of both statistics (it would be nice a distinction of the variance in both approaches too) and its properties.

Thanks o/

Answer 1

If you have a random sample of size n from a single population, it can be viewed as a set of n independent identically distributed random variables. The mean of that sample will then under certain conditions such as the existence of a moment of the distribution slightly above 2 have an approximate normal distribution (central limit theorem). If mean of sample and sample mean are both used to refer to a sample from one population they mean the same thing. If by the mean of various samples you are taking samples from more than one population then that is something altogether different and the central limit theorem would not necessarily apply in that case.