supposedSuppose we draw a sample x$x$ from a population X$X$, this sample has n$n$ random variables (x1, x2, x3... xn)$(x_1, x_2, x_3... x_n)$, the sample mean is x̄$\bar{x}$, and it's variance is v(x)$v(x)$, the whole population is X$X$, has a mean is μ$\mu$ , variance V(X) = σ$V(X) = \sigma^2$,
say this specific sample mean x̄$\bar{x}$ follows a samplesampling distribution W$W$, the variance of this x̄$\bar{x}$ is V(x̄)$V(\bar{x})$.
and the variance of this specific sample is V(x)$V(x)$,
are V(x)$V(x)$ and V(x̄)$V(\bar{x})$ exactly the same?
we know that V(x̄) = σ*σ /n$V(\bar{x}) = \sigma^2 /n$
but is V(x)$V(x)$ also equal to σ*σ /n????? $\sigma^2 /n$?
please notice V(X)$V(X)$ and V(x)$V(x)$ and V(x̄)$V(\bar{x})$ has three total different meaning:
V(X)$V(X)$ is the real variance of the whole population, V(x)$V(x)$ is the variance of one specific sample. V(x̄)$V(\bar{x})$ is the sample mean, when we change the sample, this sample mean changes too, and this changed sample mean follows a specific distribution W
