In order to learn about the distribution of a statistic in the long-run, a common practice is to construct the sampling distribution of that statistic across a number of samples, each of a certain sample size $n$. One example of this is constructing a sampling distribution of sample means by constructing a density histogram of the sample means considered across a number of random samples of size $n$.
What if, after one collects the sample mean from each of a number of random samples of size $n$, one takes the mean of these sample means ? Is this a common practice to investigate the overall population mean ? If not, why not ?
Thanks!
Best Answer
The mean of sample means would simply be the mean of all of those samples. Sure, the random variable given by the mean of means would have lower variance than just one sample mean, but you could have gotten the same result just by increasing the size of your sample in the first place.
For example, if we took the average of 3 samples each of size k:
$\frac{(x_1+x_2+...+x_k)/k + (x_{k+1}+...+x_{2k})/k + (x_{2k+1}+...+x_{3k})/k}{3}$
= $\frac{(x_1+x_2+...+x_k) + (x_{k+1}+...+x_{2k}) + (x_{2k+1}+...+x_{3k})}{3k}$
= $\frac{(x_1+x_2+...+x_{3k})}{3k}$
it would have been the same as taking a sample of size 3k in the first place.