Interpreting the sample mean, $\bar y$

I have a hypothetical population that contains the values $2,3,4,5,6,7,8,9$. I have to draw a sample of size $4$ from that given population. So I have $\binom{8}{4}=70$ ways to draw the sample. I wrote down all possible samples and computed their sample mean. I got $70$ as the sample mean ($\bar y$). Then I made a frequency table of $\bar y$, showed $\bar y$ is an unbiased estimate of population mean, $\mu_Y$ and drew a histogram of $\bar y$.

The histogram shows that the distribution of $\bar y$ is a bell shaped curve. So at the end of this work, my interpretation of the whole task is:

Though the parent population follows uniform distribution, the sampling distribution of the sample mean, $\bar y$, tends to Normal distribution.

Is that all there is to interpreting the whole task? I want to interpret most appropriately so that I can get the whole picture.

• It's unclear what you're asking, or what you were asked to do. What was the assignment? Feb 28 '14 at 11:35
• @PeterFlom The assignment is Show all possible sample of size $4$ from a population of size $8$ and compute the sample mean of each possible sample and also draw the histogram of the sample mean. And I am asking for a clear interpretation of this assignment as I think i have lacking in my interpretation . Feb 28 '14 at 11:56
• If that's the assignment, I think you've done a good job Feb 28 '14 at 11:56
• This is correct. I wounder if the Central Limit of the sample mean distribution was invented like this. Feb 28 '14 at 12:23

Another thing you can do is see how the sampling distribution of the sample mean gets narrower as the number of data in your sample increases (although that does not seem to have been a part of your assignment). For example, if you sampled only $2$ data from that population, the highest sample mean you could get is $8.5$, and the lowest is $2.5$. Consider how the range of possible sample means changes as a function of sample size:
$$2\quad 3\quad 4\quad 5\quad 6\quad 7\quad 8\quad 9 \\ \underbrace{\ 2.5\quad\quad\quad\quad\quad\quad\quad\;\ 8.5} \\ \underbrace{3\quad\quad\quad\quad\quad\quad\quad 8} \\ \underbrace{3.5\quad\quad\quad\quad\;\ 7.5} \\ \underbrace{4\quad\quad\quad\quad 7} \\ \underbrace{4.5\quad\ \ \ 6.5} \\ \underbrace{5\quad 6} \\ \underbrace{5.5}_{\mu_Y}$$ When your sample is smaller, there are a wider range of possible estimates that you can get. As your sample gets larger, the range narrows. In the end, when you have the entire population, you necessarily get the population mean. (In fancy terms, the sample mean is a consistent estimator of the population mean.)
Now that you have got the distribution of $\bar y$, you can make some statistical inference based on it. For example, you can take a sample from a population that MAY or MAY NOT be the same as the initial population. Calculate the mean of this new sample, let's call it $\bar y_j$ and compare the "value" of $\bar y_j$ with the histogram of $\bar y$, you can actually have the feeling of how likely $\bar y_j$ is drawn from the same population as $\bar y.$