Confusion about the sample mean and random variables As I understand the sample mean you just add a bunch of random variables that constitute a sample from their common distribution and divide by the number of those same random variables. 
When I apply it in actual problems I get specific numbers from the population and take their mean. This confuses me as I thought that when you add random variables it represents adding all the possible outcomes of those random variables.
So my understanding of adding random variables is that if you had random variables $X$ and $Y$, that could take on the values $1$ and $2$, $X + Y$ represents $1+1$, $1+2$, $2+1$, $2+2$. 
Thus I feel that the definition of a sample mean clashes with it's application where you just take specific values from the population and add them up.
I hope that makes sense.
 A: Suppose $X=\left.\begin{cases} 1 & \text{with probability }2/5, \\ 2 & \text{with probability }3/5, \end{cases}\right\}$ and $X_1,X_2$ are independent copies of $X$, i.e. we choose two individuals independently from this population, each time having a $2/5$ chance that the value of the variable is $1$ and a $3/5$ chance that it's $2$.
The population mean is $\operatorname{E}(X) = \dfrac 2 5 \cdot1+\dfrac 3 5\cdot 2 = 1.6$.  That is not a random variable.
The sample mean is
$$
\left.\begin{cases}
(1 + 1)/2 = 1 & \\
(1 + 2)/2 = 1.5 & \\
(2 + 1)/2 = 1.5 & \\
(2 + 2)/2 = 2 &  \end{cases}\right\}
= \begin{cases}
1 & \text{with probability }\frac 2 5\cdot\frac 2 5 = 0.16, \\[8pt]
1.5 & \text{with probability }\frac25\cdot\frac35+\frac35\cdot\frac25 = 0.48, \\[8pt]
2 & \text{with probability }\frac35\cdot\frac 35 = 0.36. 
\end{cases}
$$
The sample mean is a random variable.
The expected value of the sample mean is
$$
(1\times0.16) + (1.5\times0.48)+(2\times0.36) = 1.6.
$$
This is not a random variable.
The expected value of the sample mean is equal to the population mean.  That is expressed by saying the sample mean is an unbiased estimator of the population mean.
The variance of the sample mean is smaller than the variance of the population whenever the sample size is more than $1$, and decreases as the sample size grows.
Edit: Fixed typo
