clarification on sample mean, population mean Suppose the population size is 10 and the independent measurements pertaining to a particular interest(for example number of siblings) are as following, 1,1,2,2,3,4,4,4,5,5,5. 
So the frequency is as following,
number of siblings , frequency
1  ,  2
2 , 2
3 , 1
4 , 3
5 , 3
Suppose i collect a sample of size of 3, the sample mean is a random variable denoted as $\bar X_3$, where $\bar X_3 = \frac{X_1+X_2+X_3}{3}$, $X_1,X_2,X_3$ are identical independent random variables.
The population mean $\mu$, is $E[X_i]$, which is also equivalent to $E[\bar X_n]$.
I have some questions that i need help on.
Are my thoughts 1) and 2) below correct?
1) the random variables $X_1, X_2, X_3$ each takes the possible value of 1,2,3,4 or 5. 
The actual value that $X_i$ takes  is the number of siblings that the $i$-th person in my sample has.
2) To calculate the population mean $\mu$, i will need the probabilities associated to the frequency.
So $E[X_i] = \frac{2}{11}\cdot 1 + \frac{2}{11}\cdot 2 + \frac{1}{11}\cdot 3 + \frac{3}{11}\cdot 4 + \frac{3}{11}\cdot 5$
and this agrees to the definition of population mean, $\frac{x_1+x_2+...+x_N}{N}$, where $N$ is the population size.
 A: First, it would help to get the numbers right. Second, this might be
easier to understand with more carefully chosen notation.
Numerical computations. There are two ways to find the population mean:
$\mu = (1+1+2+2+3+4+4+4+5+5+5)/11 = 3.2727\;$ and
$\mu = [2(1) + 2(2) + 3(1) + 4(3) + 5(3)]/(2+2+1+3+3) = 36/11 = 3.2727.$
Formulas. If the individual numbers of siblings in the population are denoted $s_1 = 1,\,s_2=1, \dots, s_{11} = 5.$ then the population size is $N = 11$ and $\mu = \frac{\sum_{i=1}^N s_i}{N}.$
Also, if the $k=5$ values are $v_1 =1, v_2 = 2, \dots, v_5 = 5$ siblings with corresponding
frequencies $f_1 = 2, f_2 = 2, \dots, f_5 = 3,$ then 
$\mu = \frac{\sum_{j=1}^k f_jv_j}{\sum_{j=1}^k f_j} = \frac{\sum_{j=1}^k f_jv_j}{N} = \sum_{j=1}^k r_iv_i,$
where $r_j = f_j/N$ are called relative frequencies.
Theoretical statements. If you are sampling $n = 3$ subjects from this population at random
and with replacement, you can say that the $n$ values $X_i$ you observe
are independent and identically distributed random variables. 
The expected number of siblings on $i$th observation $E(X_i) = \mu$.
