What's the expected value of average absolute deviation from the mean of k randomly picked numbers? Say we have to randomly pick k integral numbers out of n. The numbers are from the range < a; b >. What is the expected value of average absolute deviation from the mean for that random subset of k-numbers as the number of drawings approaches infinity?
Sorry if didn't make myself clear. Could you explain the answer so that it is understandable for a not so bright high school student?
EDIT: This is not homework :) Somobody asked me to program a vizualization of Lotto lottery results and I just got curious about the statistics of that.
 A: I do not have a complete answer; I am posting this just in case someone finds it interesting or useful. I assume sampling with replacement; the “without replacement” variation seems much harder. 
We are interested in the quantity
$$
\mathbf E \left[\frac{1}{k} \sum_{i=1}^k \left| X_i - \frac{X_1 + X_2 + \cdots + X_k}{k} \right| \right]
$$
where $X_1, \ldots, X_k$ are iid and drawn from a distribution $\mathcal D$. By linearity
of expectation and symmetry, this is equal to
$$
\begin{align*}
\mathbf E \left[\left| X_k - \frac{X_1 + X_2 + \cdots + X_k}{k} \right| \right] 
&=  \mathbf E \left[\left| \frac{(k-1)X_k - (X_1 + X_2 + \cdots + X_{k-1})}{k} \right| \right] 
\\ &= \frac{k-1}{k} \cdot \mathbf E \left[\left| X_k - \frac{X_1 + X_2 + \cdots + X_{k-1}}{k-1} \right| \right] 
\\ &= \frac{k-1}{k} \cdot \mathbf E \left[\left| X_k - Y \right| \right] 
\end{align*}
$$
where $Y = \frac{X_1 + \cdots + X_{k-1}}{k-1}$ is independent of $X_k$.
A: Since this may be homework, here are some hints:


*

*When the number of drawings approaches infinity, what will the mean be close to?

*Can you work out the deviation from the mean for each of the $n = b-a+1$ possible values? 

*Can you take the absolute values of these deviations, and then take the average (i.e. add them up and divide by $n$)?


You may find yourself doing the calculations twice, once for $n$ odd and once for $n$ even.
