11. Let the set $M=\{1,2, \cdots, 2005\}$. If $X \subseteq M$, $X \neq \varnothing$, and $a_{X}$ is the sum of the largest and smallest numbers in $X$ (if the set $X$ contains only one element, then this element is both the largest and the smallest number), then, for all non-empty subsets $X$ of $M$, the average value of all $a_{X}$ is
11.2006.

For any non-empty subset $X \subseteq M$, define $X^{\prime}=\{2006-x \mid x \in X \}$. It is clear that $X^{\prime} \subseteq M$. Thus, $f: X \rightarrow X^{\prime}$ provides a one-to-one correspondence on the family of all non-empty subsets of $M$. Note that
$$
\begin{array}{l}
\max (X)+\min \left(X^{\prime}\right)=2006, \\
\min (X)+\max \left(X^{\prime}\right)=2006 .
\end{array}
$$

It follows that, if $X=X^{\prime}$, then $a_{X}=a_{X^{\prime}}=2006$;
if $X \neq X^{\prime}$, then $a_{X}+a_{X^{\prime}}=2 \times 2006$.
Therefore, for all non-empty subsets $X$, the average of all $a_{X}$ is 2006.