6. Let the set $M=\{1,2, \cdots, 1000\}$, and for any non-empty subset $X$ of $M$, let $a_{X}$ denote the sum of the largest and smallest numbers in $X$. Then, the arithmetic mean of all such $\alpha_{X}$ is $\qquad$.
6. Pair the non-empty subsets of $M$. For each non-empty subset $X \subset M$, let
$$
X^{\prime}=\{1001-x \mid x \in X\},
$$

then when $X_{1}$ is also a non-empty subset of $M$, and $X \neq X_{1}$, we have $X^{\prime} \neq X^{\prime}_{1}$. Thus, all non-empty subsets are divided into two categories: (A) $X^{\prime} \neq X$, (B) $X^{\prime}=X$.

For $X$ in (B), it must be that $\alpha_{X}=1001$. For a pair of $X$ and $X^{\prime}$ in (A), we have $\alpha_{X}+\alpha_{X^{\prime}}=1001 \times 2=2002$. From this, we can see that the arithmetic mean of all $\alpha_{X}$ is 1001.