6. Let the set $M=\{1,2, \cdots, 2020\}$. For any non-empty subset $X$ of $M$, let $\alpha_{X}$ denote the sum of the largest and smallest numbers in $X$. Then the arithmetic mean of all such $\alpha_{X}$ is $\qquad$ .
6. 2021.

Construct the subset $X^{\prime}=\{2021-x \mid x \in X\}$.
Then all non-empty subsets can be divided into two categories:
$X=X^{\prime}$ and $X \neq X^{\prime}$.
When $X=X^{\prime}$, it must be that $X=X^{\{\prime\}}=M$.
Thus, $\alpha_{X}=2021$.
When $X \neq X^{\prime}$, let $x$ and $y$ be the maximum and minimum numbers in $X$, respectively, then $2021-x$ and $2021-y$ must be the minimum and maximum numbers in $X^{\prime}$, respectively.
Thus, $\alpha_{X}=x+y$, $\alpha_{X^{\prime}}=4042-x-y$.
Therefore, $\frac{\alpha_{X}+\alpha_{X^{\prime}}}{2}=2021$.
Hence, the arithmetic mean of all $\alpha_{X}$ is 2021.