6. Let the set $A=\{1,2, \cdots, 2014\}$. For any non-empty subset $X$ of $A$, let $a_{X}$ denote the sum of the largest and smallest numbers in $X$. Then the arithmetic mean of all such $a_{X}$ is $\qquad$ .
6.2015.

Pair the non-empty subsets of set $A$. For each
$$
\begin{array}{l} 
X=\left\{x_{1}, x_{2}, \cdots, x_{k}\right\} \text {, assume without loss of generality that } x_{1}<x_{2}<\cdots<x_{k} . \\
\text { Let } X^{\prime}=\{2015-x \mid x \in X\} .
\end{array}
$$

Then $X^{\prime} \subseteq A$, and
$$
a_{X}=x_{1}+x_{k}, a_{X^{\prime}}=4030-\left(x_{1}+x_{k}\right) \text {. }
$$

If $X \neq X^{\prime}$, then $a_{X}+a_{X^{\prime}}=4030$;
If $X=X^{\prime}$, then $a_{X}=2015$.
In summary, the arithmetic mean of all $a_{X}$ is 2015.