Example 3 Let $X=\{1,2, \cdots, 100\}$, for any non-empty subset $M$ of $X$, the sum of the maximum and minimum numbers in $M$ is called the characteristic of $M$, denoted as $m(M)$, find the average of the characteristics of all non-empty subsets of $X$.
Let $A \subset X$, and let $f: A \rightarrow A^{\prime}, A^{\prime}=\{101-a \mid a \in A\} \subset X$. Then $f: A \rightarrow A^{\prime}$ is a surjective function from the set $Y$ of all non-empty subsets of $X$ to itself. Denote the non-empty subsets of $X$ as $A_{1}, A_{2}, \cdots, A_{n}$ (where $n=2^{100}-1$), then the average of the characteristics is $\frac{1}{n} \sum_{i=1}^{n} m\left(A_{i}\right) = \frac{1}{2 n} \sum_{i=1}^{n}\left(m\left(A_{i}\right)+m\left(A_{i}^{\prime}\right)\right)$. Since the sum of the maximum number in $A$ and the minimum number in $A^{\prime}$ is 101, and the sum of the minimum number in $A$ and the maximum number in $A^{\prime}$ is also 101, it follows that $m\left(A_{i}\right)+m\left(A_{i}^{\prime}\right)=202$, thus the average of the characteristics is $\frac{1}{2 n} \cdot 202 \cdot n=101$.