【Analysis】From the given conditions, we have
$$
\begin{array}{l}
10=\left|\sum_{i=1}^{10}\left[\left(x_{i}-1\right)-\left(x_{i}-2\right)\right]\right| \\
\leqslant \sum_{i=1}^{10}\left|x_{i}-1\right|+\sum_{i=1}^{10}\left|x_{i}-2\right| \leqslant 4+6=10 \\
\Rightarrow \sum_{i=1}^{10}\left|x_{i}-1\right|=4, \sum_{i=1}^{10}\left|x_{i}-2\right|=6 \\
\Rightarrow 1 \leqslant x_{i} \leqslant 2 \\
\Rightarrow \bar{x}=1+\frac{1}{10} \sum_{i=1}^{10}\left(x_{i}-1\right)=1.4 .
\end{array}
$$
Example 3 Given real numbers $x_{1}, x_{2}, \cdots, x_{10}$ satisfy $\sum_{i=1}^{10}\left|x_{i}-1\right| \leqslant 4, \sum_{i=1}^{10}\left|x_{i}-2\right| \leqslant 6$. Find the average value $\bar{x}$ of $x_{1}, x_{2}, \cdots, x_{10}$. (2012, Zhejiang Province High School Mathematics Competition)