Example 7 Given that $a, b, c, d, e$ are all real numbers, and
$$
a+b+c+d+e=4, a^{2}+b^{2}+c^{2}+d^{2}+e^{2}=4 \text {. }
$$

Find the maximum value of the real number $a$.
For the arrays $b, c, d, e$, we have
$$
\begin{aligned}
\bar{x} & =\frac{b+c+d+e}{4}=\frac{4-a}{4}, \\
S^{2} & =\frac{1}{4}\left[\left(b^{2}+c^{2}+d^{2}+e^{2}\right)-4 \bar{x}^{2}\right] \\
& =\frac{1}{4}\left[4-a^{2}-4\left(\frac{4-a}{4}\right)^{2}\right] \geqslant 0 .
\end{aligned}
$$

Simplifying, we get $a(5a-8) \leqslant 0$.
Solving, we get $0 \leqslant a \leqslant \frac{8}{5}$.
Thus, the maximum value of $a$ is $\frac{8}{5}$.