Chebyshev inequality comparison
Chebyshev's inequality states that
$$
\Pr[|x_i - \mu| &gt; c\sigma] &lt; \frac{1}{c^2}.
$$
Choosing $c = \sqrt{n}$, we deduce that with positive probability,
$$
|x_i - \mu| \leq \sqrt{n} \cdot \sigma \text{ for all } i.
$$