Suppose $n_i$ is uniformly distributed on the interval $[-1,1].$ Then \begin{align} & \operatorname{var}\left( \frac{n_1+\cdots+n_n} n \right) = \frac 1 {3n} \\[10pt] \text{and} \quad & \operatorname{var} \left( \frac{\max\{n_1,\ldots,n_n\} + \min\{n_1,\ldots+n_n\}} 2 \right) = \frac{4n}{(n+1)(n+2)^2} \end{align} and the former exceeds the latter when $n\ge 6.$
Michael Hardy
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