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This note displays an interesting phenomenon for percentiles of independentbut non-identical random variables. Let $X_1,\cdots,X_n$ be independent randomvariables obeying non-identical continuous distributions and $X^{(1)}\geq\cdots\geq X^{(n)}$ be the corresponding order statistics. For any $p\in(0,1)$,we investigate the $100(1-p)$%-th percentile $X^{(pn)}$ and provenon-asymptotic bounds for $X^{(pn)}$. In particular, for a wide class ofdistributions, we discover an intriguing connection between their median andthe harmonic mean of the associated standard deviations. For example, if$X_k\sim\mathcal{N}(0,\sigma_k^2)$ for $k=1,\cdots,n$ and $p=\frac{1}{2}$, weshow that its median $\big|{\rm Med}\big(X_1,\cdots,X_n\big)\big|=O_P\Big(n^{1/2}\cdot\big(\sum_{k=1}^n\sigma_k^{-1}\big)^{-1}\Big)$ as long as$\{\sigma_k\}_{k=1}^n$ satisfy certain mild non-dispersion property.

Non-asymptotic bounds for percentiles of independent non-identical random variables