We study the problem of estimating the common mean $μ$ of $n$ independent symmetric random variables with different and unknown standard deviations $σ_1 \le σ_2 \le \cdots \leσ_n$. We show that, under some mild regularity assumptions on the distribution, there is a fully adaptive estimator $\widehatμ$ such that it is invariant to permutations of the elements of the sample and satisfies that, up to logarithmic factors, with high probability, \[ |\widehatμ - μ| \lesssim \min\left\{σ_{m^*}, \frac{\sqrt{n}}{\sum_{i = \sqrt{n}}^n σ_i^{-1}} \right\}~, \] where the index $m^* \lesssim \sqrt{n}$ satisfies $m^* \approx \sqrt{σ_{m^*}\sum_{i = m^*}^nσ_i^{-1}}$.