Suppose to have a set of exact data $x_i, i=1,\dots,N$ and to calculate its median value $m$. Then, a sound way to estimate the error $\delta m$ on $m$ would be bootstrapping. (I think...)
But what if each $x_i$ has an associated error $\delta x_i$? In this case bootstrapping would not retain the information on the error of the single data.
Is this loss of information a common assumption when using bootstrap? Is there some other method to calculate $\delta m$ that retains the information given by the $\delta x_i$?