  Differential privacy is a restriction on data processing algorithms that
provides strong confidentiality guarantees for individual records in the data.
However, research on proper statistical inference, that is, research on
properly quantifying the uncertainty of the (noisy) sample estimate regarding
the true value in the population, is currently still limited. This paper
proposes and evaluates several strategies to compute valid differentially
private confidence intervals for the median. Instead of computing a
differentially private point estimate and deriving its uncertainty, we directly
estimate the interval bounds and discuss why this approach is superior if
ensuring privacy is important. We also illustrate that addressing both sources
of uncertainty--the error from sampling and the error from protecting the
output--simultaneously should be preferred over simpler approaches that
incorporate the uncertainty in a sequential fashion. We evaluate the
performance of the different algorithms under various parameter settings in
extensive simulation studies and demonstrate how the findings could be applied
in practical settings using data from the 1940 Decennial Census.
