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