This study explores issues related to one-sample nonparametric tests for the median of a continuous distribution when the sample is collected via size-bias of a known order. A general principle on how to construct the reference distribution of a given test statistic is presented. Following this principle, we create new bias-corrected nonparametric testing procedures. Computationally intensive, exact P-values are available for a small sample. When the sample size is large, P-values can be easily estimated by the asymptotic approximation developed here. Power functions of these tests are investigated in both small- and large-sample cases and consistency is shown to hold under fairly general conditions. The tests' performances are then compared via asymptotic relative efficiency under four theoretical distributions.