Improved estimates of a survey population parameter can be obtained by using information about a co-related auxiliary variable. The standard version of this problem concerns estimation of the population mean where ratio, difference, and regression methods for incorporating auxiliary information have been studied. Recently there has been interest in the related problem in which the median is the parameter of interest. Median versions of mean-based methods have been proposed and analyzed. Not yet considered however are methods based on the least absolute value (L₁) criterion for defining estimators. This approach is motivated by the criterion's close connection to the median. In this paper we introduce these methods and present properties in the context of the standard superpopulation model.