In this paper a distribution sampling study consisting of four major experiments is described. The L1 norm is employed in two new estimating techniques, direct least absolute (DLA) and two-stage least absolute (TSLA), and these two are compared to direct least squares (DLS) and two-stage least squares (TSLS). Four experiments testing the normal distribution case, a multicollinearity problem, a hetoroskedastic variance problem, and a misspecified model were conducted. Two small sample sizes were used in each experiment, one with N = 20 and one with N = 10. In addition, conditional predictions were made using the reduced form of the four estimators plus two direct methods, least squares no restrictions (LSNR) and another new method known as least absolute no restrictions (LANR). The general conclusion was that the L1 norm estimators should prove equal to or superior to the L2 norm estimators for models using a structure similar to the overidentified one specified for this study, with randomly distributed error terms and very small sample sizes.