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In two recent contributions, we have provided a comparative analysis of various optimization algorithms, which can be used for atomic location estimation, and suggested an enhanced version of the Steepest Descent (ESD) algorithm, which we have shown to be competitive with other distributed localization algorithms in terms of estimation accuracy and numerical complexity. Moreover, therein we have conducted a preliminary statistical characterization of the positioning error distribution, by showing that it can be well approximated by the family of Pearson distributions, as well as pointed out that its knowledge may be efficiently used to speed-up the analysis of iterative-based positioning algorithms by avoiding the need of simulating the whole location discovery algorithm, and allowing simulation at the atomic level only. In this contribution, based on the preliminary results shown in, we propose a comprehensive statistical analysis of the positioning error distribution for the ESD algorithm, by providing the parameters of the Pearson fitting distribution with respect to two important design factors for wireless sensor networks (WSNs): i) the ranging error standard deviation, which represents the input parameter for every localization algorithm, and ii) the geometric dilution of precision factor, which provides a simple parameter to account for different network topologies. In particular, we report an extensive number of simulation results that may provide important insights to the system designer: i) allow a parametric analysis to figure out the joint effect of ranging errors and network topology on the performance of the localization algorithms, and ii) give a general framework for modeling the statistics of the positioning error, which may be used for network planning, as well as for the analysis and design of the upper layers of the protocol stack.