The rate-distortion performance of conventional resolution-constrained quantization (RCQ) based on the mean-squared error criterion (MSE-RCQ) is generally compromised by the impact of distortion outliers and source mismatch. Not only the mean distortion, but also the number of distortion outliers should be considered in quantizer design. Thus, we propose the use of a design criterion that gives more importance to the tail of the source distribution, which leads to RCQ based on the second moment of distortion (SMD-RCQ). A continuous range of alternatives between MSE-RCQ and SMD-RCQ is also defined and implemented based on the weighted arithmetic-mean measure (WAM-RCQ). It can be used to control the centroid density in the tail of the source distribution. Experimental results with a Gaussian source and line spectral frequencies (LSFs) show that the proposed WAM-RCQ not only produces a similar mean distortion as conventional MSE-RCQ, but has a lower percentage of distortion outliers and a significantly reduced sensitivity to source mismatch.