Measure of Location: Comparing Means, Trimmed Means, One Step M-estimators and Modified One Step M-estimators under Non-normality

Abstract

This study examines five measures of location (means, 10% and 20% trimmed means, one step M-estimators based on Huber's Ψ and modified one step M-estimators) in terms of their Type I error rates, standard errors and significance levels in 24 empirical data sets. Twenty-four empirical data sets can be categorized into eight kinds of distributions which frequently arise in educational and psychological research- discrete mass at zero, mass at zero, extreme positive skew, extreme negative skew, bimodality, multi-modality and lumpy, digit preference, and smooth symmetric. The results show that the 20% trimmed mean, one step M-estimator and modified one step M-estimator, are good alternatives for comparing two groups based on comparing measure of location. Student's t is the least satisfactory statistic. Also, this study indicates that comparing measures of location provides information only on the typical value of the groups. This limitation is apparent in some situations considered here where none of the five measures of locations are completely satisfactory. Thus, the study recommends comparing quantiles of two groups to obtain an overall picture of the relationship between two groups.