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

集中量數：平均數、削截平均數、單階M估計值和改良式單階M估計值之比較

吳佩真(Pei-Chen Wu)

《中華心理學刊》 46卷1期 (2004/03) Pp. 29-47

https://doi.org/10.6129/CJP.2004.4601.03

摘要

本研究比較五種集中量數（平均數、10%和20%削截平均數、one step M-estimators based on Huber's Ψ、modified on step M-estimators）如何受非常態分佈的影響。由24組實際資料之檢驗結果發現：當比較二組獨立團體的集中量數時，20%削截平均數、one step M-estimator與modified one step M-estimator是較佳的選擇，而平均數是最差之選擇，因為平均數極容易受某一極端值的影響。此外，研究亦發現，集中量數僅是反應一個群體中心位置的數值。因此，若是僅以集中量數來比較二組獨立群體，將不易察覺到二組群體以複雜方式呈現不同，故本研究建議比較二組的quantiles才可完整、全盤得知二組群體間的關係。

關鍵字

集中量數 ； 平均數 ； 削截平均數 ； one step M-estimators based ； modified one step M-estimators

並列摘要

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.

並列關鍵字

measure of location ； means ； trimmed means ； one step M-estimators ； modified one step M-estimators

參考文獻

Huber, P. J.(1993).New directions in statistical data analysis and robustness.Boston:Birkhauser Verlag.

Wilcox, R. R.(1987).New statistical procedures for the social sciences: Modern solutions to basic problems.Hillsdale, New Jersey:Lawrence Erlbaum.

Emerson, J. D.,Stoto, M., A.(2000).Understanding robust and exploratory data analysis.New York:John Wiley & Sons, Inc..

Huber, P. J.(1981).Robust statistics.New York:Wiley.

Rosenberger, J. L.,Gasko, M.(2000).Understanding robust and exploratory data analysis.New York:John Wiley & Sons, Inc..