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A Note on the Estimator of the Alpha Coefficient for Standardized Variables Under Normality

Published online by Cambridge University Press:  01 January 2025

Kentaro Hayashi  and
Akihito Kamata
Kentaro Hayashi*
Affiliation:
University of Hawaii at Manoa
Akihito Kamata
Affiliation:
Florida State University
*
Requests for reprints should be sent to Kentaro Hayashi, Department of Psychology, University of Hawaii at Manoa, 2430 Campus Road, Gartley Hall 110, Honolulu, HI 96822-2294, USA. Email: hayashik@hawaii.edu
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Abstract

The asymptotic standard deviation (SD) of the alpha coefficient with standardized variables is derived under normality.The research shows that the SD of the standardized alpha coefficient becomes smaller as the number of examinees and/or items increase. Furthermore, this research shows that the degree of the dependence of the SD on the number of items is a function of the average correlation coefficients. When the average correlation approaches 1, the SD of the alpha coefficient decreases rapidly as the number of items increase, with the order of p. On the other hand, when the items are only weakly correlated, increasing the number of items decreases the SD of the alpha coefficient at a much slower rate.

Keywords

standard deviationreliabilityCronbach’s alphainternal consistency
Type
Original Paper
Information
Psychometrika , Volume 70 , Issue 3 , September 2005 , pp. 579 - 586
Copyright
Copyright © 2005 The Psychometric Society

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Footnotes

We would like to thank the associate editor and three anonymous reviewers whose comments significantly improved the quality of this paper. We are especially grateful to one of the reviewers who gave us detailed comments, including proof of Equation (10), and corrected some of our errors in earlier versions of the manuscript.

*

By “the asymptotic standard deviation of the alpha coefficient” we actually mean the asymptotic standard deviation of the “estimator” of the alpha coefficient. By the standard deviation we mean the population parameter from which we distinguish its estimator, the standard error.

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