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The Relation of Item Difficulty and Inter-Item Correlation to Test Variance and Reliability

Published online by Cambridge University Press:  01 January 2025

Harold Gulliksen
Harold Gulliksen*
Affiliation:
College Entrance Examination Board
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Abstract

Under assumptions that will hold for the usual test situation, it is proved that test reliability and variance increase (a) as the average inter-item correlation increases, and (b) as the variance of the item difficulty distribution decreases. As the average item variance increases, the test variance will increase, but the test reliability will not be affected. (It is noted that as the average item variance increases, the average item difficulty approaches .50). In this development, no account is taken of the effect of chance success, or the possible effect on student attitude of different item difficulty distributions. In order to maximize the reliability and variance of a test, the items should have high intercorrelations, all items should be of the same difficulty level, and the level should be as near to 50% as possible.

Type
Original Paper
Information
Psychometrika , Volume 10 , Issue 2 , June 1945 , pp. 79 - 91
Copyright
Copyright © 1945 The Psychometric Society

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Footnotes

*

The desirability of determining this relationship has been indicated by previous writers. Work on the present paper arose out of some problems raised by Dr. Herbert S. Conrad in connection with an analysis of aptitude tests.

†

On leave for Government war research from the Psychology Department, University of Chicago.

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