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Sampling Variability and Axioms of Classical Test Theory

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  • Donald W. Zimmerman

Abstract

Many well-known equations in classical test theory are mathematical identities in populations of individuals but not in random samples from those populations. First, test scores are subject to the same sampling error that is familiar in statistical estimation and hypothesis testing. Second, the assumptions made in derivation of formulas in test theory are not necessarily satisfied in small samples. The present study derived modified equations relating test scores and components of scores that are identities in samples of any size and that reduce to the more familiar equations when various correlations are zero. Simulations determined the accuracy of both the familiar and the modified equations when applied to samples of various sizes from populations with known reliability coefficients. The programs also determined the variability of the sample values for different parameters in the equations and for different sample sizes, as well as the means and variances of discrepancies between population and sample values.

Suggested Citation

  • Donald W. Zimmerman, 2011. "Sampling Variability and Axioms of Classical Test Theory," Journal of Educational and Behavioral Statistics, , vol. 36(5), pages 586-615, October.
    • Handle: RePEc:sae:jedbes:v:36:y:2011:i:5:p:586-615
    DOI: 10.3102/1076998610397052
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    References listed on IDEAS

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    1. Lee Cronbach, 1951. "Coefficient alpha and the internal structure of tests," Psychometrika, Springer;The Psychometric Society, vol. 16(3), pages 297-334, September.
    2. Melvin Novick & Charles Lewis, 1967. "Coefficient alpha and the reliability of composite measurements," Psychometrika, Springer;The Psychometric Society, vol. 32(1), pages 1-13, March.
    3. Louis Guttman, 1953. "Reliability formulas that do not assume experimental independence," Psychometrika, Springer;The Psychometric Society, vol. 18(3), pages 225-239, September.
    4. Leonard Feldt, 1965. "The approximate sampling distribution of Kuder-Richardson reliability coefficient twenty," Psychometrika, Springer;The Psychometric Society, vol. 30(3), pages 357-370, September.
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