The statistical theory of stepped-up reliability coefficients when a test has been divided into several equivalent parts

This paper is a presentation of the statistical sampling theory of stepped-up reliability coefficients when a test has been divided into any number of equivalent parts. Maximum-likelihood estimators of the reliability are obtained and shown to be biased. Their sampling distributions are derived and form the basis of the definition of new unbiased estimators with known sampling distributions. These unbiased estimators have a smaller sampling variance than the maximum-likelihood estimators and are, because of this and some other favorable properties, recommended for general use. On the basis of the variances of the unbiased estimators the gain in accuracy in estimating reliability connected with further division of a test can be expressed explicitly. The limits of these variances and thus the limits of accuracy of estimation are derived. Finally, statistical small sample tests of the reliability coefficient are outlined. This paper also covers the sampling distribution of Cronbach's coefficient alpha.