Creating Composite Age Groups to Smooth Percentile Rank Distributions of Small Samples

Abstract

Individually administered tests are often normed on small samples, a process that may result in irregularities within and across various age or grade distributions. Test users often smooth distributions guided by Thurstone assumptions (normality and linearity) to result in norms that adhere to assumptions made about how the data should look. Test users, however, may come across particular tests or sets of data in which the Thurstone assumptions are untenable. When users expect deviations from normality within age or grade, an alternate method is desirable. The authors present a relatively simple procedure that allows the user to treat observed raw scores as ordinal data with differently shaped sample distributions across age levels. Each age-level group is used twice to create new moving composite group distributions that replace (i.e., smooth) the original groups to reduce irregularities due to the small sample sizes. The authors present the results of a simulation study of the method, demonstrating that moving composite groups ameliorate error introduced by small samples beyond applying the normalized inverse to a score distribution. The method presented might satisfy those who question whether their data meet the strong assumptions of normality and interval-level measurement, and the simplicity might encourage smoothing by additional users.