Concentration toward the mode: Estimating changes in the shape of a distribution of student data

Abstract

When making comparisons between groups of students, a common technique is to analyze whether there are statistically significant differences between the means of each group. This convention, however, is problematic when data are negatively skewed and bounded against a performance ceiling, features that are typical of data in education settings. In such a situation, we might be particularly interested to observe group differences in the left tail, specifically among students who have room to improve, and conventional analyses of group means have limitations for detecting such differences. In this article, an alternative to these conventions is presented. Rather than comparing the means of two groups, we can instead compare how closely student data are concentrated toward the modes of each group. Bayesian methods provide an ideal framework for this kind of analysis because they enable us to make flexible comparisons between parameter estimates in custom analytical models. A Bayesian approach for examining concentration toward the mode is outlined and then demonstrated using public data from a previously reported classroom experiment. Using only the outcome data from this prior experiment, the proposed method observes a credible difference in concentration between groups, whereas conventional tests show no significant overall differences between group means. The present article underscores the limitations of conventional statistical assumptions and hypotheses, especially in school psychology and related fields, and offers a method for making more flexible comparisons in the concentration of data between groups.