Descriptive Statistics for Modern Test Score Distributions: Skewness, Kurtosis, Discreteness, and Ceiling Effects

Abstract

Many statistical analyses benefit from the assumption that unconditional or conditional distributions are continuous and normal. Over fifty years ago in this journal, Lord (1955) and Cook (1959) chronicled departures from normality in educational tests, and Micerri (1989) similarly showed that the normality assumption is met rarely in educational and psychological practice. In this paper, the authors extend these previous analyses to state-level educational test score distributions that are an increasingly common target of high-stakes analysis and interpretation. Among 504 scale-score and raw-score distributions from state testing programs from recent years, non-normal distributions are common and are often associated with particular state programs. The authors explain how scaling procedures from Item Response Theory lead to non-normal distributions as well as unusual patterns of discreteness. The authors recommend that distributional descriptive statistics be calculated routinely to inform model selection for large-scale test score data, providing warnings in the form of sensitivity studies that compare baseline results to those from normalized score scales.