Reconsidering the encoding of data in physics education research

How data are collected and how they are analyzed is typically described in the literature, but how the data are encoded is often not described in detail. In this paper, we discuss how data typically gathered in PER are encoded and how the choice of encoding plays a role in data analysis. We describe the kinds of data that are found when using short answer, multiple choice, Likert- scale, ranking task, and free response questions in terms of nominal, ordinal, interval, and ratio data. We discuss the mathematical operations that are available for each kind of data and how this affects ways that similarity and difference between student responses can be determined, a topic we discuss in terms of measures of distances and correlation. Finally, we use several papers from the literature to discuss ways in which data have been encoded and analyzed, with examples of normalized gain, factor analysis, model analysis, cluster analysis, and the investigation of epistemological agreement. We highlight both strengths and weaknesses of the data encoding approaches used in these studies. Our goal is not a comprehensive review, but one that is illustrative and can help researchers understand their own and each other's work more deeply.