SOME DIFFICULTIES OF LEARNING HISTOGRAMS IN INTRODUCTORY STATISTICS Carl Lee Central Michigan University, MI 48859, USA Carl.Lee@cmich.edu Maria Meletiou-Mavrotheris Cyprus Ministry of Education meletiu@spidernet.com.cy Keywords: statistics education research; graphical representations; histogram; bar graph; variation Abstract: The findings reported in the article came from a study where we examined over 160 students’ final examination papers, which included questions specifically designed to investigate different levels of understanding about the construction and interpretation of histograms employed to demonstrate the concept of variability. The article describes the four main difficulties in constructing and interpreting histograms identified by the study. It also briefly discusses implications for research and provides suggestions for instructional remedies to help improve students’ ability to construct and interpret histograms. 1. INTRODUCTION Research has suggested that student misconceptions are quite difficult to change (Garfield, and Ahlgren, 1988). Moreover, some of the misconceptions (such as representativeness) seem to vary with problem context (Garfield & delMas, 1990). Student difficulties in learning statistical concepts and in overcoming misconceptions may in part be due to the overlooking of some basic representations of variation and data production (Meletiou & Lee, 2002). The histogram is among the main graphical tools employed in the statistic classroom for assessing the shape and variability of distributions. Introductory statistics courses have been traditionally using the histogram both as a tool for describing data and as a means to aid students in comprehending fundamental concepts such as the sampling distribution. Because comprehension of histograms is the basis for the concepts of variability and distribution, which are the core concepts of an introductory statistics course, overlooking student difficulties with histograms might have dramatic consequences on the teaching and learning of statistical concepts. In the article, we present findings from a study specifically designed to establish better understanding of the main difficulties students encounter in the construction, interpretation and application of histograms. After providing an overview of the study design, we present findings from the study. Four main categories of mistaken beliefs about histograms identified by the study are discussed. Implications for research and instruction follow. 2. DESIGN OF STUDY 2.1 Motivation for Study Our past experience was suggesting that understanding of histograms is not as trivial as one might think. For example, in a previous semester, our college-level introductory statistics students had done extremely poorly in the following question given to them at the end of the course (Lee, 2000): When constructing a histogram for describing the distribution of salary for individuals forty years or older but not yet retired: a) What goes on the vertical axis? b) What goes on the horizontal axis? What would be the proper shape of the salary distribution? Explain why. Analysis of students’ responses suggested that most of them confused the histogram with the scatterplot of salary vs. age, thinking that ‘the graph is skewed-to-the right because as people approach retirement, their salary gradually drops’. This observation came as a surprise. Since histograms appear very frequently in the media and other contexts, one would assume that a college student would have good understanding of this important type of graphical representation. The surprising observation motivated us to conduct the current study, in order to investigate more closely student difficulties in constructing and interpreting histograms. 2.2 Context and participants The site for the study was an introductory statistics course in a mid-size Midwestern university in the United States. One of the authors, Lee, was the instructor of the course. A total of 162 students participated in this study over a three-semester period starting in the Fall 2001 semester. About 75 percent of 2003 Joint Statistical Meetings - Section on Statistical Education 2326