...three histogram comparison tasks at the end of the course to assess their abilities to identify similar means and standard deviations and to evaluate skewness as represented in histograms. Fewer than 50% of the students completed all three tasks successfully. Common errors included inferring the relative value of the mean according to the center of the x-axis rather than the center of the distribution of data, identifying histograms with greater heights as those having the greater standard deviations, and interpreting skewness as shift of the center of the distribution along the x-axis rather than an asymmetry of the distribution.

Introduction

Statistical reasoning includes interpreting numeric descriptive statistical measures and their corresponding graphic displays. Students need to develop their understanding of the information that descriptive statistical measures provide about a set of data. Students should also be able to interpret statistical graphs to assess the distribution, central tendency, and variability of data sets and be able to use these characteristics to compare data sets (Garfield & Gal, 1999).

Developing students' abilities to interpret histograms is a common goal for introductory statistics courses. First, histograms serve a role in describing the characteristics of data sets, providing visual depictions of samples, populations, and sampling distributions. Second, an understanding of histograms contributes to the understanding of statistical concepts, including descriptive statistics such as mean, standard deviation and skewness. Comprehension of inferential statistics such as t-tests and the role of p-values in hypothesis testing depending in part upon an understanding of histograms as well. Textbooks rely heavily on histograms in their presentations of statistical concepts. Third, the most commonly taught inferential techniques are based on assumptions of normality, usually described in introductory courses in terms of histogram shape. It is therefore necessary that students understand the concepts represented in histograms and that they are able to extract relative measures of central tendency, variation and symmetry of distribution in order to interpret histograms and make comparisons among them.

Although research has documented some of the difficulties students have in understanding basic statistical concepts (Garfield & Ahlgren, 1988; Pollatsek, Lima, & Well, 1981), there is little research documenting college students' interpretations of histograms. In this study, college students enrolled in an introductory statistics course were assessed on their abilities to interpret the concepts of mean, standard deviation, and symmetry of data distribution as represented in histograms. More specifically, students were asked to compare histograms showing symmetric and asymmetric distributions along with varying means and standard deviations. From these, students were directed to identify histograms displaying means and standard deviations similar to a "reference" histogram and to identify a skewed distribution.

With the goal of determining specific aspects of students' abilities to extract information from histograms at the end of the statistics course, we constructed a set of noncontextual problems that were included on the final examination and were intended to challenge students to interpret and compare histograms at a high level of abstraction. Students' responses were analyzed so that we could identify common errors and interpret these errors in a way that would inform our pedagogy.

Because of the complexity of the histogram interpretation tasks, we first ran a pilot study. Based on our analysis of pilot results, we made some modifications to the tasks to eliminate areas of confusion that appeared to be caused by the way the problems were presented. The histograms used in the pilot study did not include scale marks and numbering on the x-and y-axes, and some students' responses suggested that they might not have made the assumption that the scales were consistent for all histograms. As a result, histograms that were identical except for differing position of the center of the distribution along the x-axis could have been interpreted as having identical means but different scales when the intent was to show differing means long x-axes of the same scale. To eliminate this potential confusion, we revised the histograms by adding explicit scaling information along their x-axes. In addition, we noticed that some pilot respondents appeared to be making comparisons among all seven histograms rather than by using the reference histogram as a point of comparison for the other six. Therefore, we revised the wording to clarify the instruction that the comparisons were to be made from the reference histogram to the six choices below it.

Related Research

A review of the existing literature on graph interpretation indicates that students must integrate mathematical, statistical, and graphic representation knowledge in order to interpret histograms (Bell & Janvier, 1981; Clement, 1989; Curico, 1987; Friel & Bright, 1996; Friel, Curcio, & Bright, 2001; Leinhardt, Zaslavsky, & Stein, 1990; Meyer, Shinar, & Leiser, 1997). (1) General graph interpretation skills are required, along with recognizing this graph form as a histogram and understanding the special meanings the of the x- and y-axes in this form. Beyond this, knowledge about the...

NOTE: All illustrations and photos have been removed from this article.



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