Some insights about college students’ interpretations of histograms

Some insights about college students’ interpretations of histograms

Susan S. Gray

Abstract

The interpretation of histograms is a complex process requiring the integration of understanding about how graphs convey information with knowledge about how statistical constructs are displayed graphically. For this study, students in an introductory statistics class completed 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 a 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 ways in which histograms represent statistical information, supported by the understanding of this information, is needed. Such knowledge includes recognizing the distinction between bar graphs and histograms and knowing that histograms represent distributions of grouped data. Statistical concepts of class interval and frequency and how these are represented on the x- and y-axes are additional aspects of requisite knowledge. Finally, and of particular interest to us, statistical concept knowledge about mean, standard deviation, and the distribution of data need to be coupled with an understanding about how these concepts are represented graphically in histograms as center, spread, and shape of the data. In making comparisons among histograms, students must integrate their graphic understanding of these statistical concepts and estimate how differing degrees of center, spread, and symmetry are depicted graphically in a relative sense.

Although some of the difficulties students have in interpreting histograms appear to be particular to this type of graph (Friel & Bright, 1996; Simonsen & Teppo, 1999), a number of problems are also common to other types of graph interpretation tasks. For example, some of the findings from research on student interpretation of line graphs are relevant to our present work on histograms. In this regard, Leinhardt et al. (1990) make an important distinction between what they call “point” and “global” interpretation of line graphs. According to these researchers, point interpretation involves reading values directly from the graphs, while global interpretation, required for making histogram comparisons, focuses on patterns or trends that are either depicted or suggested by graphs. Students typically have considerable experience with point interpretation, but global interpretation can be more difficult because skills like pattern identification, estimation, and prediction are required (Curcio, 1987; Leinhardt et al., 1990). Furthermore, Leinhardt et al. and Dunham and Osborne (1991) identify a common misconception related to the interpretation of scale on Catesian line graphs, reporting that students frequently assume that the scales on the x- and y-axes of a Cartesian graph are always identical and generally numbered in single units. These assumptions are often incorrect for Cartesian line graphs, and they never apply to histograms because the axes of histograms represent two distinctly different constructs. Another interpretation challenge is presented because histograms represent frequencies and the distribution of data with areas on the surface of the graph. Carswell (1992) found that graphs requiring the interpretation of area or volume were the most difficult to interpret correctly.

Research on bar graph interpretation with younger students can illuminate our research problem as well. Curcio (1987) described three levels of graph interpretation that reflect increasing levels of complexity. The first level is reading information directly from the graph in a manner similar to the point interpretation described by Leinhardt et at. (1990). The second level of interpretation, shown to be more difficult, involves making comparisons “between the data” (Curcio, 1987. p. 384), such as quantifying differences between two or more components. Finally, the third level of interpretation requires students to infer or predict “beyond the data” (Curico, 1987, p. 384). This level of interpretation is similar to what Leinhardt et al. call global interpretation, and several studies have shown that school age students have the most difficulty with this type of graph interpretation (Curico, 1987; Friel et al., 2001; Pereira-Mendoza & Mellor, 1990).

The bar graph interpretation abilities of older students have been shown to improve in relationship to the types and levels of mathematics they are actively studying. Fisher (1992) conducted a study to compare the bar graph interpretation abilities of three groups of college students: freshmen taking an algebra course, sophomores enrolled in the second semester of calculus, and seniors and graduate students studying a calculus-based advanced statistics course. Fisher found that all three groups of students could read values from the graphs equally well. Not surprisingly, Fisher also found that the students with more mathematical and statistical experience were more successful at extracting information about trends suggested by patterns displayed in bar graphs.

In addition to what we can infer from previous investigations on graph interpretation in general, research on students’ interpretations of histograms in particular confirms that these are especially challenging graphs for students to understand. Although prior learning can enhance students’ abilities to extract meaning from histograms in some ways, students may also find that their assumptions based on previous experiences do not necessarily hold for histograms. For example, students typically have experiences with bar graphs, but this does not help them understand the concept of frequency that is represented in histograms. Further confounding the situation is the fact that histograms resemble bar graphs. Also, much of students’ previous mathematics learning related to algebra, functions, and calculus involves experiences with graphs in the Cartesian coordinate plane. Students can be confused when they are unable to shift their perceptions of the meaning of the x- and y-axes in the Cartesian plane to the concepts represented by these axes in histograms (Friel & Bright, 1996).

In a study related to our current work, Simonsen and Teppo (1999) examined preservice elementary students’ interpretations of histograms. These researchers found that their students had more difficulty interpreting frequency distributions (which represent grouped data) than they did in interpreting bar charts (which represent individual data points). Simonsen and Teppo’s students also constructed graphs to represent data that they had collected. Results showed that some students were confused about how to employ and label the x- and y-axes, and that others did not use any labels at all on one or both axes. Since the graph format was unspecified, some students constructed bar charts and others constructed frequency distributions. When asked to “grade” representative graphs that their peers had produced, students consistently gave the bar charts higher ratings than the frequency distributions, stating that the bar charts were easier to understand. The fact that students had participated in an experiment to generate their data did not seem to assist them in making sense of their data when it was reconfigured into a frequency distribution.

In discussing their findings, Simonsen and Teppo (1999) pointed out that the abstract nature of grouped data contributed to the complexity of the task. These authors further speculated that their college-aged students had not developed the mental representations that would allow them to interpret frequency distributions effectively. Because the statistical concepts of mean, standard deviation, and the symmetry of the distribution are implicitly represented in a histogram, a comprehensive interpretation of the histogram is needed to identify these characteristics as they pertain to the data set being examined graphically.

Research Questions

In view of the existing research, histograms appear to be among the more complex types of graphs to interpret. However, there is little research on college students’ abilities to interpret histograms in general or to relate histograms to common descriptive statistical concepts. In order to gain more insight about college students’ interpretations of histograms and how these students view the concepts of mean, standard deviation, and skewness as represented in historgrams, the present study investigated these research questions:

1. What are students’ success rates in identifying histograms with similar means and standard deviations?

2. What are students’ success rates in identifying a histogram with a skewed distribution?

3. What are the most common errors that students make in performing these tasks?

4. Do students’ errors suggest commonly held misconceptions about the representation of means, standard deviations, and skewness in historgrams?

Method

Subjects were 159 students enrolled in an introductory statistics course for science majors. The sample included a few first- and third-year students, but most were traditional college-age students in their second year of study. There were 35% males and 65% females in the sample. About 60% of the students in this sample had recently taken a precalculus course in college, while the remaining 40% had recently completed at least one semester of calculus.

Students in the course studied histograms and numeric descriptive statistics early in the semester and discussed these descriptive measures for a variety of data sets throughout the semester. As in common in introductory statistics courses, students constructed a few histograms from frequency tables and computed means and standard deviations with a calculator at the beginning of the term. Students then used statistical software to produce histograms and other graphic representations of data distributions and to compute means and standard deviations. Class discussion about graphic and numeric descriptive statistical analysis included observing the relationship of computed means and standard deviations to the shapes of corresponding histograms and identifying approximately normal and skewed distributions.

The histogram interpretation tasks, consisting of three comparison questions, were included on the final examination in four different semesters (See Figure 1). Because the first question required the choice of two histograms for a correct answer, responses were considered to be correct only if they included both of the appropriate histograms and no inappropriate histograms. Thus, there were no partially correct responses to any question. Correct responses received one point, and a total score for each subject was determined by adding the number of correct responses to the three questions. Types of incorrect responses were categorized and their frequencies tabulated for the purpose of further analysis.

[FIGURE 1 OMITTED]

Results

Table 1 presents each possible score along with the frequencies and percentages of students obtaining each score.

As can be seen in Table 1, almost half of the students completed all three histogram interpretation tasks correctly. At the same time, 20% scored only one or zero points, indicating that these students had difficulty interpreting basic descriptive statistical concepts as displayed in histograms.

To further analyze the results for each of the three histogram comparison tasks, percentages of correct responses and the most common incorrect responses were determined. These results are presented in Table 2.

As can be seen in Table 2, students were most successful with identifying similar means and standard deviations for data sets displayed as histograms. Students had somewhat more difficulty determining which histogram was more skewed than the one given for comparison, which will be identified as the “reference histogram” for the remainder of this discussion. A more detailed description of the responses for each question follows.

Comparison of Means

Question 1 asked students to identify the histogram(s) with the same mean as the reference histogram. There were two such histograms out of the six choices given. These were histograms E and F, both of which also showed a greater degree of standard deviation as compared to the reference histogram. This question showed a 77% success rate.

The most common incorrect response to Question 1 was C, accounting for 19, and 51%, of the errors. Histogram C has the exact same shape as the reference histogram, but is shifted to the left on the x-axis, signifying a lower mean value than the reference histogram. In this case, students selected a histogram with the same shape to represent the same mean rather than using the position of the center of the distribution on the x-axis to identify similar means.

Histogram E was one of the two correct responses to Question 1. Five students gave this answer without including F, and thus were counted incorrect. Only two students gave the response F without including E. There are several possible explanations for this result. One is that histogram E shows a lesser degree of standard deviation, and thus is closer in shape to the reference histogram. Therefore, this choice may seem like the more plausible response. A second possibility is that students thought there should be only one response, so they chose the response with the same mean that is most like the reference histogram in shape. Students may have thought that the spread of histogram F was too extreme to also possess an attribute similar to the reference histogram. It is also possible that students may have had a correct understanding of how histograms represent means, but simply missed identifying histogram F after deciding that E was the correct response. Finally, in choosing E or F alone, students may have recognized similar means on the x-axis, but also may have believed that the shape of the distribution should be a factor when comparing means.

Other incorrect responses to Question 1 took a variety of forms showing frequencies of one or two and are not included in Table 2. One of these was the response BDEF, which occurred twice. In this case, all four of these histograms are similar in that they show symmetic distributions. However, the means of B and D are different from those of E and F. The two students who gave this response may be interpreting histograms with similar shapes as having similar means.

Comparison of Standard Deviations

Question 2 asked students to choose the histogram(s) showing the same standard deviation as the reference histogram. The correct response rate was 80% for this question. Of the choices given, only histogram C was positioned farther to the left on the x-axis, indicating a lower mean than was shown on the reference histogram.

Almost 50% of the errors for Question 2 were the responses EF or E. Ten students chose both E and F, while four students chose E alone. Students who responded with EF or E selected one or both of the histograms with a mean similar to the reference histogram rather than the one histogram with a similar standard deviation (and a lower mean). These students appeared to interpret the position of the center of the data on the x-axis as an indication of the standard deviation rather than using the shape of the distribution to estimate the relative standard deviation. The incorrect responses EF and E suggest that students were confused about how the mean and standard deviation are reflected in the shape of the histogram and its position along the x-axis. Essentially, students identified histograms with similar means rather than similar standard deviations as compared to the reference histogram. There were also several other types of incorrect responses to Question 2 that occurred with frequencies of one or two, and three students left this question blank.

Comparison of Responses to Questions 1 and 2

Some students’ incorrect responses to either Question 1 or 2 conflicted directly with their correct response to the other question. For example, thirteen students incorrectly chose histogram C as their response for Question 1, while correctly choosing C for Question 2 as well. These students either did not notice or were not bothered by the conflict created by choosing the same histogram as the response for both questions. These students did not appear to make a distinction between the way the mean and the standard deviation are represented in histograms. Rather, they chose the histogram that matched the shape of the reference histogram and did not seem to consider the different locations of the centers of the data distributions on the x-axis.

Another group of eight students answered Question 1 correctly with EF, but also gave EF as their response to Question 2. These students appeared to focus on the fact that histograms E and F both have the same center of distribution as the reference histogram. These students may also have been influenced by the scale mark that indicates the center of the x-axis, or by the fact that the centers of all three of these histograms (E, F, and the reference) are the exact center of the x-axis as it is drawn and scaled from 0 to 100.

Finally, a group of five students indicated their interpretation of mean and standard deviation by choosing histogram C for Question 1 and histogram EF for Question 2, when the correct responses for these two questions were directly reversed. This error suggests that these students believed that similar means would be represented by similar shapes of data distributions, while similar standard deviations would be represented by similar centers of data on the x-axis.

Comparison of Skewness

Question 3 asked students to choose the histogram that is more skewed than the reference histogram. The correct response was histogram A, and 67% of students responded correctly. Of all the histograms, only A shows an asymmetric distribution. Histogram A also has a shift in mean to the left as compared to the reference histogram. All other histograms showed symmetric distributions with differing standard deviations.

The response ABCD was the most frequent error for Question 3, appearing 24 times, or as 46% of all errors. In this case, it seems that students had selected the four histograms with means or centers different from the reference histogram rather than the one histogram showing an asymmetric distribution. At the same time, the only two histograms that were not chosen by students making the ABCD error were E and F. The reference histogram and choices E and F all showed symmetric distributions and similar means. The ABCD response may be the result of students viewing a shift in the relative position and center of the histogram on the x-axis as an indication of skewness rather than an indication of a different mean. These students may not be aware that skewness is defined by an asymmetry of distribution, or they may be confused about the relationship between skewness and the position of the mean on the x-axis.

The second and third most frequent errors for Question 3 were responses AD, which occurred nine times, and ABD, which occurred five times. These errors also suggest that students are associating skewness with a change in the relative position of the center of the distribution on the x-axis. Histograms A and D show the most extreme shift to the left in the center of the data as compared to the reference histogram. However, A has an asymmetric distribution while D does not. In this case, students selected both A and D regardless of the fact that the distributions of data in histograms A and D are symmetrically dissimilar. Histograms A, B, and D show both a shift in the center of the data along the x-axis and different degrees of variation in their distributions as compared to the reference histogram. Again, it appears that students were confused by the distinction between the position of the histogram on the x-axis and the degree of asymmetry of the distribution as a measure of skewness. The remaining incorrect responses to Question 3 were quite varied, suggesting that this questions generated a considerable amount of uncertainty overall.

Discussion and Implications

A substantial number of students in this study exhibited misunderstandings about histogram interpretation even after completing an introductory statistics course. Fewer than 50% were able to complete all three interpretation tasks successfully. The task that presented the most difficulty was identifying a skewed distribution.

Our findings add to the existing research examining histogram interpretation by revealing several types of errors that college students made when attempting to extract information about means, standard deviations, and skewness as represented in histograms. The most common error in identifying similar means was choosing histograms with the same shapes, regardless of the position of their centers along the x-axis. Conversely, the most common error in identifying similar standard deviations was choosing histograms with similar means, regardless of the shape of the distributions. And, the most common error in identifying a skewed distribution was choosing histograms possessing different means as compared to the reference histogram, regardless of the degree of symmetry of the distribution.

Previous research suggests that the histogram interpretation tasks employed in this study would be expected to be challenging for many students because these tasks required students to estimate and make comparisons among global interpretations of statistical concepts represented graphically (Curcio, 1987; Hollands & Spence, 1992: Leinhardt et al., 1990; Pereira-Mendoza & Mellor, 1990). It is also likely that other aspects of histograms, such as differences in the scaling and meaning of the x- and y-axes as compared to line and bar graphs, may have contributed to the misinterpretations documented by this study (Dunham & Osborne, 1991; Friel & Bright, 1996; Leinhardt et al., 1990; Simonsen & Teppo, 1999). Factors such as students’ levels of understanding and previous learning about mathematics, graphs, and statistics are likely to be influencing their histogram interpretations as well (Curcio, 1987; Fisher, 1992).

In order to assess students’ abilities to interpret statistical features of histograms, it is important to recognize that some of their difficulties may be attributed to factors related to the general interpretation of graphs in addition to those ideas related specifically to the concepts taught in the statistics course. Therefore, we cannot determine to what extent the misinterpretations uncovered in this study are related to problems with graph interpretation in general, difficulties understanding the concepts represented by the x- and y-axes of histograms, differences in visual perceptions (Carpenter & Shah, 1998; Friel et al., 2001), other aspects of prior learning, or additional factors. For example, Leinhardt et al. (1990) have pointed out that students often allow their intuitions to influence their interpretations of graphs. One interpretive error that may underlie a number of the incorrect responses is related to the scale mark located midway between 0 and 100 on the x-axis of the histograms. This mark might reinforce or correspond with students’ intuition that the mean should be located visually in the “middle” of the graph. A few student commented about having this perception when discussing these histogram interpretation tasks with their instructor after completing them.

A number of researchers have noted that interpretation of statistical information may be influenced by the context of the task (Carpenter & Shah, 1998; Friel et al., 2001; Gal, 1998; Lajoie, 1999). Although context can make the task more meaningful for students, it is also possible that misunderstandings about contextual settings can have a detrimental influence on the interpretation of the statistical or graphical information (Friel, et al., 2001). For this reason, we decided to present the histogram interpretation tasks without situational contexts or numerical values for the means and standard deviations. Our goal was to gather information about students’ notions of mean, standard deviation, and skewness as represented in histograms without including context as a task variable. It may be that context contributes to histogram interpretation, either positively or negatively, but we decided not to focus on this aspect of investigation in this particular research project. Setting similar histogram interpretations tasks in different contexts could very well produce different results, and this is a line of research worth pursuing.

This study reports on data collected from students’ written responses to a type of “multiple-choice” testing instrument, and students’ reasons for their responses were not documented on the testing instrument. However, in subsequent informal conversations with their instructor, several students stated that they viewed skewness as a shift to the left from the midpoint on the x-axis, and also that they considered the scale mark as the mean or midpoint of the data even though the data spread, as indicated by the bars on the histogram, did not extend from 1 to 100. Further research is needed to explain the reasons for the histogram interpretation errors documented by this study and to interpret the conflicting responses some students gave for Questions 1 and 2.

Establishing appropriate connections between students’ understandings of the statistical concepts of mean, standard deviation, and normality or asymmetry of data distribution and how these are represented graphically has always been an important instructional goal. As a result of the insights we gained from conducting this research, we have made changes in our instructional practice to focus more specifically on addressing the areas of difficulty we identified. Among these changes are spending more time making comparisons among the numeric values of descriptive statistics and the corresponding graphic representations, placing emphasis on the meanings and scales of the x- and y-axes of histograms and how these differ from bar graphs and functional Cartesian graphs (Bright & Friel, 1998), and focusing more attention on the definition of skewness and the relative nature of asymmetric distributions. Instructional activities that require students to make estimates and comparisons related to the global aspects of histogram interpretation should assist students in developing these skills and encourage them to use these skills more consciously in their work with histograms. Further assessment to evaluate the effectiveness of these practices is ongoing.

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Susan S. Gray

University of New England

Cary Moskovitz

Duke University

(1) In existing research, the terms “comprehension” and “interpretation” are typically used to refer to the process of using graphs to extract information and answer questions about that information. For the current

paper, the interpretation of histograms refers to the extraction and expression of satistical meanings from histograms.

Table 1. Score Frequencies and Percentages

Score Frequency Percent of Sample

0 4 2%

1 28 18%

2 53 33%

3 74 47%

Table 2. Frequencies and Percentages of Correct and Incorrect Responses

Question 1 Question 2 Question 3

Mean Standard Deviation Skewness

Correct Response EF C A

Number Correct 122 127 106

Percent Correct 77% 80% 67%

Incorrect Response C EF ABCD

Number of Errors 19 10 24

Percent of Errors 51%* 31%* 46%*

Incorrect Response E E AD or ABD

Number of Errors 5 4 14

Percent of Errors 14%* 13%* 27%*

*Percentage of errors do not total to 100% in each column because only

the most commonly occurring incorrect responses are included.

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COPYRIGHT 2008 Gale, Cengage Learning