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Parts (a) and (b) Use categorical data that have been summarized in a bar graph to answer questions about the data distribution of a categorical variable

Most students provided correct responses to parts (a) and (b) of this question. Responses that were not scored as essentially correct generally made one of three common mistakes.

Some students did not combine the choices made by girls and the choices made by boys in order to determine the most popular and least popular choices for the class as a whole. Often these students provided separate assessments for boys and for girls or only addressed one gender in their response. For example, consider the following three student responses. Each of these responses describes the most or least popular for girls and for boys, but does not combine the given information to make a statement about the class. These responses were scored as partially correct for parts (a) or (b).

The student response below is representative of those that did not answer the question in terms of the class and also did not address both genders in the response. This student response only answers the question for girls and also does not provided the requested justification. This student response was scored as incorrect for both parts (a) and (b).

Other students provided a justification that was judged to be weak or that did not make it clear that the answer was about the entire class. For example, the following two student responses were scored as only partially correct. In the first response (part (a)), the justification for Thin Mints in part (a) is “Thin mints because it has the largest bar.” In this response, it is not clear that the choices for girls and the choices for boys were combined. In the second student response (parts (a) and (b)), the justifications only states that the selected cookie type was “the most chosen option” or “the least chosen option.” As with the first response, there is not a clear indication that the counts for the girls and the counts for the boys were combined.   

To be considered essentially correct for either part (a) or part (b), the justification needs to make it clear that the counts for girls and the counts for boys were combined in order to reach an answer based on data for the entire class. This was sometimes a difficult call to make when scoring this question. For example, consider the following two responses to part (a) that are quite similar. The first was scored as essentially correct for part (a) because of the last part of the answer: “so altogether they were the most popular.” The second was scored as only partially correct, because it does not make it clear that information from the girls and from the boys was combined in order to reach an answer.

Finally, some students were not able to use the information provided to make a correct determination of the most and least popular types of cookies for the class. For example, consider the following three student responses below. In the first response, which is a response to part (a), the student is not able to determine which type of cookie is most popular and indicates that all three types were the most popular. The second and third responses below are responses to part (b) and both responses incorrectly identify Tagalongs as the least popular type of cookie.

Part (c)  Recognize and describe variability in the distribution of a categorical variable

Many students struggled with part (c) of this question and clearly had difficulty in thinking about the concept of variability in the context of categorical data. Many student responses received a score of incorrect for part (c).

By far the most common error was for the student to interpret the frequencies (heights of the bars) as if they were numerical data values and to try to describe variability in the frequencies rather than variability in the actual data. This represents a conceptual error in thinking about the underlying data, which is categorical. This is illustrated in the following five student responses. Each of these responses make it clear that the student is looking at the category counts and think that the responses for boys are more consistent because the counts for the three different cookie types are quite similar. But for categorical data, this is actually an indication of variability in the data. For a data on a categorical variable, the least amount of variability possible occurs when all of the responses are in the same category.

A different type of student error when responding to part (c) is illustrated by the following student response. In this response the student interpreted the question as one that asked for a comparison of the response distributions for boys and girls. This response was scored as incorrect for part (c) because it does not address variability in the data.

Some students made correct observations about the choices made by girls and the choices made by boys, but never answered the question about whether boys or girls were more consistent in their choices. This is illustrated by the following student response.

There were also a number of student responses that were scored as only partially correct because the justification provided was incomplete. For example, the following student response was scored as partially correct because the justification provided does not include any reference to the boys.