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Part (a) Understand the mean as the balance point of a data distribution.

Students struggled with part (a) and not many student responses demonstrated an understanding of the mean as the balance point of a data distribution. Of those students who did try to get at the idea of the mean as a balance point, many provided only a weak explanation that did not directly appeal to deviations from the mean or distances from the mean. The following two student responses are responses that illustrate such a description.

Many students did not really address the idea of balance point at all and just used the data presented in the dot plot to verify that the mean was equal to 3, as illustrated in the two responses that follow.

Another common mistake was to focus on the effect of outliers on the sample mean rather than on the idea of mean as a balance point of a data distribution. The student response below is typical of responses that made this error.

Part (b)  Interpret the mean absolute deviation (MAD) in context

There were two common errors in answering part (b). Some students did not realize that interpretations should always be in context. This resulted in responses that were scored as only partially correct, even though they may have included a correct generic interpretation of the MAD. Each of the following three responses were scored as partially correct because of this error, but could have been scored essentially correct if they had been in context.

Many students indicated a lack of understanding of the MAD as a measure of variability and provided incorrect interpretations of the MAD in their responses. Each of the four student responses below illustrates a different incorrect interpretation of the MAD. These incorrect interpretations indicate that students did not have an idea of what the MAD measures and that they may not have worked with the MAD as a measure of variability. This may improve with the inclusion of the MAD as part of the 6th grade curriculum under the Common Core.

Part (c)  Compare the variability in two data distributions given dot plots of the two distributions

In part (c) students needed to assess the variability in two data sets that have been displayed in dot plots, indicate which data set had the greater variability, and justify their choice. There were two common errors that students made in part (c) that resulted in scores of partially correct or incorrect for this part.

One common error was made by students who correctly chose Team B as the data set with the greater variability, but provided a weak explanation for the choice. The student response below is a response that was scored as only partially correct because the explanation was considered weak.

The second common student error was to select Team A as the data set that exhibited the most variability. This error could result in a score of partially correct or incorrect, depending on the explanation provided. The student response below is an example of one that received a score of incorrect for part (c). The student has selected Team A and the justification for the choice is incorrect. The range is equal to 8 for both data sets, and the statement about the ratio of number of goals addresses center rather than spread in the data sets.

There was one way that a student could justify a choice of Team A and receive a score of partially correct for part (c). In each of the two student responses that follow, the student supports the choice of Team A by appealing to the fact that Team A scored more different numbers of goals (0,1,2,3,5,6,8 for Team A and 0,1,4,5,7,8 for Team B). Responses that were based on this justification for selecting Team A were scored as partially correct for part (c).