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Parts (a) and (b) Represent patterns of association in bivariate numerical data using a scatterplot; distinguish between positive and negative relationships in bivariate numerical data; distinguish between weak, moderate and strong relationships in bivariate numerical data

Responses that were not scored as essentially correct for parts (a) and (b) generally made one of five common mistakes. Some students did not know how to construct a scatterplot and instead tried to create some other type of graphical display. Some students created displays that are appropriate for univariate data (such as a histogram or a dot plot) rather than a display that is appropriate for bivariate data. Because of the univariate nature of these graphical displays, they were unable to represent the type of relationship between two variable stated in the question. This is illustrated in the following three student responses to part (a), each of which was scored as incorrect.

Some students did try to convey that there were two variables, but did not successfully crate a scatterplot. While the two student responses below do convey some understanding of what it means for a relationship to be positive (the first response below, which was a student response to part (a)) and of what it means for a relationship to be negative (the second response below, which was a different student’s response to part (b)), each of these responses was considered incorrect.

A second relatively common student error was incorrectly representing the direction of a relationship. Some students did not realize that for a positive relationship between two variables, larger values of one variable tend to be paired with larger values of the second variable and that for a negative relationship, larger values of one variable tend to be paired with smaller values of the second variable. Some students reversed positive and negative, and provided a scatterplot that exhibited a negative relationship in part (a) where the stated relationship was positive and a positive relationship in part (b) where the stated relationship was negative. Other students appeared to think that the pattern in the scatterplot would always slope upwards in moving from left to right for both positive and negative relationships.

This error is illustrated in the following four student responses. The first scatterplot below is a response to part (a) (which described a moderate positive relationship), but the relationship represented in this scatterplot is clearly a negative relationship. The next three scatterplots were all student responses to part (b), which asked for a representation of a strong negative relationship. Two of these student responses clearly show a positive relationship and the final one shown is more consistent with no relationship than a strong negative relationship.

A third common student misconception is related to how the strength of a relationship is represented in a scatterplot. In constructing a scatterplot in part (a) to represent a moderate positive relationship, many students drew scatterplots that did not include enough scatter around the pattern in the scatterplot, often depicting very strong relationships. Others constructed scatterplots that would best be described as representing a very weak relationship. These responses did not demonstrate a clear understanding of the strength of a relationship. For example, consider the following two student responses to part (a). While each of these two scatterplots represents a positive relationship, the relationship would be described as weak rather than moderate.

The following two student responses to part (a) illustrate the opposite error with respect to strength. The relationships represented in these two scatterplots are very strong, and would not be considered to be moderate positive relationships.

Similarly, in part (b), some students created scatterplots that would not be considered to represent a strong relationship. For example, the following student response to part (b) was considered to be incorrect because the direction of the relationship is not correct and the scatter of the points in the scatterplot is not consistent with a strong relationship.

A final type of misunderstanding related to representing strength of relationships is illustrated in the following student response to parts (a) and (b). In these responses, the scatterplot in part (a) (the moderate relationship) shows a stronger relationship than the one shown in part (b) (the strong relationship). This student also makes the mistake of failing to include axis labels.

The fourth type of student error that was fairly common in responses to all parts of this question was to create scatterplots that included more than or fewer than 10 data points. This is illustrated in the following two student responses to part (b). This error is most likely the result of failure to read the question carefully than of a more fundamental misunderstanding of scatterplots.

The fifth, and by far the most common student error was failure to include labels on the axes. This error is illustrated in the sample student response above. While this does not represent a conceptual misunderstanding, it does indicate that the student does not understand the importance of always including labels and scales on the axes in graphical displays.

Part (c) Understand that when there is no relationship between two numerical variables, there will be no apparent pattern in a scatterplot of the data

In answering part (c), the only common student error (other than the omission of axis labels) was to create a scatterplot in which there was a notable linear or nonlinear pattern. These students did not recognize that if there is no relationship between two numerical variables, there should be no discernible pattern in the scatterplot of the data. For example, consider the following two responses to part (c). Both of these scatterplots show a pattern and neither is compatible with the description of no relationship. The second plot below also makes the error of not labeling the axes.