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Part (a): Calculate the median and quartiles for a given data set.

The majority of students were able to correctly calculate the median and the quartiles for the given data set. Responses that were not scored as essentially correct could be classified as falling into three categories: (1) those that demonstrated an understanding of what quartiles were but made an error in the calculation of one or both quartiles; (2) those that confused the quartiles with the minimum and maximum of the data set; and (3) those that did not know what quartiles are or how to calculate the quartiles of a data set.

The following response is typical of those falling into the first category. This student made an error in calculating the upper quartile and part (a) was considered to be only partially correct. However, part (b) for this student was scored as essentially correct, even though the upper end of the box was placed at 94 rather than 96 because this was consistent with the student’s answer to part (a). 

Some students also confused Q1and Q3. This is illustrated in the following student response, which was scored as partially correct for part (a). Students making this mistake were generally able to produce a correct boxplot in part (b). Note that this student has also made a minor error in calculating the upper quartile to be 97 rather than 96 or 96.5.

A surprising number of responses fell into the second category described above and reported the values of the minimum and maximum for the quartiles. Students making this mistake generally had difficulty producing a reasonable boxplot in part (b).

Finally, it was clear that some students did not know how to calculate quartiles, and so were only able to calculate the median in part (a). While such responses were considered partially correct for part (a), students in this situation were rarely able to produce a reasonable boxplot in part (b) and so tended not to score well on the other parts of this question.

Part (b): Construct a box plot.

Responses that were not scored as essentially correct for part (a) generally made one of three common mistakes, two of which indicate a lack of understanding of box plots and how the minimum, lower quartile, median, upper quartile and maximum are used to construct a box plot.

Some students just made minor errors in representing the values in the five number summary using the scale provided. This was not considered a serious error, as these students demonstrated an understanding of how a boxplot is constructed. This is illustrated in the following student response, which was scored as partially correct for part (b) because the placement of the median, upper quartile and maximum are not correct.

The following response illustrates partial understanding of how a boxplot is constructed. In this response the placement of the box (representing the lower quartile, the median and the upper quartile) is correct. However in adding in the “whiskers” the student did not use the minimum and maximum from the given data set and instead extended the whiskers to the minimum and the maximum for the other boxplot that was shown.

Finally, a number of students mimicked the look of the boxplot that was given for African countries and drew something that looked similar but that was unrelated to values of the five number summary for the South American data set. The following student response is typical of those that fall into this category. This student did not calculate the quartiles correctly in part (a), but the boxplot is not even consistent with the values given in part (a). This student response was scored as incorrect for part (b).

Part (c): Use information provided by box plots to compare two data distributions.

Students who were able to produce a boxplot in part (b) were generally able to provide good answers to part (c). To be considered essentially correct, responses needed to provide two different comparative statements about the distribution of percent of the population with access to clean water. Some students overlooked the request for two statements and only provided a single comparison. This is illustrated in the following student response, which was considered as partially correct for part (c).

Other students actually wrote two statements, but the statements were really making the same comparison. This is illustrated in the following two student responses. In the first response, the statement “South America has more access…” is the same as the statement “Africa has less access…”  In the second response, both statements are saying that there is more variability in the data for African countries. Both of these student responses were scored as partially correct for part (c) because they did not provide two comparisons that were different.

In order to be considered essentially correct for part (c), the statements made needed to be comparative in nature. Some students made statements based on the boxplots but did not involve a comparison of the two distributions. For example, the following response was scored as only partially correct for part (c) because only the first statement is one that compares the two distributions. The second statement is only about the data for African countries.

Finally, some students made statements that are not correct—or at least are not statements that can be justified by the given data and boxplots. The following two responses illustrate responses making this error. In the first response, while the second statement was considered as a correct comparison, the first statement does not follow from the given data and boxplots. In the second response, the statement given also does not follow from the given data.