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Part (a): Calculate and compare differences in medians.

Responses that were not scored as essentially correct for part (a) generally made one of two common mistakes. The most common student error was made by students who calculated the differences in medians correctly, but then failed to comment on the differences between the two grades. Most of these students commented instead on the differences between boys and girls, which did not address the question asked in part (iii) of part (a). This error is illustrated by the following two student responses. Both of these responses were scored as partially correct for part (a).

Some students either did not actually calculate the difference in medians for each of the grades or made errors in calculation. This is illustrated by the following two student responses. In the first, the difference in medians is not calculated, but the statement made in part (iii) is correct. In the second response, an error is made in the calculation of the difference in medians for 8th graders. The statement in part (iii) is correct based on the calculated differences, so this response was considered to be partially correct for part (a).

Of more concern were student responses that made incorrect statements in part (iii). Such responses were scored as partially correct if the differences in medians was calculated correctly for each grade and incorrect if the difference in medians was not calculated correctly for each grade. For example, consider the following three student responses, each of which makes an incorrect statement in part (iii) that indicates a lack of understanding.

In the response below, the student says that “as you get older you start to use the internet less and less.” This is not supported by the summary measures or the boxplots, which have similar centers for the two grade levels. This indicates that the student is confusing center and variability.

In the following response, the student says that “girls for both grades are similar to each other: and that the same goes for boys". This does not address the difference in medians and like the previous response indicates that the student is focused only on center as the variability is noticeable different for the two grades.

The following response includes an incorrect statement that illustrates another misconception. In this response, the student indicates that the times for 7th graders and the times for 8th graders have the same distribution. Here the student believes that if the medians are the same, the distribution must be the same, but this is clearly not the case as the two distributions differ with respect to variability.

Parts (b): Use boxplots and summary statistics to compare the variability in two or more data distributions.

Students who did not receive a score of essentially correct for part (b) generally made one of three common student errors. The most common error was made by students who did not use a measure of variability, such as the range or the interquartile range (IQR), to justify statements about how the variability in times of 7th graders compared to the variability in times of 8th graders. This is illustrated by the following student responses.

A second common student error in answering part (b) was not recognizing that the question asks about variability. The following student response is typical of those making this error.

A more serious student error in thinking is illustrated by the following student response. This student appears to think that greater variability indicates that 7th graders use the internet more, even though the medians for the two grades are equal.

Part (c): Informally assess the degree of overlap of two numerical data distributions with similar variability. Assess the strength of evidence for a difference in medians in a way that takes variability into account.

Part (c) was designed to address two of the more challenging standards in the Common Core State Standards (7.SP.3 and 7.SP.4). Developing an understanding that informal inferences about differences in populations need to take variability in the data distributions into account presents a challenge for teachers who have not previously had to include statistics in their curriculum.

Students generally found part (c) difficult to answer. Many students reached an incorrect conclusion, and some who did reach a correct conclusion provided an explanation that included incorrect statements.

For example, consider the following two student responses. These students are have chosen 7th grade girls and justify this choice based on the fact that this group includes the greatest Internet use times. These students do not compare the strength of evidence for the two grades and confuse strength of evidence with the maximum. 

The following response makes a similar error in reasoning, selecting 7th grade because the upper quartiles and maximums are higher for the 7th grade distributions.  

Other student responses included incorrect statements, as illustrated by the following three student responses. In the first response, the student states that the 8th grade boxplot is more centrally located around 160. However the box plot for 7th grade girls is also centered at 160. In the second response below, the statement that 8th grade girls use more internet than 8th grade boys because there are no outliers is not correct. Finally, in the third response below, the statement that the difference between boys and girls is bigger for the 8th grade is not accurate because the difference in medians is the same for both grades (and this student computed these differences correctly in part (a)).

Responses that did not include an explanation were scored as incorrect, even if they stated that the evidence was stronger for 8th graders. This error is illustrated by the following student response, which was scored as incorrect for part (c).