EXAMPLE 3 SAT Mathematics Scores by State

Each year, more than 1 million high school seniors take the SAT standardized test, which has three parts: Mathematics, Critical Reading, and Writing. We sometimes see individual states rated or ranked by the average SAT scores of their high school seniors. However, this is misleading because the mean SAT score is explained largely by what percentage of a state’s graduating seniors take the SAT. For example, the scatterplot in Figure 6.5 shows a negative association between the mean score on the Mathematics section and the percentage of test-takers for the class of 2013. Each dot represents a particular state.

The form of Figure 6.5 is a bit irregular, but there are two distinct clusters of states. In each state in the lower-right cluster, a majority or near-majority of high school graduating seniors take the SAT, and the mean scores are low. In the upper-left cluster’s states, 35% or fewer of seniors take the SAT—and these states have higher mean scores. Clusters in a graph suggest that the data describe several distinct kinds of individuals, and the two clusters in Figure 6.5 indeed describe two distinct sets of states.

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There are two common college entrance examinations, the SAT and the ACT, and each state tends to prefer one or the other. In ACT-dominant states (the left cluster in Figure 6.5, where a smaller fraction of those states’ seniors take the SAT), most students who do take the SAT are applying to selective, out-of-state colleges. This select group performs well. In SAT-dominant states (the right cluster), a higher percentage of seniors take the SAT, and this broader group has a lower mean score.

The relationship in Figure 6.5 also has a clear direction: States in which a higher percentage of students take the SAT tend to have lower mean scores. This is true both between the clusters and within each cluster. That is, there is a negative association between the two variables.

There are no clear outliers in Figure 6.5, but each cluster does include at least one state whose mean SAT Mathematics score is lower than we would expect from the percentage of its students who take the SAT. In the cluster of ACT-dominant states, this occurs with West Virginia (WV). In the cluster of SAT-dominant states, this occurs with the District of Columbia (DC)—which is actually a federal district, not a state— and Maine (ME), Delaware (DE), and Iowa (IA).