the average tenth-grade test score increased for a decade after the introduction of reforms that have no effect on growth in student achievement. These results are admittedly somewhat counterintuitive. They arise from the fact that tenth-grade achievement is the product of gains in achievement accumulated over a 10-year period. The average tenth-grade test score is, in fact, exactly equal to a 10-year moving average of school performance. This stems from the simple assumption that school performance is identical at different grade levels in the same year. The noise introduced by this type of aggregation is inevitable if school performance is at all variable over time.

The problem of aggregation of information that is grossly out of date also introduces noise into the comparisons of different schools at the same point in time. The degree to which noise of this type affects the relative rankings of school depends on whether the variance over time in average achievement growth is large relative to the variance across schools in achievement growth. To illustrate this point, Figure 10.5 considers the consequences of aggregation over time and grade levels for two schools that are identical in terms of school performance in the long-term. In the short-term, however, school performance is assumed to vary cyclically. For school 1, performance alternates between 10 years of gradual decline and 10 years of gradual recovery. For school 2, performance alternates between 10 years of gradual improvement and 10 years of gradual decline. These patterns are depicted in Figure 10.5(b). The correct ranking of schools, based on school performance, is noted in the graph. Figure 10.5(a) depicts the associated levels of average tenth-grade achievement for the two schools. The rankings of schools based on this indicator are also noted. The striking aspect of Figure 10.5 is that the average tenth-grade test score ranks the two schools correctly only 50 percent of the time. In short, the noise introduced by aggregation over time and grade levels is particularly troublesome if comparing schools that are roughly comparable in terms of long-term performance. On the other hand, the problem is less serious for schools that differ dramatically in terms of long-term average performance. It is also less serious if cycles of decline and improvement are perfectly correlated across schools—an unlikely phenomenon.

Example Based on Data from the National Assessment of Educational Progress

To consider whether the average test score exhibits these problems in real-world data, consider average mathematics scores from 1973 to 1986 from the National Assessment of Educational Progress (NAEP) (see Table 10.2). Unfortunately, the NAEP is not structured in such a way that it is possible to construct a value-added measure of school performance,25 so we compare average test scores with the simple average growth in achievement from one test period to the next for the same cohort of students. This measure is typically referred to as a gain

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