> In the 1980s, the Reagan administration seized on a report called A Nation at Risk, which claimed that the US was on the verge of collapse due to its falling SAT scores.
Suppose that low-income individuals start to take the SAT in 1980 whereas they didn't in 1970. The wrong way to analyze SAT scores is to evaluate:
sum over cohorts P(SAT Score | cohort, Y)
where Y is the year. For instance, you might compare the total average score in 1980 vs. 1970. Doing so will show a decrease in SAT score because of the increase in low-income individuals taking the SAT, not because the high-income individuals are doing worse. (This assumes that low-income people have less access to SAT training materials, and those training materials affect the score).
The correct way is to only compare scores within a cohort:
P(SAT Score | cohort, 1980) > P(SAT Score | cohort, 1970)
That is, did the same cohort do better in 1980 vs. 1970?
(There might still be some differences between the cohorts in 1980 vs. 1970. Maybe the low-income individuals who took it in 1970 had high confidence in school, whereas the 1980s kids were from a broader background.)