Here is the approach they take: " Our primary source of data is University of Oregon student transcripts, covering all undergraduate classes administered from fall quarter of 1999 through winter
quarter of 2007. ... We combine these data with readily available reports of the football team’s win-loss records ... Over our sample period, the winning percentage is 69.7 percent, on average, and varies from 45.5 percent to 90.9 percent." Because the researchers have data on individual students, they can make a statistical comparison of how the grade point average for an individual student changes from year to year, and see if it is correlated with the winning percentage of the football team. They can also do a number of other calculations, like adjusting for a time trend so that grade inflation is taken into account, as well as looking at how responses differ by gender, by income level (measured by which students are receiving financial aid), and by test scores before entering the university.
There's one additional element of complexity here: In most college classes, grades are given according to some explicit or implicit "curve": that is, even if the academic performance of all students was worse one fall in absolute terms, if a certain percentage of students get As, Bs, Cs, and so on, then grade point average might not show the drop in absolute level of performance. This suggests two kinds of comparisons: 1) male students in their data are more likely to watch football than female students, so one can look at the how the grade gap between male and female students is related to the winning percentage of the football team; and 2) one can compare fall grades when the football team is playing to winter/spring term grades. Lindo, Swensen, and Waddell summarize their results this way:
"That is, our preferred estimates are based on considering how a student’s grades deviates from his or her own average grades as the winning percentage varies from its average, and then how this response varies across gender. With our analysis we show that male grades fall significantly with the success of the football team, both in absolute terms and relative to females. There is also pronounced heterogeneity among students, suggesting that the impact is largest among students from relatively disadvantaged backgrounds and those of relatively low ability. ...After comparing fall and winter academic terms, "only in the quarter we associate with football—the fall quarter—is there movement in the gender gap in academic performance that varies systematically with athletic success."
"Relative to females, males report being more likely to increase alcohol consumption, decrease studying, and increase partying around the success of the football team. Yet, both male and female students report that their behavior is responsive to athletic success. This suggests that female performance is likely affected by the performance of the football team as well, but that this effect is masked by grade curving. ... [A] 25 percentage point increase in the football team’s winning percentage will increase the gender gap in GPAs ... by 8.5 percent."
I'll spare you a homily on the "true purpose" of higher education, and the extent to which big-time sports supports or undermines that purpose. But for those who hold the misapprehension that college sports provide a financial subsidy to the academic programs of these institutions, Lindo, Swensen, and Waddell toss out one cold fact: "In 2010, 211 out of 218 Division I athletics departments at universities subject to open records laws received a subsidy from their student body or general fund. These subsidies are substantial and rapidly growing. From 2006 to 2010, the average subsidy increased 25 percent, to $9 million."