The New York Yankees have had the highest payroll in Major League Baseball for sixteen years. (Sixteen! This year, the Los Angeles Dodgers finally unseated the Yankees in that distinction.) Naturally, the assumption is that spending all that money on talent will translate to wins, and wins will translate to revenue for the franchise. And sure, the Yankees played in the World Series in six out of those sixteen years. But looking at the results for one highly-paid team can only tell us so much. Are wins correlated with team payroll, if we look at some more data?
Um... sorta? The scatter plot shows each team's regular season wins in the most recent full season vs its payroll for that season. Wins are most closely correlated with payroll in the NBA, with r = 0.535. Payroll also seems to predict the highest gain in wins for the NBA; that trendline's slope shows that an NBA team can expect 0.688 more wins for every extra million dollars they spend. Major League Baseball has the weakest correlation with r = 0.314, and also the shallowest slope; baseball owners can only expect 0.084 more wins for every million spent on player salaries. (Consider these comparisons in context, as each league plays a different number of regular season games. One more win in the NFL, for example, means much more than one more win in MLB, where teams play far more games in one season.)
It might be more useful to talk in terms of how much it would cost for one extra win, which would just be the reciprocals of those values. An NBA team can expect to spend 1/0.688 = $1.45 million to bump their W column by one, and an MLB team should expect to spend 1/0.084 = $11.9 million for one measly win.
Several of these leagues (NHL, NBA, and NFL) have a salary cap in place, which is meant to prevent wealthier teams from hoarding all the best players. It's in each league's interest that games and seasons are reasonably competitive... games would be no fun to watch if the winners were essentially predetermined! If the concerns that lead to salary caps were justified, we'd expect to see a strong correlation between wins and payroll with no cap in place, and a weaker correlation in leagues with salary caps. Interestingly, we are looking at the opposite outcome here. The league with no salary cap (MLB) has the weakest correlation! Even though this is only one season's worth of data, it is not compelling evidence for the argument that salary caps are effective.
Of course, the goal of major league franchise owners isn't necessarily to win games. Owners also want to make money. Implicit in our original question was the assumption that wins are valuable, and to an owner, "value" means they result in a team making more money. But are wins actually correlated with revenue? Here, for instance, is the relationship between revenue and regular season wins for each team in the NFL in 2012-2013:
This correlation is terrible! The equation on this bad boy is y = 0.154x + 285.24 with a correlation coefficient of r = 0.008. Apparently in the NFL, how much money a team makes has very little to do with how many games it wins.
Throughout all of this, it's important to keep in mind that correlation tells us there is a particular kind of association (or not) between variables, and it can tell us something about the strength of that association, but it doesn't say anything about whether a change in one variable actually causes a change in the other. Even if there were a strong correlation between revenue and wins (which there isn't), that wouldn't necessarily mean that more wins will necessarily lead to more revenue.
There are many more factors that contribute to a team's success than what we've looked at here! Is your high school math class full of sports fans? Particularly ones who are bitter about how much the Yankees can spend on payroll? They might enjoy this conversation and apply some sophisticated statistical ideas in the process. Check out our new lesson, Win at Any Cost.