- Ted Fraser

# The Surprising Relationship Between Spending and Success

As part of the IB SL Math curriculum, each student is required to do a “Math Exploration”. The goal of the assignment is to “give IB students the opportunity to appreciate a wider range of mathematics, as well as applying mathematical concepts to real life situations.”

I settled on the question of “Can we predict the success of a team solely from their spending?” I defined success as a team’s league-wide placement. I decided to examine this question because it involved topics that interest me – sports and economics – all while using relatively rudimentary math – Pearson’s correlation coefficient and linear regression.

I decided to use data from two leagues, Major League Baseball (MLB) and the National Hockey League (NHL). The reason for this is that the MLB has no salary cap (a maximum amount of money a team can spend on its players’ salaries) and the NHL has a salary cap.

This ensured that a broad, diverse range of data was examined. Additionally, the purpose of a salary cap is to eliminate any kind of financial advantage, so I presumed the relation in the MLB, where a team can spend as much as they possibly can, would be more pronounced.

The results of the investigation were extremely interesting. As established through Pearson’s correlation coefficient, the correlation between a team’s spending and their league-wide placement in the MLB was -0.39, while in the NHL it was -0.409. This suggests that, in both cases, there is a weak, yet indisputable, correlation between success and spending. (The reason it's a **negative** correlation is that in this case teams want to place* lower*, not higher, thus creating a negative correlation).

This isn’t that big of a surprise; you would think that the LA Dodgers, who spent $243,000,000 in 2014, might fare better than the Miami Marlins, who spent roughly $55,000,000. Yet, the correlation between the two variables -- team salary expenditure and league-wide placement -- is classified as “weak” (that is, a relation between -0.11 and -0.5). Additionally, there was actually a* greater* correlation in the NHL, where there’s a limit put in place to prevent the financial advantages discovered in this investigation.

Using linear regression, formulas for each league were developed to observe if it was possible to accurately predict where a team would place league-wide, solely using their salary expenditures. For the MLB, I used the salary expenditures and placement of my favourite team, the Toronto Blue Jays; they placed 14th and spent $99,413,019. After plugging and chugging, the equation predicted that the Jays would place 17th (16.98 was actual number).

What this means is that based on the Blue Jays’ salary expenditures, they were projected to place 17th league-wide. In reality, they placed 14th. For the NHL, I used my beloved Boston Bruins. They placed a dismal 17th league-wide and spent $61,319,470. After inputting these numbers, an answer of 17.345, or 17th place, was produced.

Using only how much the Bruins spent *before the season even began*, the formula was able to predict *almost exactly* how well they would perform.

The data seems to support yet also challenge traditional thinking. It challenges in that the effects of a team’s expenditures in relation to season success are observable. However, the two correlation coefficients displayed “negative, weak” correlations, as described by the IB Mathematics curriculum. For one to have any faith in the findings or attribute any sort of verdict, a moderate correlation is needed at the absolute very least (0.5 < r < 0.87). Yet, after using linear regression, it was revealed that, although not perfectly, using solely a team’s salary expenditures, one could estimate where they would place league-wide. This seems to suggest that, contrary to the weak correlation coefficient, salary expenditures have some kind of observable (although not significant) effect.

These findings present questions we should consider. For example, should teams who spend the most be penalized in the form of a point deduction at season’s end? Is there a point of even having a salary cap if the correlation between spending and success is the same, no matter what league you’re in? To make sports fairer, should all teams only be allowed to spend the exact same amount of money, in the process ensuring increased competition?

All of these questions and observations stemmed from challenging one of the most oft-quoted “truisms” in sports: the more a team spends, the better they’ll be. Or so we think.