WIN AT ANY COST? How should sports teams spend their m oney to win more games?

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1 Mathalicious 2014 lesson guide WIN AT ANY COST? How should sports teams spend their m oney to win more games? Professional sports teams drop serious cash to try and secure the very best talent, and the dough can really pile up. But is all that money well spent? Conventional wisdom says that teams with higher payrolls ought to perform better than those with more modest means. Some leagues have even instituted limits on spending in order to make games more competitive. But do higher- spending teams really do better, and do salary caps actually level the playing field? In this lesson, students look at data for four major pro sports leagues and try to answer the question: Can you buy wins? Primary Objectives Use technology to calculate linear regressions for wins versus team salary for major professional sports Compare relative strengths of linear associations using correlation coefficients Interpret the slope of a linear regression function in context Discuss the effects of salary caps on the relationship between wins and team salary Given regression functions and correlation coefficients, examine the relationship between wins and several other variables Discuss the difference between correlation and causation Content Standards (CCSS) Statistics & Probability ID.6, ID.7, ID.8, ID.9 Mathematical Practices (CCMP) Materials MP.3, MP.5 Student handout LCD projector Computer speakers Graphing technology Before Beginning This lesson focuses on interpreting linear regression functions and correlation coefficients in the context of sports salaries. Students should be able to use technology to compute correlation data for two quantitative variables.

2 2 Preview & Guiding Questions Begin by showing the clip from a 2005 episode of 60 Minutes where Andy Rooney talks about the disparity between the amount of money different Major League Baseball teams spend on player salaries and how well they actually perform. For instance the Yankees who traditionally have a very large budget didn t do as well as many people expected that season, especially with all the talent they could afford. And some other teams with much leaner payrolls performed pretty well. People love to get upset about organizations like the Yankees because they think it s unfair for a team to buy wins by simply outspending competitors. Major League Baseball has even instituted a salary cap to try and curb that problem. But is it even a problem? The goal of the Preview discussion is to get students thinking about the question implicit in Andy Rooney s analysis of the 2005 MLB season: Does spending more money on players actually result in more wins? This is what students will examine in detail over the course of the lesson by looking at some recent data across major professional sports leagues in the U.S. Why do people love to hate the Yankees, according to Andy Rooney? Does it seem like they have an unfair advantage? Why or why not? Do you think teams with large payrolls tend to do much better than other teams? Does the information in the video seem to support your claim? Why or why not? What kind of information would you like to have in order to make a better judgment? Act One In Act One students are given single- season data for the total payroll and number of wins for each team in the four major professional sports to see whether there is a relationship between the amount of money a team spends on players and the number of games it wins. They calculate linear regression functions and correlation coefficients for each league and figure out how much additional money spent on players corresponds to a single additional win. Finally, students discuss the proposition that salary caps limit the effect of spending on wins and whether or not the data supports that claim. Act Two In Act Two, students consider how other variables are related to victories. They examine how the salary of a team s highest paid player, previous performance, and total revenue are correlated with number of games won. They use correlation coefficients to compare the relative strengths of each explanatory variable and discuss which factor(s) likely have the greatest impact on winning. To close the lesson, students discuss the difference between correlation and causation the idea that a pattern of association isn t enough to establish that a change in one variable necessarily causes a change in the other and what that might imply for team owners trying to boost their records.

3 use 3 Act One: Hey, Big Spender 1 Many people assume that the more a team spends on players, the more it will win. The graph below shows the wins vs. total payroll data for MLB, NBA, NFL, and NHL teams. Pick a league, and calculate the linear regression equation, as well as the correlation coefficient, r. (For the remaining leagues, gather this from your classmates.) Based on this, would you conclude that total payroll is strongly related to the number of wins? Explain. Regular Season Wins NFL NBA MLB NHL Team Payroll (millions). MLB NBA NFL NHL Lin. Reg. y = 0.084x y = 0.688x 4.89 y = 0.103x 2.96 y = 0.276x r r = r = r = r = Students are given the tabular data they need in order to run the linear regression, but it s always a good idea to plot the data first. After all, the shape of a plot can be an important indicator of what type of function is likely to fit the data. In this case, the plots alone give some important information about what the regression results should look like. For instance, all of the best- fit lines have positive slope, so the correlation coefficients should all be positive as well. It s important for students to understand that the correlation coefficient and the slope of the best- fit line must have the same sign, but their magnitudes aren t related. The r- value only tells you about the strength of the linear association, not the particular details of the association. Students may notice that the NBA s plot has both the highest slope and highest r- value, but this is only a coincidence. The NFL and NHL show a counterexample: the NFL has a higher correlation coefficient, but a lower slope than the NHL. What can you tell about all of the correlation coefficients just by looking at the graph? Which plot do you think should have the highest correlation coefficient? Do your results agree? Does a higher correlation coefficient necessarily mean a steeper slope? If you were to switch the axes, what about your regression would change, and what would stay the same? (The slope and intercept would change in fact the new slope would be the reciprocal of the previous one but the correlation coefficient would stay the same.)

4 4 2 For each league, calculate how much on average a team has to spend to get an additional win. For which sport is this amount the highest, and why do you think this is? MLB: $11.9 million NBA: $1.45 million NFL: $9.7 million NHL: $3.62 million Major League Baseball has the highest cost per additional win, followed closely by the NFL. Both sports have a lot of players on the field at one time, so it might cost more to have an impact on winning. This question asks students to contextualize the slope of the best- fit lines, but requires one extra step in reasoning. The slopes here give the rate of change in terms of wins per million dollars spent on players, but a more natural question in this context is to ask how much each additional win costs in terms of payroll. In other words, if your goal as a team owner is to get more victories, it s reasonable to ask how much additional money corresponds to a single additional win. Mathematically, this means we re really asking students to interpret the reciprocal of the slope. To see why, notice that we re interested in millions of dollars per win, which is just the inverse of the current units. If students have trouble with this reasoning, it might be helpful to start with the NFL s line of best fit, which has a slope of just about 1/10. That means each additional million dollars corresponds to roughly 1/10 of a win, and so an increase of (about) $10 million corresponds to one whole win. MLB and the NFL have the highest cost per additional win, far ahead of the other two leagues. One potential reason is that there are a lot more players on the field at one time in each of those sports, so it s reasonable to think that it costs more money to get enough high quality players on the field to really affect wins and losses. Compare this with basketball, for instance, where only a handful of players (five or six) are almost entirely responsible for the outcome of any given game. What units does the slope of each best- fit line have? What units should your answer be in? What do you notice about those two sets of units? The NFL s line of best fit has a slope of about 1/10. What does the 1/10 mean in this context? So, how much additional money spent corresponds to one additional win? Which sport has the highest cost per additional win? Which sport(s) have the most players? The fewest? Why might that be important? In which league would one additional win be the most meaningful? (In the NFL, teams play a very small number of games, compared to other leagues. The outcome of one game has the biggest impact in the NFL.)

5 5 3 To prevent wealthier teams from being able to buy success, the NFL, NHL, and NBA have salary caps: limits on total payroll. (MLB does not.) Do you think this is an effective way to make leagues more competitive? Explain. Answers will vary. Students may or may not be familiar with the idea of a salary cap, so it might be helpful to spend a minute discussing why some leagues institute them. In theory, teams with more money can spend more money to get top players, which could lead to a situation where the league is no longer competitive: the richest teams might consistently beat up on the poorest teams, which wouldn t be much fun for the fans to watch. Salary caps attempt to level the playing field by ensuring that rich teams can t just hoard the best players. If those fears were justified (i.e., if spending a lot more money on players led to a lot more wins), then we d expect to see a strong correlation between wins and total payroll. We d also expect to see a weaker correlation in leagues with salary caps than in leagues without them. But we re actually faced with the opposite situation here: the MLB which doesn t have a cap has the lowest correlation coefficient of all four leagues. And none of the leagues show a particularly strong relationship. Of course we re only looking at a single season s worth of data, but it s certainly not very good evidence that salary caps are effective. Who has heard of a salary cap? Why might some leagues think salary caps are a good idea? What might happen if teams could spend any amount of money they wanted to on players? If spending a lot of money leads to a lot of wins, what kind of correlation would you expect to see? If salary caps in fact help level the playing field, would you expect to see stronger or weaker correlation between wins and payroll in leagues with a cap? Does the data support the claim that salary caps prevent teams from buying success? Why or why not? How else might leagues encourage more competitive play, besides salary caps? (The draft is another mechanism leagues use, for example. The lowest ranked teams from the previous season typically get the first pick of new players, which helps to spread talent around a bit.)

6 6 Act Two: The Owner s Box 4 In addition to its total payroll, there may be other factors that are correlated to how many wins a team gets. The tables below show the details for two other relationships. For each one, what do the linear regressions and correlation coefficients tell you about the strength of the relationship, and is this what you d expect? Explain. Wins vs. Salary of Highest Paid Player (mill.) MLB NBA There doesn t seem to be a very strong correlation here, though it s interesting that the NBA is the highest, which makes sense, since getting a single superstar could have a huge impact on winning. It s also interesting that MLB actually has a negative correlation. Maybe high paid pitchers aren t in the game enough or don t have a big enough individual impact to dramatically affect a team s wins. y = x r = NFL y = 0.127x r = y = 1.364x r = NHL y = 0.914x r = Wins vs. Previous Year s Wins MLB NBA This is the highest correlation so far, except for the NFL. It makes sense that a team who has been successful in the past would likely continue to be successful in the following season. Maybe the NFL is so low because they play so few games in a season: the difference between being 8-8 and 5-11 for the season might be only a few plays. y = 0.594x r = NFL y = 0.178x r = y = 0.863x r = NHL y = 0.487x r = Certainly a team s total payroll is only one factor related to how many games they win in a season. Maybe having a superstar is helpful, and so we might look at the highest paid player s salary alone. Or maybe success breeds success. In fact, there seems to be some evidence for that: the correlation between wins and previous season wins is the strongest we ve seen so far except for the NFL. Which variable would you say is more strongly related to wins, and why? Does this match what you would expect? Why or why not? Can you think of any other factors that might be correlated with number of wins? (Answers will vary. Responses might include the team's average height or weight, or the team's average sprinting speed or vertical jump distance.)

7 7 5 The following graph shows the relationship between wins and total revenue for all NFL teams in (Total revenue represents all the money a team makes from ticket sales, concessions, broadcasting rights, apparel, etc.) If you were the owner of a professional sports team, how do you think you would spend your money and why? Revenue (millions) Regular Season Wins MLB y = 1.317x r = NFL y = 0.154x r = NBA y = 1.226x r = NHL y = 0.812x r = For one thing, the association between total revenue and wins doesn t seem to be very strong at all (and in the NFL it s almost nonexistent). That suggests that having more revenue isn t necessarily related to winning games. But there s a more important factor at play here as well. Throughout the entire lesson we ve been looking at relationships between different variables and the number of games a team wins in a season, but we also need to consider the nature of those relationships. Correlation tells us that there is a particular kind (linear) of association between variables, and it can even 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. That s an important distinction for students to make: even if there were a strong correlation between total revenue and wins (which there isn t), that doesn t necessarily mean that increasing revenue will lead to more wins. In fact, the causal relationship might run the other way: having a team that wins more games might lead to higher revenue. Or neither of those might be true: there could be a third factor that affects both wins and revenue. In other words, it s hard to say what a team owner ought to spend his or her money on. These relationships alone don t give enough information to figure out what causes teams to win more games. And certainly there are a lot more factors that contribute to a team s record than we ve looked at here. At a minimum, we can draw the conclusion that having more money, or spending more money on salaries, doesn t necessarily improve a team s record. How would you describe the relationship between total revenue and wins? Do you think having higher revenue causes teams to win more games? Do you have enough information to make that kind of statement? What s the difference between correlation and causation? What do you think this means for how a team owner should spend his or her money? Wins are not highly correlated with revenue, but can you think of factors that might be? (Answers will vary but might include: the region's median income or the size of the media market.)

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