Mathletics

Mathletics: How Gamblers, Managers, and Sports Enthusiasts Use Mathematics in Baseball, Basketball, and Football

WAYNE L. WINSTON
Copyright Date: 2009
Pages: 400
https://www.jstor.org/stable/j.ctt7sj9q
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  • Book Info
    Mathletics
    Book Description:

    Mathleticsis a remarkably entertaining book that shows readers how to use simple mathematics to analyze a range of statistical and probability-related questions in professional baseball, basketball, and football, and in sports gambling. How does professional baseball evaluate hitters? Is a singles hitter like Wade Boggs more valuable than a power hitter like David Ortiz? Should NFL teams pass or run more often on first downs? Could professional basketball have used statistics to expose the crooked referee Tim Donaghy? Does money buy performance in professional sports?

    InMathletics, Wayne Winston describes the mathematical methods that top coaches and managers use to evaluate players and improve team performance, and gives math enthusiasts the practical tools they need to enhance their understanding and enjoyment of their favorite sports--and maybe even gain the outside edge to winning bets.Mathleticsblends fun math problems with sports stories of actual games, teams, and players, along with personal anecdotes from Winston's work as a sports consultant. Winston uses easy-to-read tables and illustrations to illuminate the techniques and ideas he presents, and all the necessary math concepts--such as arithmetic, basic statistics and probability, and Monte Carlo simulations--are fully explained in the examples.

    After readingMathletics, you will understand why baseball teams should almost never bunt, why football overtime systems are unfair, why points, rebounds, and assists aren't enough to determine who's the NBA's best player--and much, much more. In a new epilogue, Winston discusses the stats and numerical analysis behind some recent sporting events, such as how the Dallas Mavericks used analytics to become the 2011 NBA champions.

    eISBN: 978-1-4008-4207-0
    Subjects: Statistics

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Table of Contents
    (pp. vii-x)
  3. PREFACE
    (pp. xi-xii)
  4. ACKNOWLEDGMENTS
    (pp. xiii-xiv)
    Wayne Winston
  5. List of Abbreviations
    (pp. xv-xviii)
  6. Part I. Baseball
    • 1 BASEBALL’S PYTHAGOREAN THEOREM
      (pp. 3-10)

      The more runs a baseball team scores, the more games the team should win. Conversely, the fewer runs a team gives up, the more games the team should win. Bill James, probably the most celebrated advocate of applying mathematics to analysis of Major League Baseball (often called sabermetrics), studied many years of Major League Baseball (MLB) standings and found that the percentage of games won by a baseball team can be well approximated by the formula

      $\frac {{runs scored^2}}{runs scored^2 + runs allowed^2} = estimate of percentage of games won (1)$

      This formula has several desirable properties.

      The predicted win percentage is always between 0 and 1.

      An increase in runs scored increases predicted...

    • 2 WHO HAD A BETTER YEAR, NOMAR GARCIAPARRA OR ICHIRO SUZUKI? The Runs-Created Approach
      (pp. 11-16)

      In 2004 Seattle Mariner outfielder Ichiro Suzuki set the major league record for most hits in a season. In 1997 Boston Red Sox shortstop Nomar Garciaparra had what was considered a good (but not great) year. Their key statistics are presented in table 2.1. (For the sake of simplicity, henceforth Suzuki will be referred to as “Ichiro” or “Ichiro 2004” and Garciaparra will be referred to as “Nomar” or “Nomar 1997.”)

      Recall that a batter’s slugging percentage is Total Bases (TB)/At Bats (AB) where

      TB=Singles + 2×Doubles (2B) + 3×Triples (3B) + 4×Home Runs (HR).

      We see that Ichiro had...

    • 3 EVALUATING HITTERS BY LINEAR WEIGHTS
      (pp. 17-29)

      In chapter 2 we saw how knowledge of a hitter’s AB, BB+HBP, singles, 2B, 3B, and HR allows us to compare hitters via the Runs Created metric. As we will see in this chapter, the Linear Weights approach can also be used to compare hitters. In business and science we often try to predict a given variable (called Y or the dependent variable) from a set of independent variables$x_1,x_2,…,x_n$. Usually we try to find weights B1, B2,…Bn and a constant that make the quantity

      $ \text{Constant} + \text{B}1\text{x}_1 + \text{B}2\text{x}_2 + \dots \text{Bnx_n} $

      a good predictor for the dependent variable.

      Statisticians call the search for the...

    • 4 EVALUATING HITTERS BY MONTE CARLO SIMULATION
      (pp. 30-40)

      In chapters 2 and 3 we showed how to use Runs Created and Linear Weights to evaluate a hitter’s effectiveness. These metrics were primarily developed to “fit” the relationship between runs scored by a team during a season and team statistics such as walks, singles, doubles, triples, and home runs. We pointed out that for players whose event frequencies differ greatly from typical team frequencies, these metrics might do a poor job of evaluating a hitter’s effectiveness.

      A simple example will show how Runs Created and Linear Weights can be very inaccurate.¹ Consider a player (let’s call him Joe Hardy...

    • 5 Evaluating Baseball Pitchers and Forecasting Future Pitcher Performance
      (pp. 41-51)

      In chapters 2–4 we discussed three methods that can be used to evaluate the performance of a baseball hitter: Runs Created, Linear Weights, and Monte Carlo simulation. Let’s turn our attention to evaluating the performance of baseball pitchers. As we will see, evaluating their performance is no easy matter.

      Until recently, the most frequently used technique for evaluating the performance of pitchers was earned run average (ERA). Let’s consider a pitcher, again named Joe Hardy. Consider all the runners Joe allows to reach base. Any of the base runners who score or would have scored if Joe’s team made...

    • 6 BASEBALL DECISION-MAKING
      (pp. 52-63)

      During the course of a season, managers make many crucial decisions, including the ones listed below.

      With a man on first and nobody out should we attempt a sacrifice bunt to advance the runner to second base?

      With a man on first and one out should we attempt to steal second base?

      We are the home team and the score is tied in the top of the ninth inning. The opposing team has a man on third base and none out. Should we play the infield in? This increases the chance of a hit (most people think bringing the infield...

    • 7 EVALUATING FIELDERS Sabermetrics’ Last Frontier
      (pp. 64-70)

      Surprisingly, until the late 1990s little progress was made in determining how to evaluate the effectiveness of fielders and the relative importance of fielding (as compared to batting and hitting). Until recently the prevailing wisdom in baseball was that you had to have “strength up the middle” (good fielding at second base, shortstop, catcher, and center field) to have a good team. We will see that in most cases, the differences in player fielding abilities are not significant enough to be a major factor in team performance. As the saying goes, the exception proves the rule and we will see...

    • 8 PLAYER WIN AVERAGES
      (pp. 71-78)

      Anyone associated with a baseball, football, or basketball team would probably say that a player’s objective is to help his team win games. Therefore, it seems reasonable to measure how much a professional athlete’s efforts help his team win or cause his team to lose games. As we will see in later chapters, for basketball and football this is a very difficult task. For baseball, however, Eldon Mills and Harlan Mills (Player Win Averages) came up with a simple yet elegant way to measure how a baseball player changes the chance that his team will win a game. To illustrate...

    • 9 THE VALUE OF REPLACEMENT PLAYERS Evaluating Trades and Fair Salary
      (pp. 79-83)

      In this chapter we will learn how to use the Player Win Averages discussed in chapter 8 to evaluate trade offers and calculate a player’s fair salary (based on his previous year’s performance).

      The key tool involved in our analysis will be the Value of a Replacement Player Points (VORPP), which was developed by Keith Woolner, formerly of theBaseball Prospectusand now an executive for the Cleveland Indians. You may recall in several earlier chapters we compared a player’s batting, pitching, and/or fielding performance to that of an average player. Although such comparisons are interesting, they do not really help...

    • 10 PARK FACTORS
      (pp. 84-86)

      During the 2006 season right fielder Brad Hawpe of the Colorado Rockies had a basic Runs Created rating of 5.04 runs per game. During the 2006 season San Diego Padre second baseman Josh Barfield had a basic Runs Created rating of 4.21 runs per game. On the surface, this would seem to indicate that Hawpe had a much better hitting season than did Barfield. Most baseball fans, however, realize that the Rockies play in Coors Field, which is notorious for being a hitter’s park because the air is thin (the ball carries farther) and the park is not that big....

    • 11 STREAKINESS IN SPORTS
      (pp. 87-101)

      We have all heard Marv Albert tell us that Dirk Nowitzki is “on fire” or Jack Buck tell us that Albert Pujols is “red hot” and nobody can get him out. We also hear announcers tell us the Spurs are on a hot streak, are unbeatable, and so forth. Is it true that athletes and teams encounter hot streaks, or are the observed patterns of player and team performance just randomness at work?

      Let’s first examine how a random sequence of 162 wins and losses appears. Let’s suppose a team wins 60% of their games. To generate a random sequence...

    • 12 THE PLATOON EFFECT
      (pp. 102-105)

      For most right-handed pitchers, the curve ball is an important part of their pitching repertoire. A right-handed pitcher’s curve ball curves in toward a left-handed batter and away from a right-handed batter. In theory, when facing a right-handed pitcher, a left-handed batter has an edge over a right-handed batter. Similarly, when a left-handed pitcher is on the mound, the right-handed batter appears to have the edge. Managers take advantage of this alleged result by platooning batters. That is, managers tend to start right-handed batters more often against left-handed pitchers and start left-handed batters more often against right-handed pitchers. Ignoring switch...

    • 13 WAS TONY PEREZ A GREAT CLUTCH HITTER?
      (pp. 106-109)

      Tony Perez played first base for the “Big Red Machine” during the 1960s and 1970s and had a lifetime batting average of .279. Such an average does not often lead to a Hall of Fame selection, but in 2000 Perez was elected to the Hall of Fame while some of his contemporaries who have similar statistics (such as Dave Parker) have not yet been elected. One reason Perez made it to the Hall of Fame was that his manager, Sparky Anderson, said that Perez was the best clutch hitter he had ever seen. Is there an objective way to determine...

    • 14 PITCH COUNT AND PITCHER EFFECTIVENESS
      (pp. 110-112)

      In October 2003 the Red Sox were leading the Yankees 5–2 after seven innings of the seventh and deciding game of the American League Championship Series. Pitcher Pedro Martinez was cruising along and had allowed only two runs. At the start of the eighth inning Martinez got the first batter out, but then Derek Jeter hit a double. Red Sox manager Grady Little went to the mound and talked to Martinez, and then left him in the game. The Yankees promptly tied the game and went on to win in the eleventh inning on a dramatic walk-off home run...

    • 15 WOULD TED WILLIAMS HIT .406 TODAY?
      (pp. 113-115)

      In 1941 Ted Williams hit .406. If he were in his prime today (say, the 2006 season), could he still hit around .400? Across the United States arguments similar to the following take place every day: Could Bill Russell dominate Shaq? Who was better: Peyton Manning or Joe Montana? Of course we can’t know for sure the answers to these questions. We can, however, use mathematics to determine whether today’s players are superior to players from an earlier time.

      Let’s examine how hitters from the 1940s through the 1980s compare to the hitters in 1941. We will define the level...

    • 16 WAS JOE DIMAGGIO’S 56-GAME HITTING STREAK THE GREATEST SPORTS RECORD OF ALL TIME?
      (pp. 116-122)

      In a beautifully written article, the late paleontologist and lifelong baseball fan Stephen Jay Gould argues that Joe DiMaggio’s 56-game consecutive hitting streak is the greatest sports record of all time.¹ In this chapter we will use basic probability and statistics to determine how likely it is that a 56-game hitting streak would ever occur.

      In June 1938 Johnny Vander Meer pitched consecutive no-hitters. This has never been done by anyone else. Is this the greatest sports record of all time? After making some reasonable assumptions, basic probability and statistics can help us determine that the occurrence of a 56-game...

    • 17 MAJOR LEAGUE EQUIVALENTS
      (pp. 123-124)

      Major league general managers must decide every year whether a promising minor league player is ready to be brought up to the major league team. Of course, the minor league player faces inferior pitching in the minors, so he is not expected to duplicate his minor league statistics when he is brought up to the majors. In 1985 Bill James developed Major League Equivalents to help major league front office personnel determine whether a minor leaguer is ready for the majors.

      The Excel file mle.xls gives the OBP for a set of hitters whose last minor league year was played...

  7. Part II. Football
    • 18 WHAT MAKES NFL TEAMS WIN?
      (pp. 127-131)

      NFL teams want to win games. Is it more important to have a good rushing attack or a good passing attack? Is rushing defense more important than passing defense? Is it true that turnovers kill you? During the early 1960s statistician Bud Goode studied what makes a team win. He found that passing yards per attempt (PY/A) on both offense and defense were the most important factors in predicting an NFL team’s success. This is intuitively satisfying because PY/A is more of a measure of efficiency than total yards passing. Since we divide by pass attempts, PY/A recognizes that passing...

    • 19 WHO’S BETTER, TOM BRADY OR PEYTON MANNING?
      (pp. 132-137)

      Most American men spend a good deal of time arguing about who are the best quarterbacks in the NFL. For example, is Tom Brady better than Peyton Manning? The NFL quarterback rating system works as follows.

      First one takes a quarterback’s completion percentage, then subtracts 0.3 from this number and divides by 0.2. You then take yards per attempts subtract 3 and divide by 4. After that, you divide touchdowns per attempt by 0.05. For interceptions per attempt, you start with 0.095, subtract from this number interceptions per attempt, and then divide this result by 0.04. To get the quarterback...

    • 20 FOOTBALL STATES AND VALUES
      (pp. 138-142)

      In chapter 8 we discussed how the inning, score margin, outs, and runners on bases were sufficient data when trying to determine whether a baseball team would win a game (assuming two equal teams were playing.) For example, if a team is down by three runs in the top of the seventh inning with two outs and the bases loaded, it has a 15% chance of winning the game. We call the inning, score margin, outs, and runners on bases the state of the baseball game. Once we know the state of the game and have evaluated the chance of...

    • 21 FOOTBALL DECISION-MAKING 101
      (pp. 143-150)

      During the course of a football game, coaches must make many crucial decisions, including the following:

      It is fourth and 4 on the other team’s 30-yard line. Should we kick a field goal or go for a first down?

      It is fourth and 4 on our own 30-yard line. Should we go for a first down or punt?

      We gained 7 yards on first down from our own 30-yard line. The defense was offside. Should we accept the penalty?

      On first and 10 from their own 30-yard line our opponent ran up the middle for no gain. They were offside....

    • 22 A STATE AND VALUE ANALYSIS OF THE 2006 SUPER BOWL CHAMPION COLTS
      (pp. 151-157)

      By winning the Super Bowl, the 2006 Indianapolis Colts brought great joy to the Hoosier state. In this chapter we will use the state and value approach described in chapter 20 to answer many interesting questions about the Colts’ offense, such as the following:

      On any down and yards to go situation, is running more or less effective than passing? For example, on first and 10, is running more effective than passing?

      Are runs more or less effective than passes overall?

      Was Joseph Addai a more effective runner than Dominique Rhodes?

      Who is better: Marvin Harrison or Reggie Wayne?

      Is...

    • 23 IF PASSING IS BETTER THAN RUNNING, WHY DON’T TEAMS ALWAYS PASS?
      (pp. 158-164)

      In football the offense selects a play and the defense lines up in a defensive formation. Let’s consider a very simple model of play selection in which the offense and defense simultaneously select their play:

      The offense may choose to run or pass.

      The defense may choose a run or pass defense.

      The number of yards gained is given in table 23.1, which we call a payoff matrix for the game. We see that if the defense makes the right call on a run, the opposing team loses 5 yards, and if the defense makes the wrong call, the team...

    • 24 SHOULD WE GO FOR A ONE-POINT OR TWO-POINT CONVERSION?
      (pp. 165-171)

      Since 1994, when the NFL began allowing teams to go for a two-point conversion after a touchdown, it has become important for NFL coaches to determine whether to go for one or two points after a touchdown. The success rate for a one-point conversion is over 99%, so we will assume that there is a 100% chance that a one-point conversion will be successful. The success rate for two-point conversions is around 47%.¹ On average, a one-point conversion try earns one point and a two-point conversion attempt earns 0(.6) + 2(.47) = .94 points. So, on average, a one-point conversion...

    • 25 TO GIVE UP THE BALL IS BETTER THAN TO RECEIVE The Case of College Football Overtime
      (pp. 172-174)

      In college football tie games are resolved with overtime. The winner of a coin toss chooses whether to start with the ball or to give the ball to his opponent. The first team with the ball begins on the opponent’s 25-yard line and keeps going until they attempt a field goal, score a touchdown, or lose possession. Then the other team gets the ball on their opponent’s 25-yard line and keeps going until they attempt a field goal, score a touchdown, or lose possession. The team that is ahead at this point is the is the winner. If the score...

    • 26 WHY IS THE NFL’S OVERTIME SYSTEM FATALLY FLAWED?
      (pp. 175-179)

      When an NFL game goes into overtime a coin toss takes place and the team that wins the coin toss has the choice of kicking off or receiving. Since the overtime is sudden death, the team winning the coin toss invariably chooses to receive so they have the first chance to score and win the game. During the 1994–2006 seasons the team that received the kickoff in overtime won 60% of the games. It seems unfair that in NFL overtime the team winning the coin flip should have such a huge edge. In attempt to lessen the impact of...

    • 27 HOW VALUABLE ARE HIGH DRAFT PICKS IN THE NFL?
      (pp. 180-184)

      The NFL is generally thought to exhibit more parity than other leagues. This means that it appears easier for a bad NFL team to improve from season to season than for a bad NBA or MLB team to improve from one season to the next. We will investigate the truth of this matter in chapter 41.

      Most NFL fans believe that the major equalizer from year to year is the structure of the NFL draft. Teams draft in each round in inverse order of performance with the worst team getting the first pick and the best team getting the last...

  8. Part III. Basketball
    • 28 BASKETBALL STATISTICS 101 The Four-Factor Model
      (pp. 187-194)

      For each player and team NBA box scores track the following information:

      two-point field goals made and missed

      three-point field goals made and missed

      free throws made and missed

      personal fouls committed

      assists

      offensive and defensive rebounds

      blocked shots

      turnovers

      steals

      minutes played

      How can we use this data (on a per game or per season basis) to break down what makes an NBA team perform well or poorly?

      Many coaches and players currently evaluate their shooting by looking at their field goal percentage. For example, suppose in a Dallas Mavericks–New York Knicks game the Mavericks make 45 out...

    • 29 LINEAR WEIGHTS FOR EVALUATING NBA PLAYERS
      (pp. 195-201)

      In chapter 3 we discussed the use of Linear Weights to evaluate MLB hitters. We found that by determining appropriate weights for singles, walks, doubles, triples, home runs, outs, stolen bases, and caught stealing we can do a pretty good job of estimating the Runs Created by a hitter.

      Given the wealth of information in an NBA box score, many people have tried to come up with Linear Weights formulas that multiply each box score statistic by a weight and equate the weighted sum of player statistics as a measure of the player’s ability. In this chapter we will discuss...

    • 30 ADJUSTED+/−PLAYER RATINGS
      (pp. 202-223)

      Basketball is a team game. The definition of a good player is somebody who makes his team better, not a player who scores 40 points per game. My favorite story about what defines a good player is told by Terry Pluto in his excellent bookTall Tales.¹ The late great Celtics coach Red Auerbach said that whenever the Celtics practiced, KC Jones’s team always won. This must mean he was a good or great player. Yet during his peak years his PER rating was around 10, indicating he was a poor player. KC must have done some great things that...

    • 31 NBA LINEUP ANALYSIS
      (pp. 224-227)

      In chapter 30 we described the methodology for creating Adjusted +/− ratings. These are helpful to teams in making decisions involving players such as trades and salaries. During the season, however, few players are traded and a team’s major concern is how to win more games with their current roster. The most important decisions coaches make during the season are which lineups to play when. For example, should a team try to go small against the 2006–7 Suns’ “small ball,” or should they go big and push the ball inside?

      On average a team plays 300–600 different lineups...

    • 32 ANALYZING TEAM AND INDIVIDUAL MATCHUPS
      (pp. 228-232)

      Among other things, a successful coach must be a master psychologist who can motivate players to play for the team rather than themselves (there is “noIin team”; “the whole is greater than the sum of its parts”). A great coach must have sound offensive and defensive strategies and get the players to buy into executing his strategic concepts. Great coaches also have excellent insight as to which players to put in a game at a given time to best match up with the opponent’s lineup. Recall from chapter 30 that we have Adjusted +/− ratings for each NBA...

    • 33 NBA PLAYERS’ SALARIES AND THE DRAFT
      (pp. 233-236)

      In chapter 9 we determined salaries for baseball players based on how many wins a player generated over and above the number of wins that would be achieved with a team of “replacement players.” Using the WINVAL point ratings we can use the same approach to come up with an estimate of a fair salary for an NBA player.

      During the 2006–7 season the average team payroll was $66 million. The minimum player salary was around $400,000. We will define the point value of a “replacement player” as −6. This is the usual point value for a player in...

    • 34 ARE NBA OFFICIALS PREJUDICED?
      (pp. 237-241)

      The sports page of the May 2, 2007, edition of theNew York Timescontained the headline: “Study of NBA Sees Racial Bias in Calling Fouls.”¹ The article was based on a study by Cornell professor Joseph Price and Wharton professor Justin Wolfers.² Price and Wolfers (PW) claim that “more personal fouls are called against players when they are officiated by an opposite-race refereeing crew than when officiated by an own-race crew.” In this chapter we discuss their insightful analysis of the referee bias question.

      An NBA officiating crew consists of three officials. The ideal way to determine whether the...

    • 35 ARE COLLEGE BASKETBALL GAMES FIXED?
      (pp. 242-243)

      In 2006 Wharton professor Justin Wolfers created a stir by claiming that around 5% of all college basketball games are fixed by players who intentionally slacken their effort (often called point shaving). Wolfers argued that for games in which the favorite is favored by S points, we should find that the probability of the favorite winning by between 1 and S−1 points should equal the probability of the favorite winning by between S+1 and 2S−1 points. This follows because statisticians usually find that forecast errors about an unbiased prediction (like a point spread) should be symmetrically distributed, like a normal...

    • 36 DID TIM DONAGHY FIX NBA GAMES?
      (pp. 244-247)

      In July 2007 shockwaves rippled through the sports world when NBA referee Tim Donaghy was accused of fixing the outcome of NBA games. If bettors attempt to fix a game, after the opening betting line is posted the line would move substantially as the “fixers” put their bets down. A move of two or more points in the line is generally considered highly unusual. For example, on November 14, 2007, when the Toronto Raptors played the Golden State Warriors, the opening line on total points (referred to hereafter as the Total Line) was 208 points. This means that if you...

    • 37 END-GAME BASKETBALL STRATEGY
      (pp. 248-252)

      In this chapter we will consider the optimal strategy for two important situations that can occur at the end of a close basketball game:

      In game 1 of the first round of the 2001 Eastern Conference playoffs the Philadelphia 76ers led the Indiana Pacers by two points. The Pacers had the ball with five seconds to go. Should the Pacers have attempted a two-pointer to tie or a three-pointer to win?

      During game 6 of the 2005 Western Conference semifinals, the Dallas Mavericks led the Phoenix Suns by three points with five seconds to go. Steve Nash is bringing the ball...

  9. Part IV. Playing with Money, and Other Topics for Serious Sports Fans
    • 38 SPORTS GAMBLING 101
      (pp. 255-261)

      In this chapter we will review (largely through a question-and-answer format) the basic definitions and concepts involved in football, basketball, and baseball gambling.

      In the 2007 Super Bowl the Colts were favored by 7 points over the Bears and the predicted total points for the game was 48. How could someone have bet on these odds? Theoretically the fact that the Colts were favored by 7 points meant the bookies thought there was an equal chance that the Colts would win by more than 7 or less than 7 points (in the next chapter we will see this may not...

    • 39 FREAKONOMICS MEETS THE BOOKMAKER
      (pp. 262-265)

      Recall from chapter 38 that if a bookmaker gives 11–10 odds on NFL point spread bets and sets a line so that half the money is bet on each side, then the bookmaker is guaranteed to make a riskless 4.5% profit.

      Steven Levitt ofFreakonomicsfame showed that bookmakers can exploit bettor biases to make an expected profit exceeding 4.5% per dollar bet. Levitt obtained bettor records for 20,000 bettors during the 2001 NFL season. He found that much more than 50% of all money is bet on favorites and less than 50% on underdogs. When the home team...

    • 40 RATING SPORTS TEAMS
      (pp. 266-282)

      Most gamblers believe that when bookies set point spreads their goal is to have half the money bet on each team. If I bet $10, for example, on a 7.5-point favorite to cover the spread, I win $10 if the team covers but I lose $11 if the favorite does not cover. If the favorite covers the points spread half the time, on average each $10 bet results in an expected profit of (1/2)($10)+(1/2)(−$11) = $−.50. Thus a bettor loses on average $0.50/$10.50 or $1/21 per dollar bet. Assuming we bet the same amount on each game, to break even...

    • 41 WHICH LEAGUE HAS GREATER PARITY, THE NFL OR THE NBA?
      (pp. 283-286)

      Sports fans love the NFL because it seems like there is always a surprise team that wins the Super Bowl or challenges for the championship. For example, who expected Tampa Bay to win the Super Bowl in 2002? NBA fans complain the same teams (such as Detroit and San Antonio) are always on top. It is easy to show that the NFL does indeed exhibit more parity and unexpected team performances than does the NBA.

      If a league has a great deal of parity you would expect it to be difficult to predict a team’s performance based on their previous...

    • 42 THE RATINGS PERCENTAGE INDEX (RPI)
      (pp. 287-289)

      During the college basketball season hoop fans anxiously anticipate the selection and seeding of teams for the NCAA tournament. The NCAA selection committee wants an accurate view of the teams’ relative abilities, but, like the BCS selection committee, the NCAA tournament selection committee wants to use only a team’s win-loss record and not the score of their games to rank teams. The NCAA believes that including game scores in the ranking and seeding process would cause the top teams to try and run up the score on lesser opponents. In chapter 40 we explained how to use a logistic regression-based...

    • 43 FROM POINT RATINGS TO PROBABILITIES
      (pp. 290-297)

      In chapter 40 we learned how to calculate “power ratings,” which allow us to estimate how many points one team is better than another.¹ In this chapter we will show how to use power ratings to determine the probability that a team wins a game, covers a point spread bet, or covers a teaser bet. For NBA basketball, we will see how to use power ratings to determine the probability of each team winning a playoff series. At the end of the chapter we’ll look at how power ratings can be used to compute the probability of each team winning...

    • 44 OPTIMAL MONEY MANAGEMENT The Kelly Growth Criteria
      (pp. 298-302)

      Suppose we believe we have an almost sure bet on Colts −12. We believe the Colts have a 90% chance of covering the spread. This would probably never happen, but let’s assume that such a bet really exists. What fraction of our capital should we allocate to this bet? If we bet all our money many times on bets with a 90% chance of winning, eventually we will be wiped out when we first lose a bet. Therefore, no matter how good the odds, we must be fairly conservative in determining the optimal fraction of our capital to bet

      Edward...

    • 45 RANKING GREAT SPORTS COLLAPSES
      (pp. 303-310)

      With seventeen games left in the 2007 baseball season the New York Mets held a seemingly comfortable seven-game lead in the National League East over the second-place Philadelphia Phillies. The Mets collapsed and the Phillies won the division. This collapse inspired Fox News sportswriter Todd Behrendt to write an article ranking the “all-time great sports collapses.”¹ In this chapter we will make some simple assumptions and then use basic probability to try and determine the probability of each collapse occurring. The “greatest collapse” would then be the collapse that had the smallest probability of occurring.

      We now describe the great...

    • 46 CAN MONEY BUY SUCCESS?
      (pp. 311-318)

      We all know money can’t buy love or happiness. In professional sports, can spending more money on players buy a team more success? Let’s analyze this question for the NFL, NBA, and MLB.

      In chapter 40 we learned how to calculate offensive and defensive power ratings for NFL teams. For example, an offensive team rating of +3 means a team (after adjusting for the strength of opposition) scores 3 points more than average and a defensive rating of −5 means that (after adjusting for the strength of opposition) that a team gives up 5 fewer points than average. For the...

    • 47 DOES JOEY CRAWFORD HATE THE SPURS?
      (pp. 319-320)

      In March 2007 NBA official Joey Crawford ejected Spurs star Tim Duncan during a game against the Dallas Mavericks. The NBA then suspended Crawford for the rest of the 2007 season. They said that he “was unfair to the Spurs.” Is it possible to determine whether Joey Crawford’s officiating harmed the Spurs’ performance during Spurs games in which he officiated?

      If Joey Crawford was biased against the Spurs, then we would expect the Spurs to have played significantly worse than expected during the games in which Crawford officiated. To determine an expected level of performance by the Spurs, we looked...

    • 48 DOES FATIGUE MAKE COWARDS OF US ALL? The Case of NBA Back-to-Back Games and NFL Bye Weeks
      (pp. 321-323)

      “Fatigue makes cowards of us all” is a famous anonymous quote popularized by the late, great Green Bay Packers coach Vince Lombardi. The idea, of course, is that if you are tired you cannot perform at peak performance level. In this chapter we use the following two types of game situations to show that fatigue does indeed have a significantly deleterious impact on team performance:

      NBA teams that play back-to-back games perform significantly worse than expected during the second game of the back-to-back and perform even worse

      The week after an NFL team has a bye or an open date...

    • 49 CAN THE BOWL CHAMPIONSHIP SERIES BE SAVED?
      (pp. 324-330)

      As every college football fan knows, since 1998 the Bowl Championship Series (BCS) has selected two college football teams to play for the national championship in early January. In this chapter we will explain how the BCS currently (2007 season) ranks teams and chooses the two teams that play for the championship. We will also discuss two commonly suggested alternatives to the BCS: an eight-team playoff or a “plus-one” system that chooses the two teams that get to play for the championship after the New Year’s Day bowl games.

      Starting in 1997, teams were ranked using the following four factors:...

    • 50 COMPARING PLAYERS FROM DIFFERENT ERAS
      (pp. 331-334)

      In chapter 15 we tried to determine whether it would be likely that Ted Williams would hit .400 if he were to play today. Our analysis required that we compared the pitching and fielding abilities of players from different eras. In chapter 15 we used a fairly simplistic approach and found that it was unlikely that Ted Williams would hit .400 today. In this chapter we use our WINVAL ratings to determine whether the players in the NBA have improved or declined in quality since 2000. The end of the chapter summarizes the results of Berry, Reese, and Larkey, who...

    • 51 CONCLUSIONS
      (pp. 335-340)

      We have covered a lot of material in this book. We have shown how applying mathematics can improve the performance of baseball, football, and basketball teams. We have learned how to evaluate teams and players and determine the probabilities of interesting events such as consecutive nohitters. We have also gained an understanding of how sports gambling works. In this chapter we will review the important mathematical tools that we have used in our analysis.

      Throughout the book, we have used regression to try to understand how various team statistics impact team performance. Often we tried to predict a dependent variable...

  10. EPILOGUE TO THE PAPERBACK EDITION
    (pp. 341-354)

    In this epilogue we will cover some interesting developments in the mathematics of sports that have occurred since the publication of Mathletics in 2009.

    The Web site Fangraphs.com makes it a snap to keep up on the latest advanced baseball statistics. You can search for statistics by player or team. Of particular interest to Mathletics readers will be the fact that Fangraphs.com contains excellent advanced fielding statistics (see chapter 7) and a version of Player Win Averages (see chapter 8).

    The key fielding statistic is UZR (Ultimate Zone Rating). In table E.1 we show the UZR for members of the...

  11. LIST OF DATABASES
    (pp. 355-356)
  12. ANNOTATED BIBLIOGRAPHY
    (pp. 357-366)
  13. INDEX
    (pp. 367-372)