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Sabermetrics The Art and Science of Quantifying an Athlete’s Value Mark Rogers April 2, 2010

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SABR The Society for American Baseball Research The Society for American Baseball Research Founded in 1971 in Cooperstown, New York Founded in 1971 in Cooperstown, New York “To foster the research, preservation, and dissemination of the history and record of baseball” “To foster the research, preservation, and dissemination of the history and record of baseball” This is certainly made easier by being located in town alongside the National Baseball Hall of Fame. This is certainly made easier by being located in town alongside the National Baseball Hall of Fame. 6,700 members 6,700 members Mostly statisticians, sports writers, and former players and officials Mostly statisticians, sports writers, and former players and officials

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A Universe of Statistics Baseball fans are particularly fond of using statistics to measure a player’s ability, for several reasons. Baseball fans are particularly fond of using statistics to measure a player’s ability, for several reasons. – Low scoring: limits most other quantitative measurements of the only thing affecting the outcome, the points scored. – No clock: games can have indeterminate length and scoring chances, so a “fair number of chances to score” can vary – Consistency: the game is played under almost the exact same rules (and using mostly the same strategy) as it was when it premiered in the 19 th century, unlike other sports. The modern “live-ball era” of baseball: 1920—present The modern “live-ball era” of baseball: 1920—present Football: the forward pass altered the way games were played Football: the forward pass altered the way games were played Basketball: the modern “shot clock era” accelerates scoring Basketball: the modern “shot clock era” accelerates scoring Hockey/soccer: hey, until recently, who cared? Hockey/soccer: hey, until recently, who cared? This consistency allows current players to be readily compared to almost any other player from the past, unlike most other sports. This consistency allows current players to be readily compared to almost any other player from the past, unlike most other sports.

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The Pioneer The first great baseball statistician was Henry Chadwick (1824—1908). The first great baseball statistician was Henry Chadwick (1824—1908). – An English transplant to Brooklyn, where he followed cricket, rounders, and their American cousin, baseball – Wrote summaries of games for New York newspapers, and included a summary table of the game’s major statistics, the box score. – For his contributions to the legacy of the game, Chadwick was elected to the Baseball Hall of Fame in 1938.

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The Original Baseball Statistics Chadwick’s early box scores focused primarily on tabulating easily observable aspects of the game. Chadwick’s early box scores focused primarily on tabulating easily observable aspects of the game. – Hits/H: any ball hit in fair territory, but not easily retrievable by the fielders, resulting in a player safely reaching base – Walks (abbreviated BB, for “reached base on balls”) – Strikeouts (usually abbreviated K, or occasionally SO) – At-bats/AB: # of batting chances not resulting in a walk, since those “ball” pitches are deemed unhittable – Stolen bases/SB: safe advances made between hit balls – Runs/R: # of times a player scores/crosses home plate – Runs batted in/RBI: # of players crossing home plate because of that player’s at-bats, and not other issues

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The Original Baseball Statistics Chadwick’s early box scores focused primarily on tabulating easily observable aspects of the game. Chadwick’s early box scores focused primarily on tabulating easily observable aspects of the game. – Errors/E: mental or physical lapses resulting in a player being safe who should have theoretically been thrown out, or “running themselves out” by overrunning – Single/double/triple (1B/2B/3B): the # of bases safely reached by a player immediately following their hit, not counting any fielding error – Home runs/HR: player scores on their own hit, not due to fielder’s error; counts as 4 bases – Total extra-base hits/XBH: 2B + 3B + HR – Total bases/TB: combined # of bases reached by a player on their own hits, not counting any fielding errors, for an entire game (1B + 2×2B + 3×3B + 4×HR)

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The Original Baseball Statistics Errors and other issues “outside the player’s control” are often used to distinguish batting performances worthy of credit from those less so. Errors and other issues “outside the player’s control” are often used to distinguish batting performances worthy of credit from those less so. – A player who reaches base solely due to a fielding error is not credited with a hit or an official at-bat. – Hits followed by an error are scored as the type of hit an errorless version of the play would have resulted in, plus “advanced (or thrown out) on error.” – Walks/BB: appearance not counted as an official at-bat – Hit by pitch/HBP: player awarded first base as a result, but not given credit for a hit or an at-bat – Fielder’s choice/FC: player reached base only because a fielder chose to throw out a runner closer to scoring; thus, the player is not given credit for a hit – Double play/DP: batter causes multiple runners to be thrown out; deemed so terrible as to not warrant RBI credit even if a run scores!

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Raw Data vs. Average Data Since baseball games are of varying length, the number of at-bats (and therefore, the number of “expected” hits, runs, etc.) can vary widely. Since baseball games are of varying length, the number of at-bats (and therefore, the number of “expected” hits, runs, etc.) can vary widely. – Therefore, measuring only the raw batting data is not the fairest measure of who is the “best” player. – Chadwick devised several alternative statistical methods by calculating averages based on the ratio between the number of achievements made (in batting, pitching, or fielding) and the number of opportunities for them.

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The Hitting Averages Batting average: a measure of the rate of fair hits made in appropriate batting opportunities: Batting average: a measure of the rate of fair hits made in appropriate batting opportunities: – Chadwick viewed this as superior to the cricket average, which compared the number of runs to the number of outs. – “Situational” batting averages can also be measured, such as the batter’s average with “runners in scoring position” (RISP), or one factoring in the # of times they grounded into a double play (GIDP). Slugging percentage: a measure of the player’s power, which counts extra-base hits extra, but for the same number of ABs: Slugging percentage: a measure of the player’s power, which counts extra-base hits extra, but for the same number of ABs:

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The Hitting Comparisons In addition to the “portion of a whole” ratios of the BA, SLG, OBP, etc., ratios can be measured comparing one type of statistic to another. – Walk-to-strikeout ratio (BB/K): measures the hitter’s ability to maximize one and minimize the other – Ground-ball-to-fly-ball ratio (G/F): ditto – At-bats per home run (AB/HR): measures the rate of home runs, by using its easier-to-work-with reciprocal Mark McGwire holds the all-time career record, with an AB/HR of (having hit a home run in 9.4% of his official at-bats). The league average for AB/HR in 2009 was 32.9 (the average players hit a home run in 3.0% of their at-bats).

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The Triple Crown Traditionally, the HR and RBI counts, along with the popularly followed race for the batting-average title, were deemed to be the “major” hitting categories. Players who lead the league in all three are said to have won the Triple Crown of hitting. However, such a feat is difficult because of the wide gap between a power hitter “swinging for the fences” at the cost of many strikeouts and someone hitting “for average,” aiming for numerous hits even if they were “only” singles. – The most recent Triple Crowns were: American League: Carl Yastrzemski (Boston Red Sox), 1967 National League: Joe “Ducky” Medwick (St. Louis Cardinals), 1937

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A Problem with the System One problem with the use of the BA as a gauge of a player’s ability is that it makes no distinction between singles and “bigger” extra-base hits. – Thus, a “power hitter” who makes fewer hits, but scores more RBI with a more powerful selection of hits, would be deemed a worse player. – Example: Ryan Howard, 2008 (the year he finished 2 nd in MVP voting) His 48 home runs and 146 RBI led the league (with 331 total bases in 610 official at-bats), but he had 199 strikeouts to go with them, which helped lower his BA to just.251. He got a hit ¼ of the time, but his TB makes it look as if he did so ½ of the time. One solution is to use the slugging percentage (SLG), which gives “extra credit” for these bigger hits. – Howard’s SLG for 2008 was.543, much closer to the league-best.

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Another Problem with the System Both the BA and the SLG also fail to count walks as an official at-bat, which fails to give credit to a player with “good eyes” who is able to avoid strikeouts long enough to draw a walk. – Example: Pete Rose, 1974 In an “off” year, he had a.284 BA, but also a career-best 104 walks. A solution to this problem is to use the on-base percentage (OBP), which counts hits and walks (as well as getting hit by a pitch, which can also have tactical advantages) and uses something more closely approximating the total “plate appearances” instead of “at-bats”: – SF: sacrifice flies – Rose’s 1984 OBP was.385, close to the league lead.

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To Count, Or Not To Count? Some baseball occurrences vary as to which statistical categories they will count toward. Some baseball occurrences vary as to which statistical categories they will count toward. – Bunt/sacrifice hit/SH: a deliberate attempt to hit the ball so as to allow a runner to advance, at the expense of the batter Like a reverse fielder’s choice, the bunt does not count as a hit, and the attempt does not count as an official at-bat. Like a reverse fielder’s choice, the bunt does not count as a hit, and the attempt does not count as an official at-bat. Assuming the runner is indeed thrown out before reaching base, it will also not count towards the OBP. Assuming the runner is indeed thrown out before reaching base, it will also not count towards the OBP. However, if the runner is deemed to primarily be trying to reach first, it may be scored as a single or out instead. However, if the runner is deemed to primarily be trying to reach first, it may be scored as a single or out instead. If the batter bunts toward a runner on third (to draw away the third baseman), this squeeze play will result in an RBI credited. If the batter bunts toward a runner on third (to draw away the third baseman), this squeeze play will result in an RBI credited.

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To Count, Or Not To Count? Some baseball occurrences vary as to which statistical categories they will count toward. Some baseball occurrences vary as to which statistical categories they will count toward. – Sacrifice fly/SF: a fly ball hit with less than two outs, that is caught far enough from the infield to allow a runner to score Like a sacrifice hit (bunt), the sacrifice fly does not count as a hit, and the attempt does not count as an official at-bat. Like a sacrifice hit (bunt), the sacrifice fly does not count as a hit, and the attempt does not count as an official at-bat. However, unlike the sacrifice hit, it will count towards the OBP, as the play is considered more accidental and less a tactical decision. However, unlike the sacrifice hit, it will count towards the OBP, as the play is considered more accidental and less a tactical decision. The maneuver’s primary benefit is allowing a runner on third base to score while the caught ball is relayed; thus, the batter is credited with an RBI if successful. The maneuver’s primary benefit is allowing a runner on third base to score while the caught ball is relayed; thus, the batter is credited with an RBI if successful.

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The Best of Both Worlds Some armchair statisticians have argued that a measurement fusing the advantages of the SLG and the OBP would be ideal. “On-base plus slugging”: OPS = OBP + SLG The OPS has become one the primary statistical benchmarks used for hitters in the modern-day game. However, it is not without its share of controversy. – The “equal mixture” blends two measurements that normally have very unequal numbers; typically, SLG > OBP, weighting it preferentially. – It also has no intrinsic meaning in game-play terms, unlike the BA (the frequency of getting a hit) or OBP (the frequency of reaching base).

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Try 1 from Column A, 1 from Column B Each of these statistics can also be added to or subtracted from each other for a variety of results. Isolated power/IsoP: a measure of the hitter’s power effects without the influence of the number of hits: IsoP = SLG − BA Secondary average/SecA: a measure of the hitter’s number of bases attained without the influence of the number of hits: – Including any gained through walks and stolen bases – CS: # of times caught stealing (presumably, additionally subtracted so as to highlight the difference between two players who achieve the same number of stolen bases in a very different number of attempts)

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So…What’s “Good”? The variety of measurements necessitate some sort of benchmark for what would rank a player among the league’s best in each category. Typical career averages (modern “live-ball era,” since 1920): BAOBPSLGOPS “Average” “Great” “Elite” All-time record.366 (Ty Cobb).482 (Ted Williams).690 (Babe Ruth) (Babe Ruth) Active leader.334 (Albert Pujols).427 (Todd Helton/ Albert Pujols).628 (Albert Pujols) (Albert Pujols)

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Seeing the Numbers In addition to studying the numbers themselves, we can also visualize them using a scatterplot, searching for a presumed correlation between the OBP (the ability to get on base using more than just hits) and the SLG (the ability to get past first base with one’s hits). The red lines represent the league average for each statistic. – Upper-left: + power, − average – Lower-right: − power, + average – Lower-left: weaker in both – Upper-right: stronger in both

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Seeing the Numbers We can also plot the best-fit trend line for this scatterplot (shown here in light blue), showing the expected link between hitting for average (OBP) and hitting for power (SLG) that certain players exceed and others trail. – Anyone above this line is hitting for more power than their OBP would have suggested. – Anyone below this line is hitting for less power than their OBP would have suggested (or possibly getting on base more often than their SLG would have suggested). Once again, there is a clear sign of which current player excels at both of these critical areas. – Albert Pujols, St. Louis Cardinals

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Pitching Statistics Much as the batting average (BA) represents the number of “official” at-bats for a hitter, and PA the actual number of “plate appearances,” the pitcher’s performance can be measured by how many batters were faced. Much as the batting average (BA) represents the number of “official” at-bats for a hitter, and PA the actual number of “plate appearances,” the pitcher’s performance can be measured by how many batters were faced. – The equivalent of the PA is the # of “batters faced” (BF). – The equivalent of the BA is the “opponents’ batting average” (OBA), which similarly subtracts any plate appearance not counted as an at-bat for the hitter: CI: Catcher Interference with the play (½ times per year per team) CI: Catcher Interference with the play (½ times per year per team)

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The Pitching Averages Earned run average/ERA: a measure of how many earned runs a pitcher would be expected to give up on average over a full 9-inning game, regardless of the actual number of innings pitched (IP): Earned run average/ERA: a measure of how many earned runs a pitcher would be expected to give up on average over a full 9-inning game, regardless of the actual number of innings pitched (IP): – “Earned” runs/ER: those directly or indirectly due to the pitcher’s actions, including: Runners who score due to the pitcher’s actions (not any fielders’) Runners who score due to the pitcher’s actions (not any fielders’) Runners left behind by that pitcher (who is “responsible” for them) who later score when the relief pitcher allows a hit Runners left behind by that pitcher (who is “responsible” for them) who later score when the relief pitcher allows a hit But not including runners who score only because a player’s earlier error gave the team an “extra out” to drive them in But not including runners who score only because a player’s earlier error gave the team an “extra out” to drive them in – “Good” ERAs vary widely depending on the era played or less in “pitchers’ eras,” 4.00 or more in “hitters’ eras” 2.00 or less in “pitchers’ eras,” 4.00 or more in “hitters’ eras”

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The Other Triple Crown For pitchers, the three statistical categories deemed to be the most important have traditionally been wins, strikeouts, and ERA. – Like the hitting categories, these were deemed the easiest to follow. Players who lead the league in all three are said to have won the Triple Crown of pitching. Because the skill sets involved for pitchers are not as disparate as for hitters, some (particularly “power pitchers” excelling in strikeouts as a means to an end) can find winning it easier. – The most recent pitching Triple Crowns were: American League: Johan Santana (Minnesota Twins), 2006 National League: Jake Peavy (San Diego Padres), 2007

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The Pitching Averages Much like the batting average “levels the field” of batting statistics between players with different numbers of at-bats, a variety of pitching averages attempt to balance pitchers who have a varying number of innings pitched (IP), in the same manner as the ERA (by dividing the raw data by the number of innings pitched and multiplying by 9). Much like the batting average “levels the field” of batting statistics between players with different numbers of at-bats, a variety of pitching averages attempt to balance pitchers who have a varying number of innings pitched (IP), in the same manner as the ERA (by dividing the raw data by the number of innings pitched and multiplying by 9). – Hits per 9 innings pitched: “H/9” = H ÷ IP × 9 – Strikeouts per 9 innings: “K/9” = K ÷ IP × 9 – Walks per 9 innings: “BB/9” = BB ÷ IP × 9 Since excess walks and hits can still exhaust a pitcher (who must then be replaced) even if they do not result in runs, one popular modern average combines these “trivial” slip-ups. Since excess walks and hits can still exhaust a pitcher (who must then be replaced) even if they do not result in runs, one popular modern average combines these “trivial” slip-ups. – Walks plus hits per inning pitched (WHIP): (BB + H) ÷ IP

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The Pitching Comparisons Many of the ratios used to measure hitting prowess can be inverted to measure pitching prowess. – Strikeout-to-walk ratio (K/BB): measures the pitcher’s ability to maximize one and minimize the other – Fly-ball-to-ground-ball ratio (F/G): ditto The ERAs can be adjusted as well for certain situations, including the “catcher’s ERA” (CERA), the average ERA of the team with a particular catcher playing. – A measure of the catcher’s ability to control the game – Thus, it is more of a fielding statistic.

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Fielding Statistics Putouts (PO): the number of outs directly caused by a fielder Assists (A): the number of outs in which a player was indirectly involved Total (fielding) chances: TC = PO + A + E – The total number of opportunities to make a defensive play Fielding percentage: – Typically 98.5% (0.985) or better for most players, or slightly lower for difficult defensive positions (third base and shortstop) Range factor: RF = (PO + A) ÷ IP × 9 – A proportional extrapolation of a full game, the fielding equivalent of the ERA; used to gauge the amount

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The Prophet As general manager of the St. Louis Cardinals and the Brooklyn Dodgers, Branch Rickey profoundly altered baseball, from the integration he pioneered with Jackie Robinson to the “farm system” he invented to find untested players and train them for major-league success in the minor leagues. As general manager of the St. Louis Cardinals and the Brooklyn Dodgers, Branch Rickey profoundly altered baseball, from the integration he pioneered with Jackie Robinson to the “farm system” he invented to find untested players and train them for major-league success in the minor leagues. – With both teams struggling when he arrived, he worked to maximize player value by signing them for the lowest cost and training them to their fullest potential. – From Ken Burns’ film Baseball: “Nobody knew how to put a dollar sign on the muscle better than Branch Rickey.”

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The Prophet Rickey was asked by LIFE magazine if there was a “formula” for baseball success. Skeptical at first, he worked for six months to come up with what he thought might hold the key. – Rickey argued that this carefully balanced formula did an excellent job of approximating the final standings at season’s end, even if it violated many long-held beliefs about what was important for a team to win. – Hall of Famer George Sisler: “I still don’t believe it, but there it is.”

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Focusing on What Really Matters Rickey’s formulas were the precursors to the OBP and SLG— new ways of measuring the effectiveness of a hitter in hopes of finding a better match for the only baseball statistic that really counts, the number of runs scored. – Research suggests that a player’s batting average correlates with the team’s run-scoring success only 75% of the time. – OPS (OBP + SLG), on the other hand, does so 90% of the time. A perfect solution would be to calculate the number of runs each player is personally responsible for, but since scoring runs in baseball is such a communal effort, this is difficult.

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The Visionary In 1977, baseball writer and statistician Bill James coined the term sabermetrics (based on the acronym SABR) to refer to the statistical analysis of baseball data. In 1977, baseball writer and statistician Bill James coined the term sabermetrics (based on the acronym SABR) to refer to the statistical analysis of baseball data. For years, James wrote a self-published baseball statistical abstract after publishers deemed its subject matter too esoteric for a mainstream audience. For years, James wrote a self-published baseball statistical abstract after publishers deemed its subject matter too esoteric for a mainstream audience. His work found an obsessive audience of writers, fans, and baseball officials, and was soon published nationwide. His work found an obsessive audience of writers, fans, and baseball officials, and was soon published nationwide. – It would inspire an entire field of study and copycat publications. – 2006: Bill James named one of Time’s 100 Most Influential People

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Creating the Runs James created a number of new statistical categories. – Range factor (RF) – Pythagorean expectation: an estimate of how many games the team should have won, based on the total runs scored and allowed: The Pythagorean expectation has a strong correlation with the number of games the team actually goes on to win, although it can be improved still further by using an exponent of 1.82 instead. – Win shares: like the winnings divided up by a championship team, this formula divides up 3w “shares” of w wins among the players according to the amount each is entitled to, depending on their performance. Incorporates hitting, pitching, fielding, and even other “intangible” issues However, it is very difficult to calculate; its description in James’s book is 84 pages.

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Creating the Runs James created a number of new statistical categories. – Runs created (RC): a general category of formulas combining an on- base factor A, an advancement factor B, and an opportunity factor C: – This formula can vary widely depending on how those three categories are defined (or refined). One basic formula is: – Adjustments can be made to this formula by adding in other factors, or by weighting the various factors with coefficients to make them more or less important:

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Creating the Runs James created a number of new statistical categories. – One particularly elaborate version incorporates many small tweaks: James also developed formulas to help predict a pitcher’s performance, based on both the ERA and other acts within the game, all carefully balanced to create the “Game Score.” Using these very technical formulas, James became one of the leading experts on predicting outcomes in baseball. – Even he, though, cautioned against overdependence on their use.

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The Game Changes For a century, professional baseball players’ salaries were restricted by the reserve clause, which guaranteed teams the right of first renewal even after contracts expired. – In theory, it meant a form of job security; in practice, it was a “non- compete” clause, which meant that players could not choose to pursue higher salaries (or a better team) elsewhere. – League officials had been given an anti-trust exemption by Congress. Following the first MLB strike in 1972, players won raises and, more importantly, binding arbitration on salary issues. – An arbitrator soon struck down the reserve clause, allowing players whose contract with a team ended after 6 years to declare themselves “free agents,” who could sign for whatever the open market allowed. – The result was an explosion in player salaries, which forced many teams to take a hard look at where their money could best be spent.

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The True Believer In 1995, new owners inherited the Oakland A’s, and ordered general manager Sandy Alderson and assistant GM Billy Beane to slash spending on player salaries. – In 1998, Beane became the GM, “running the team” at just age 35. – The A’s had been one of the most successful teams of the previous two decades, winning 6 pennants and 4 World Series in a small city. Unable to spend freely to acquire talent, Beane was forced to find undervalued players, and used sabermetrics to do so. – The front office began to emphasize statistics such as OBP, SLG, and fielding ability rather than the traditional favored stats of BA and RBI. – In 2001 and 2002, the A’s were one of the best teams in baseball, winning over 100 games each year despite the second-lowest payroll. – Michael Lewis’s book Moneyball chronicles the struggles of the “Beane counters” to convince the team’s skeptical old-guard scouts.

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The True Believer Since all teams were required to spend a minimum amount on salaries, and all teams won at least 50 games, the key issue was how much extra each team paid for each extra victory. – Oakland had excelled in this area, paying $500,000 for each “extra” victory on their way to the division title (one of just two teams to spend less than $1 million per extra win), while richer but poorer- performing teams were paying $3 million or more for each of theirs. – To avoid the high costs of free agency, the A’s were also forced to get as much mileage as possible out of these undervalued stars’ contracts before their success attracted the attention of the big-market teams. – Before the 2001 season, Oakland had lost Jason Giambi, Johnny Damon, and Jason Isringhausen, three All-Stars whose new $33 million combined annual salaries were as much as the A’s entire team.

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The Apostles Following the success of Billy Beane, several other teams hired young general managers who used sabermetrics (rather than a long career as a professional baseball scout) as an integral part of analyzing potential player acquisitions. Their success rate varied widely. – Paul DePodesta Named GM of the Los Angeles Dodgers at age 31 Fired after just his second season, the Dodgers’ second-worst since moving to L.A. – Theo Epstein Named GM of the Boston Red Sox at age 28, the youngest in history Hired Bill James as a sabermetric adviser Two years later, the Red Sox broke “The Curse,” winning their first World Series in 86 years (as well as another one, three years later).

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A Very Serious Subject Several professors now teach sabermetrics courses. Several professors now teach sabermetrics courses. – Jim Albert, Bowling Green State University www-math.bgsu.edu/~albert/ www-math.bgsu.edu/~albert/ – Andy Andres, Tufts University – Steven J. Miller, Williams College Typically, the courses are elective follow-ups to a standard statistics course, created by baseball fans. Typically, the courses are elective follow-ups to a standard statistics course, created by baseball fans. – Student projects often involve designing a series of mathematical models to perform their own analysis.

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Play Ball. PowerPoint slides available at Support your team! Season Opener: New York Yankees at Boston Red Sox Sunday, April 4, 6:00 p.m., ESPN2 Opening Day: St. Louis Cardinals at Cincinnati Reds Monday, April 5, 11:10 a.m., ESPN

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Bibliography Jim Albert and Jay Bennett, Curve Ball: Baseball, Statistics, and the Role of Chance in the Game. Jim Albert and Jay Bennett, Curve Ball: Baseball, Statistics, and the Role of Chance in the Game. Baseball: A Film by Ken Burns. Baseball: A Film by Ken Burns. Baseball Reference: Baseball Reference: ESPN statistics page: espn.go.com/mlb/statistics ESPN statistics page: espn.go.com/mlb/statistics David Grabiner, The Sabermetric Manifesto. (www.baseball1.com/bb-data/grabiner/manifesto.html) David Grabiner, The Sabermetric Manifesto. (www.baseball1.com/bb-data/grabiner/manifesto.html)www.baseball1.com/bb-data/grabiner/manifesto.html Bill James’ new website: Bill James’ new website: Dan Lewis, “Lies, Damn Lies, and RBIs.” National Review, March 31, (available online at old.nationalreview.com/ weekend/play-ball/pb-lewis shtml) Dan Lewis, “Lies, Damn Lies, and RBIs.” National Review, March 31, (available online at old.nationalreview.com/ weekend/play-ball/pb-lewis shtml) Michael Lewis, Moneyball: The Art of Winning a Unfair Game. Michael Lewis, Moneyball: The Art of Winning a Unfair Game.

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Bibliography MLB Official Rules: mlb.mlb.com/mlb/official_info/official_rules/foreword.jsp MLB Official Rules: mlb.mlb.com/mlb/official_info/official_rules/foreword.jsp Branch Rickey, “Goodby to Some Old Baseball Ideas.” LIFE Magazine, August 2, (available online at y_to_old_idea.htm or “scanned” at Google Books) Branch Rickey, “Goodby to Some Old Baseball Ideas.” LIFE Magazine, August 2, (available online at y_to_old_idea.htm or “scanned” at Google Books) SABR: SABR: Alan Schwarz, The Numbers Game: Baseball’s Obsession with Statistics. Alan Schwarz, The Numbers Game: Baseball’s Obsession with Statistics. THE print almanac: John Thorn & Pete Palmer, Total Baseball. THE print almanac: John Thorn & Pete Palmer, Total Baseball. Tom M. Tiger, Mitchel Lichtman, & Andrew Dolphin, The Book: Playing the Percentages in Baseball. (insidethebook.com) Tom M. Tiger, Mitchel Lichtman, & Andrew Dolphin, The Book: Playing the Percentages in Baseball. (insidethebook.com)

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