A Quick History of Offensive Baseball Statistics: Which Is Top of the Pops?

Sabermetrician and founder of STATS, Inc. Richard “Dick” Cramer wrote the article “Average Batting Skill Through Major League History,” in which he demonstrated that the average major-league batter has gotten better over time.1 This article hopes to chart a parallel improvement of the average batting statistics over the course of baseball history.

This is a tiny, little survey of offensive statistics and the minds that created them, and where possible, some interactions. It ties each statistic to the publication year of an article or book. Sometimes, there was no formal publication found, and a time was estimated from the data at hand. It also might be the case that some of these statistics were developed long before the publication date.

Some of these folks invented multiple statistics. For simplicity’s sake, this covers just one each.

Because kids like baseball cards and often get their first introduction to statistics from the back of the cards, this article uses baseball cards. Each section includes a baseball card of a sabermetrician, a statistic they invented, the formula of the statistic, a linear regression analysis and the R² of that stat.

A Linear Regression takes two things, like batting average (BA) and runs (R), and measures how closely they relate to each other. The R² (R-squared), i.e. the Coefficient of Determination, is the number from the linear regression that tells you how much they relate.

On each card, look for the R² in the linear regression:

Think of R² as a sort of “batting average” for a statistic. Like BA, the higher an R² is, the better, i.e. the stronger the relation. Unlike BA, .300 is really bad—no statistic here is “batting” less than .500. The highest R² possible is 1.0, which is perfect. The closer to 1.0 a variable is, the better a predictor it is for the other variable.

Regressions are on the American League (AL) and National League (NL) team data for the years 1970 to 1977 from the Teams table of the Lahman database. That period was chosen for two reasons. First, because the data prior to 1970 are largely missing the HBP (hit by pitch), LWR (liner weights runs), and SF (sacrifice fly) data needed for calculating wOBA (weighted on-base average). Second, because Pete Palmer and John Thorn have not published any weights for LWR for any eras after the Expansion Era (1961–77). Because when scoring lots of runs, most runs are the collective actions of the team. And since runs are what wins games, each statistic is measured against Team Runs Scored to see how well it shows how well they did at scoring runs.

To learn more about how Team Runs leads to Team Wins, please feel free to check out Bill James’ Pythagorean Winning Percentage, and the Cook, Soolman, and Smyth variations on Win Percent.

The linear regressions and graphs were created using Python, Anaconda, and the Scikit, Pandas, NumPy, Seaborn, and Matplotlib libraries.

Each card has dotted lines for scissors, like the Hostess cards from boxes of Twinkies in the 1970s and 1980s. Please feel free to cut them out and collect them all.

 

 

Henry Chadwick and Hits Per Game (HPG)

In 1856, a theater and cricket reporter for the New York Times named Henry Chadwick watched a baseball game between the Eagle and Gotham clubs of New York at Elysian Fields in Hoboken, New Jersey.2 According to Thorn and Palmer in The Hidden Game of Baseball, in 1868, Chadwick wrote “It was not long before I was struck with the idea that base ball was just the game for a national sport for Americans…as much so as Cricket in England.”3

By 1857, he was writing for the New York newspaper The Clipper, and started shifting his focus to baseball.4 Because Chadwick’s writings both helped popularize baseball and introduced many of the statistics he concocted, Chadwick has been called “The Father of Baseball.”5 Or, as Alan Schwarz jokes, “baseball’s Frankenstein,” for the dizzying number of statistics Chadwick produced.6

There’s an award named after him—many of those in this article have won this award.7

In 1867, Chadwick had started his own publication called The Ball Players’ Chronicle, and introduced a number of statistics, including Hits Per Game, which is calculated as:

HPG=H/G

Looking at the regression on the Chadwick card, HPG has an R² of .670—so let’s just say it is “batting” over .500. Not bad.

• Strengths
  • For an audience to a new sport, familiarity with Cricket gave a sense of what it meant.
• Weaknesses
  • A player might get into a game and never come to bat.
  • Not expressed in terms of Runs (R).
  • All types of hits have the same values.
  • Does not account for:
    Bases on Balls (BB)
    Stolen Bases (SB)
    Caught Stealing (CS)
    Sacrifice Hits (SH)
    Sacrifice Flies (SF)
    Hit by Pitch (HBP)

 

 

H.A. Dobson and Batting Average (BA)

Hervie Alden “H.A.” Dobson had served in the Union army during the Civil War, losing a leg. A baseball fan, he had been involved with the Flour City baseball club in Rochester, New York.8 Dobson had also served as an umpire in an 1871 game for the Boston Red Stockings and Washington.9

Curiously, Dobson corresponded with President Teddy Roosevelt about chipmunks and pigeons:

My dear Dr. Dobson:

I am interested in your remark. I may be in error about chipmunks invariably hibernating. I thought that they did because I did not see them in the winter. I am sure that the chipmunks occasionally hibernate because I have unimpeachable testimony to that effect.

I am also much interested in what you tell me about the pigeons.

Sincerely yours,
Theodore Roosevelt10

More importantly, Dobson was the Washington, DC correspondent for The Clipper, the same paper as Chadwick also covering theater and baseball.11

On March 11, 1871, Dobson published a letter in The Clipper in which he defines this formula: “[a player’s] average is found by dividing his total ‘times first base on clean hits’ by the total number of times he went to the bat.”12

Basically, he suggests taking HPG and swapping out games for at-bats in the denominator—the Batting Average everyone knows today.13

Comparing the new formula to HPG, Chadwick preferred and adopted the new statistic and wrote, “One is erroneous, one is right.”14 Alan Schwarz notes, “Ironically, this Batting Average, one of the few early statistics not invented by Henry Chadwick, remains to this day baseball’s most famous.”

BA is calculated as:

BA=H/AB

Looking at the R² on the Dobson card, BA is “batting” .690. A step up from HPG.

• Strengths
  • Since it includes hits, it more accurately measures an individual’s contribution.
  • Most baseball fans know it.
• Weaknesses
  • Not expressed in terms of Runs (R).
  • All types of hits have the same values.
  • Does not account for:

    Bases on Balls (BB)
    Stolen Bases (SB)
    Caught Stealing (CS)
    Sacrifice Hits (SH)
    Sacrifice Flies (SF)
    Hit by Pitch (HBP)

 

 

F.C. Lane’s System

From 1912 to 1937, Ferdinand Cole “F.C.” Lane was an editor and writer for a New York City-based monthly magazine called Baseball Magazine.16 He then worked as editor of the Little Red Book of Baseball until 1948. In 1925, Lane published his only book on baseball, Batting, reprinted by SABR in 2001.17

Schwarz tells us Lane encouraged his audience to join the search for better stats, and quotes Lane: “There are many fans, no doubt, who have thought deeply upon [statistics], and will be able to make valuable suggestions.”18

Pete Palmer and John Thorn are fans of F.C. Lane. In the preface of the 2015 edition of The Hidden Game of Baseball, they wrote: “How might we have approached Hidden Game differently—say, if we were to write it afresh today? We would say a good deal about F.C. Lane, whom we unfairly neglected.”19

In the March 1916 issue of Baseball Magazine, Lane describes his system in a couple examples in his article, “Why the System of Batting Averages Should Be Changed:”

Jake Daubert made 120 singles. The value of a single is 30 per cent. of a run; the value of the 120 singles is 36 runs. He made 21 doubles at 60 per cent., equalling [sic] 12 runs. He made 8 triples at 90 per cent., 7 runs. He made two home runs at 1.15 [sic] per cent., 3 runs. Jake batted, according to revised figures, for a total of 58 runs.20

Now, for the method of computing averages. Instead of dividing times at bat by the number of hits made, [sic] as at present, divide them by the total value of hits, as outlined above. To illustrate. Daubert was at bat 544 times. His batting, assigning to each hit its proper worth, was approximately 58 runs; his batting average, therefore, according to this suggested new system, would be .106 (dividing times at bat by total value of hits).21

Rewriting that as a formula:

AB / ((1B x 0.3) + (2B x 0.6) + (3B x 0.9) + (HR x 1.15))

Lane refined those numbers over time: a single was worth .457 runs, doubles worth .786, triples 1.15, and home runs 1.55. In The Numbers Game, Schwarz notes: “These values are almost exactly the same as those that Pete Palmer, working some 60 years later on a method he called Linear Weights, came up with through sophisticated computer regression analysis.”22

Using the original formula, where singles are weighted .3, a linear regression of BA to Runs on team statistics for 1970–77 gives an R² of 0.78, or 78%. An improvement over BA.

Note that the angle of the regression on the baseball card goes down left-to-right. This would seem to mean an increasingly negative co-relation. But, Lane’s equation uses AB as the numerator in the formula, where BA uses it as the denominator. If we switch AB to the denominator in Lane’s equation, does it flip the co-relation?

((1B x 0.3) + (2B x 0.6) + (3B x 0.9) + (HR x 1.15)) / AB

That gets things going in the right direction:

Note also that Lane didn’t give his formula a proper name.

Looking at the R² on the Dobson card, BA is “batting” .780. A noticeable improvement.

• Strengths
  • Gives more appropriate values (weights) to types of hits.
• Weaknesses
  • Does not account for:

    Bases on Balls (BB)
    Stolen Bases (SB)
    Caught Stealing (CS)
    Sacrifice Hits (SH)
    Sacrifice Flies (SF)

 

 

Allan Roth and On Base Average (OBA)

Just before Opening Day, 1947, Allan Roth arrived in Brooklyn to join the Dodgers—the same day as Jackie Robinson.23 Born in 1917 in Montreal, by the time he was 13, Roth was keeping statistics for the International League, where his home team was the Montreal Royals, the top Dodgers’ farm team.24

In 1944, during spring training at Bear Mountain, New York, Branch Rickey was dining at a restaurant when Roth showed up. Roth showed Rickey some of the non-standard stats he’d been unofficially keeping for the Dodgers, like RBI percentage and batters vs. left-handed and right-handed pitching. Rickey was impressed and hired him (though he had to wait until 1947, when he could get a visa).25

Schwarz wrote, “Roth was the Dodgers’ numbers man, as vital to Brooklyn boss Branch Rickey as Robin was to Batman.”26 He was the first full-time statistician hired by a major league team. In newspaper headlines, he was called a “Human Univac [sic].”27

Together, they collaborated on many new statistics to help the Dodgers. In the August 2, 1954, issue of Life magazine, Rickey published the article “Goodby [sic] to Some Old Baseball Ideas,” where he lays out the formula for On Base Average (OBA):28

OBA = (H+BB+HBP) / (AB+BB+HBP)

Looking at the R² on the Roth card, OBA is “batting” .800. Moneyball, here we come!

• Strengths
  • Thanks to the movie Moneyball, everyone knows it.
  • But, by its other name—On Base Percentage (OBP).
• Weaknesses
  • All types of hits have the same value: singles, doubles, triples, home runs.

 

 

Earnshaw Cook and Scoring Index (DX)

In 1964 Earnshaw Cook published Percentage Baseball and in 1972 the follow-up, Percentage Baseball and the Computer. Cook was a retired metallurgist who had played college baseball and was a distant relative of former major-league pitcher George Earnshaw. He also spoke in a fake British accent.29

His writing is dry, but not totally humorless. Cook cracks a joke when he describes baseball’s place in history: “During the past century baseball became the first classic example of a spectator sport developing into a substantial business since the Romans fed wild beasts to the Christians in the Coliseum.”30

Cramer is a fan of the Cook book. Cramer cracks a joke in the same vein, describing Cook’s spot in baseball history: “Considering its relation to the phenomenon that baseball analytics has become, Cook’s work might be metaphorically analogous to Leif Erikson’s voyages or John the Baptist’s prophecies.”31

Thorn also places Cook’s work within the history of Sabermetrics: “An important area that has been little studied is the relationship of runs scored and allowed to wins and losses,” and he adds, “The initial published attempt on this subject was Earnshaw Cook’s Percentage Baseball, in 1964.”32 In other words, can you use hits, runs, etc. to figure out how much a team might win?

Did Cook collaborate with anyone here? According to Schwarz, Palmer wrote “10-page harangues, critiquing their theories on an atomic level and suggesting tweaks throughout” to several proto-sabermetricians, including Cook.33

Cook replied, “I regret that I just do not have the strength of character to argue all the details with you.”34

Cook cooked up the Scoring Index (DX), as defined in Percentage Baseball and the Computer:

DX = (p.OB.1 – q.OB1)(p.TB + p.SB + p.SF)

Where OB.1 is the probability of being On Base, OB₁ is the number of times out while on base, and TB (total bases), SB (stolen bases), and SF maintain their standard definitions. The p stands for probability and the q means Out.

Looking at the R² on the Cook card, DX is “batting” .930. Whoa! First place so far.

Will any stat ever again reach such immortal heights?

• Strengths
  • Thorn and Palmer: “At the time of its introduction, the DX was the most accurate measure of total offensive production yet seen and the first to combine ability to get on base in all manners; to move baserunner around efficiently through extra-base hits; and to gain extra bases through daring running.”
  • Values hits using TB
• Weaknesses
  • TB isn’t the most accurate way to value hits (though that’s offset by counting it once in OBA).
  • Too dry for most folks’ tastes.

 

 

Richard Cramer and On-Base Times Slugging (OXS)

From 1981 to 1996, Richard “Dick” Cramer was co-founder and owner of STATS, Inc. (Sports Team Analysis and Tracking System), and one of their products was Playball. Cramer says, “In 1991, Playball became essentially the only source of major league box scores when STATS also signed up Associated Press, which supplies the box scores in most daily newspapers.”37

Edge 1.000, the first STATS product, was a collaboration between Cramer and Pete Palmer. Cramer programmed the Apple II client in Pascal, while, Cramer said, “Pete Palmer agreed to write the FORTRAN that the PDP-10 would run to generate those splits for broadcasters.”38 It was used to give interesting facts during broadcasts of Oakland A’s, Chicago White Sox, and New York Yankees games.

Back in the late 1960s Cramer worked on a program at Harvard to “teach synthetic chemistry to a computer.”39 This gave him access to a PDP-1 Minicomputer, something relatively few people had access to back then.

Cramer used that microcomputer to run baseball simulations and discovered something: “If runs per plate appearance is graphed against the simple expression on-base average times slugging percentage (OXS), the result is a remarkably clean straight line.”40 That sounds like a linear regression.

Cramer added, “Baseball is a game whose stately pace facilitates discussion, with ‘Who’s better?’ a major topic. My OXS discovery seemed to me the best tool that anyone had ever had to answer this type of question for batters…”41

OXS is calculated as:

OXS = OBA * SLG

Looking at the R² on the Cramer card, OBP is “batting” .910, putting it in second place for now.

• Strengths
  • Easy to understand.
  • Very accurate.
• Weaknesses
  • Cramer: “As [Offense Plus Slugging] OPS’s inventor Pete Palmer observes, addition is a lot easier than multiplication—especially when the multiplication required slide rules or pencil and paper!”42 In other words: too much math?

 

 

Bill James and Runs Created (RC)

From 1977 to 1981, Bill James wrote and self-published the Bill James Baseball Abstract, and continued to write it and have it professionally published from 1982 to 1988. In The Numbers Game, Alan Schwarz describes it as a mix of “statistics, analysis, and wit into one potion that statistically-minded fans guzzled like beer in the bleachers.”43

James first learned baseball statistics as a kid cutting out and collecting baseball cards from Post cereal boxes—which by 1961 included statistics.44 Much later, James spearheaded Project Scoresheet to encourage fans to a shared collection of game data. He also helped relaunch Cramer’s STATS, Inc. after it ran aground.45

In between, James wrote the Baseball Abstracts, which, according to Schwarz, “consumed James’s every waking hour, particularly his 6pm to midnight minimum-wage shift as night watchman at the Stokely-Van Camp factory in Lawrence.”46 With James’ books, sabermetrics exploded in popularity, to a point that there were so many new numbers flying around that James regretted what he called a “Chernobyl of statistics.”47

The 1979 edition of Baseball Abstract introduced the idea of Runs Created (RC). The formula for RC has had many variations over the years, but back in the good old days of the 1979 Baseball Abstract, it was:

(H + W – CS) * (TB + 0.7 * SB) / (AB + W + CS)

As James’ wrote in the 1979 Baseball Abstract, “70 is about an ‘average’ run created total for a full-time player.”48

For the 1970-1977 season, RC batted an R² of .930. Tied for tops so far, with Cook’s DX.

• Strengths
  • Simple formula.
  • Includes Caught Stealing (CS) as a bad thing.
  • Includes a decrease in value for Stolen Bases (SB).
  • Uses Total Bases (TB) to weight the values of hits.
• Weaknesses
  • Uses Total Bases (TB) to weight the values of hits, so some are overvalued.

Looking at the R² on the Cramer card, RC is “batting” .930.

RC is literally in a statistical tie with DX for Greatest of All-Time (GOAT).

 

 

Pete Palmer and Linear Weights Runs (LWR)

In 1984, Pete Palmer and John Thorn published the first edition of The Hidden Game of Baseball: A Revolutionary Approach to Baseball and Its Statistics. The pair also wrote for and edited numerous editions of Total Baseball: The Official Encyclopedia of Major League Baseball.

Palmer “collected Bowman bubble gum cards and, with his obsession for completeness forming early, filled every set from 1948 to 1952.”49 Bowman baseball cards did not include statistics, though. Statistics didn’t find their way onto baseball cards until 1952, when Sy Berger of Topps decided to add them to the back of their cards to “grab the interest of kids.”50

Like Cramer, Palmer had access to a mainframe computer in the 1960s, working for a think tank employed by the US Air Force. To Palmer, “It was great, because I had the use of this million-dollar mainframe that I could put all my stats on!”51

As Schwarz describes The Hidden Game: “Whereas Bill James’ Baseball Abstract series was then at its height of giving fans oodles of new statistical gizmos, The Hidden Game was a more comprehensive manifesto, looking at all of baseball’s traditional statistics and examining possible replacements.”52

In the book Palmer describes Linear Weights Runs, which uses different weights for each category by time period, covering familiar eras of baseball. From page 65 of the 1985 edition of Thorn’s and Palmer’s The Hidden Game of Baseball:

 

 

For fun, this table includes F.C. Lanes’ values, with “n/a” for outcomes Lane didn’t consider.

Remember when Schwarz wrote, “These values are almost exactly the same as those that Pete Palmer, working some 60 years later on a method he called Linear Weights”?53 Lane was in the ballpark.

The Linear Weights Runs (LWR) equation, using values from the Expansion column:

Runs = (0.45)1B + (0.77)2B + (1.00)3B + (1.42)HR + (0.33) (BB + HB) + (0.19)SB – (0.32)CS – (0.25)(AB – H) –0.50(OOB)54

I opted to exclude Outs On Base (OOB) and opted to use the following formula in the regression on the card, substituting the variables for the values from the “Expansion 1961–77” column:

Runs = .045*1B + 0.77*2B + 1.00*3B + 1.42*HR + 0.33* (BB + HBP) + 0.19*SB – 0.32*CS – 0.25*(AB – H)55

Let’s call that difference the “slop factor.” It might have something to do with the results.

Looking at the R² on the Palmer card, LWR “batted” .860. Even with the “slop factor” that’s still in there with the modern stats. Including OOB, it might be George Brett’s 1980 season.

• Strengths
  • Uses weights for each offensive outcome based on values calculated for an era.
• Weaknesses
  • Lumps weights into eras, instead of calculating for each season.

 

 

Tom Tango and Weighted On-Base Average (wOBA)

When published in 2007, The Book: Playing the Percentages in Baseball, authors Tom Tango, Michael G. Lichtman, and Andrew Dolphin start with the idea that when baseball manager play baseball “by the book,” there are a set of unproven strategies managers base their decisions on to help their team win. They set out to test these strategies through statistics. They acknowledge that their work builds on sabermetricians before them. Particularly, Pete Palmer.

For example, when they introduce one of their tools, called Run Expectancy, they display a table of values they have calculated for the 1999–2002 seasons, then add, “Pete Palmer produced a similar chart, but based on a different run environment, in his classic book, The Hidden Game of Baseball, which we recommend you pick up at your local library or favorite online store.”56 They are Palmer fans.

Palmer himself wrote the foreword and clearly places The Book in its spot in history. Palmer mentions how when he started out in the 1960s, there were no play-by-play data. Palmer talks about how proto-sabermetricians had to find play-by-play data on their own: George Lindsey gathered his own data for 300 games between 1956 and 1960, Cook used game simulations, while Eldon and Harlan Mills had actual computerized data from the Elias Sports Bureau for the 1969 and 1970 seasons but did not analyze strategy.57

After James started Project Scoresheet Gary Gillette carried it into the 1990s. Dave Smith started Retrosheet in 1989 and he and Gary combined their data with other sources, like STATS, Inc.58 By the 2000s there were plenty of play-by-play data available to analyze and The Book takes advantage of that.

In The Book, Tango, Lichtman, and Dolphin introduce the Weighted On-base Average (wOBA) with a bit of an apology which echoes of James’ “Chernobyl of statistics”:

Do we really need another statistic? Yes, we do. Instead of trying to take two statistics (OBP, SLG) and combine and correct their flaws in the hopes of getting one number, we prefer to start from scratch.59

As listed in the book The Book the calculation for wOBA is:60

 

 

Where NIBB is Non-Intentional Base-on-Balls, RBOE is Reached Base On Error, and the others are the others. The actual numbers are called the wOBA Constants. The Book has information on calculating those, but they can be found precalculated online.61

Also in The Book, Tango, Lichtman, and Dolphin offer some leeway for the calculation of plate appearances, when they write, “Depending on the specific analysis, the PA term (plate appearances) may exclude (sacrifice) bunts, IBB, and a few of the more obscure plays.”62

Instead of RBOE, I opted to use this modified calculation for the regression on the Tango card:

 

 

Looking at the R² on the Tango card, wOBA “batted” .860. Again, even with the “slop factor,” it is tied even with LWR—so they are both Brett’s 1980 season.

• Strengths
  • Scaled like OBA, so it is familiar to many.
  • Values for offensive outcomes are calculated each season.
• Weaknesses
  • This is where things may start getting complicated.

CONCLUSION

To summarize:

 

Allowing for the slop factor in the wOBA and LWR regressions, it does look like the average average has averaged up over time.

By the regressions done here, James’ RC and Cook’s DX are tied for top of the pops with .930, Cramer’s comes in next-to-tops at .910, while Palmer’s LWR and Tango’s wOBA are tied for third-to-top.

Given the slop factor resulting in incomplete calculations for wOBA and LWR, it may well be those two should each have a stronger R² than resulted here. But please, keep in mind that this research was done by a Mathephobe for other Mathephobes, to hopefully help us get over at least some measure of Mathephobia.

And of course, this was just a tiny, little history of offensive baseball statistics up to a point.

Things like WAR (Wins Above Replacement) are best left to professionals.

WILL MacLEAN once interviewed for the role of director of software engineering, baseball systems for the Chicago Cubs. He didn’t get the gig, but now he can joke he had a major-league tryout. He is actually a Southside fan anyway.

 

Acknowledgments

Thanks to Plastic Crimewave for the baseball card artwork and Rick Reiff for his help in understanding F.C. Lane. And to Sean Lahman for the baseball database and the generous donation of the database to SABR.

The freeware fonts Check Book and Komica were used in additional to the original artwork:

https://www.fontspace.com/checkbook-font-f745,
https://www.1001fonts.com/komika-font.html

 

Abbreviations

• AL for American League
• BA for batting average
• BB for base on balls, walks
• CS for caught stealing
• DX for Scoring Index
• HBP for hit by pitch
• HPG for hits per game
• HR for home runs
• LWR for Linear Weights Runs
• NIBB for non-intentional base on balls
• NL for National League
• OB for on base
• OBA for On Base Average
• OBP for On Base Percentage
• OOB for outs on base
• OXS for On Base Percentage Times Slugging
• R for runs
• RBOE for reached base on error
• RC for Runs Created
• R2 for R-Squared
• SB for stolen bases
• SF for sacrifice flies
• SH for sacrifice hits
• TB for total bases
• W for walks, base on balls
• WAR for Wins Above Replacement
• wOBA for Weighted On Base Average
• 1B for single
• 2B for double
• 3B for triple

 

Sources

In addition to the sources cites in the Notes, the author consulted Baseball Reference, Baseball Almanac, and MLB.com.

 

Notes

1. Dick Cramer, “Average Batting Skill Through Major League History,” SABR Baseball Research Journal, 1980. https://sabr.org/journal/article/average-batting-skill-through-major-league-history.

2. Andrew Schiff, “Henry Chadwick,” SABR BioProject. https://sabr.org/bioproj/person/henry-chadwick.

3. John Thorn and Pete Palmer, The Hidden Game of Baseball: A Revolutionary Approach to Baseball and Its Statistics (New York: Doubleday & Company, 1984), 10.

4. Alan Schwarz, The Numbers Game (New York: St. Martin’s Press, 2004), 7.

5. Schwarz, 7–8.

6. Schwarz, 18.

7. “Henry Chadwick Award,” SABR. https://sabr.org/awards/henry-chadwick, accessed November 24, 2024.

8. “Sgt Hervie Alden Dobson,” Find a Grave. Accessed on November 24, 2024, https://www.findagrave.com/memorial/43222231/hervie-alden-dobson.

9. Bob LeMoine, “May 5, 1871: Red Stockings win first regular-season game in Boston baseball history,” SABR Games Project. https://sabr.org/gamesproj/game/may-5-1871-red-stockings-win-first-regular-season-game-in-boston-baseball-history.

10. Theodore Roosevelt, “Letter from Theodore Roosevelt to Hervie A. Dobson,” Theodore Roosevelt Center, November 1, 1907. Accessed on December 24, 2024, https://theodorerooseveltcenter.org/Research/Digital-Library/Record/ImageViewer?libID=o200601.

11. “Sgt Hervie Alden Dobson,” Find a Grave. Accessed on November 24, 2024: https://www.findagrave.com/memorial/43222231/hervie-alden-dobson.

12. H.A. Dobson, “The Professional Club Secretaries’s Meeting,” New York Clipper, March 11, 1871. Reprinted in John Thorn, “Chadwick’s Choice: The Origin of the Batting Average,” (ourgame.mlblogs.com), September 18, 2013. Accessed on December 26, 2024: https://ourgame.mlblogs.com/chadwicks-choice-the-origin-of-the-batting-average-e8e9e9402d53.

13. John Thorn. “Chadwick’s Choice: The Origin of the Batting Average,” (ourgame.mlblogs.com), September 18, 2013. Accessed on December 26, 2024: https://ourgame.mlblogs.com/chadwicks-choice-the-origin-of-the-batting-average-e8e9e9402d53.

14. Schwarz, 11.

15 Schwarz, 11.

16. Schwarz, 11.

17. Fred Ivor-Campbell, “F.C. Lane,” SABR BioProject. Accessed December 26, 2024: https://sabr.org/bioproj/person/f-c-lane/.

18. Schwarz, 37.

19. John Thorn and Pete Palmer, The Hidden Game of Baseball: A New Edition of the Baseball Classic that Ignited the Sabermetric Revolution (New York: Doubleday & Company, 1984), xvi.

20. F.C. Lane, “Why the System of Batting Averages Should Be Changed,” Baseball Magazine, March 1916: 41–47. Accessed December 23, 2025: https://courses.edx.org/c4x/BUx/SABR101x/asset/Batting_Aveage_Changes_1916_FC_Lane.pdf.

21. Lane.

22. Schwarz, 36.

23. Schwarz, 56.

24. Schwarz, 56.

25. Schwarz, 56.

26. Schwarz, 54.

27. Schwarz, 58.

28. Branch Rickey, “Goodbye to Some Old Baseball Ideas,” Life, August 2, 1954: 80. Accessed on January 20, 2025: https://books.google.ca/books?id=9FMEAAAAMBAJ&lpg=PP1&pg=PA78&hl=en#v=twopage&q&f=false.

29. Schwarz, 76.

30. Earnshaw Cook, Percentage Baseball and the Computer (Baltimore: Waverly Press, Inc., 1971), 12.

31. Richard D. Cramer, When Big Data Was Small (Lincoln: University of Nebraska Press, 2019), 26–27.

32. John Thorn, “Runs and Wins,” Our Game, July 21, 2014. Accessed on January 5, 2025: https://ourgame.mlblogs.com/runs-and-wins-2f94b7fd4351.

33. Schwarz, 162.

34. Schwarz, 162.

35 Cook, 31.

36 Thorn and Palmer, 26-46.

37. Cramer, When Big Data Was Small, 154.

38. Cramer, When Big Data Was Small, 86.

39. Cramer, When Big Data Was Small, 35.

40. Cramer, When Big Data Was Small, 41.

41. Cramer, When Big Data Was Small, 49.

42. Cramer, When Big Data Was Small, 42.

43. Schwarz, 112.

44. Schwarz, 113.

45. Schwarz, 112.

46. Schwarz, 115-116.

47. Schwarz, 112.

48. Bill James, The 1979 Baseball Abstract (Lawrence, Kansas: Bill James, 1979), 30.

49. Schwarz, 162.

50. Schwarz, 60.

51. Schwarz, 164.

52. Schwarz, 162.

53. Schwarz, 36.

54. Thorn and Palmer, 66.

55. “Understanding Linear Weights,” FanGraphs, March 9, 2011. Accessed January19, 2025: https://triplesalley.wordpress.com/2011/03/09/understanding-linear-weights/.

56. Tom Tango, Michael G. Lichtman, Andrew E. Dolphin, The Book: Playing the Percentage in Baseball (Dulles: Potomac Books, Inc., 2007), 19.

57. Tango, 12.

58. Tango, 12–13.

59. Tango, 30.

60. Tango, 30.

61. “wOBA and FIP Constants,” FanGraphs, https://www.fangraphs.com/guts.aspx?type=cn. Accessed November 23, 2024.

62. Tango, 30.