Weighted WAR: Schedule Length Adjustment and Third Base Rankings

Resuming the discussion of the weighted WAR system, we start with (hopefully) the least controversial modification of the WAR totals from individual seasons: schedule length.

Schedule length is an obvious problem when comparing players across seasons. Fortunately, using estimates of wins provided by a player (e.g. WAR) allows for a fairly systematic approach here. If a player posts a 3-WAR season, it will have a much bigger impact on an average team over a 60-game schedule (improving winning percentage by .050) than over a 125-game schedule (.024) or a 160-game schedule (.019). But a straight-line adjustment doesn’t quite work, as the standings will generally have a wider spread after 60 games than after 160. What you’d want to do is estimate how much impact an individual win has on the standings at various schedule lengths.

So that’s what I did. A little over a decade ago, I entered the records of every team for nearly 50 full MLB seasons at the end of every (roughly) 10 games through their schedule, then used this data to produce a curve estimating the standard deviation in winning percentage across various schedule lengths. The equation for this curve is as follows, where N is the number of games:

(I previously published this curve in more detail on Baseball Think Factory, which now appears to be defunct; if there’s interest, I can go into more detail on how it was developed.)

Multiply this value by N to get the expected standard deviation expressed in wins rather than winning percentage. In order to translate a WAR total in a shortened season to the equivalent total in the now-standard 162-game campaign, multiply by the ratio in the standard deviations for the two schedule lengths. So Manny Machado’s 3.2 WAR in the COVID-shortened 2020 season doesn’t translate to 8.6, as it would with a straight-line adjustment, but it still works out to 6.9, which is a noteworthy improvement.

For a vast majority of team seasons (including every single team-season from both the AL and NL), the calculation is exactly that straightforward. There is, however, a special case that arises at times in both 19^th-century baseball and the Negro Leagues: what happens when a team folds midseason?

On the surface, the question seems simple. If a team folds after 40 games of a 120-game season, their schedule length is 40 games and that’s what the adjustment should be based on. However, once the team folds, the players on the team might find employment elsewhere, and effectively be credited with more than a full season of play within the same year.

Since we’re going through the top 100 third basemen by this method, we’ll use the oldest member of the list as an example: Davy Force. In 1872, Force played for two different teams in the National Association – 25 games for the Troy Haymakers, and 19 for the Baltimore Canaries. As you might expect for a midseason team change, the Canaries played 58 total games, with Force participating in roughly a third of them. The Haymakers, meanwhile, played a total of… 25 games, meaning Force played in all of them.

What I ended up doing to adjust for this is as follows: Take the median number of team games for each league season (in modern leagues, this will generally be the standard schedule length, but there’s a lot more variation in older, less stable leagues). If a team plays in at least 80% of this total, you take their schedule adjustment at face value. If they are below 80% of the median, you use the schedule length adjustment of the lowest team above 80% of the median (or the median itself, whichever is less). This generally provides a reasonable adjustment for leagues with more instability… unless there’s a LOT more instability, which was the case in the 1872 NA. (The adjustment cutoff that year ends up being 24 games, so Force’s 25-game stint for the Haymakers is played straight.)

Force’s situation is uncommon for two reasons. First, most teams that play severely shortened schedules will have their adjustments more aggressively modified. And second, most teams that play severely shortened schedules will do so because they’re not good enough to draw fans, and therefore they may not have any players anywhere near a positional top 100 to begin with.

On that note, let’s move in the direction of our other topic, the top 100 third basemen by weighted WAR. Here are the groups most affected by schedule-length adjustment (note that some of the other adjustments are applied before schedule length, which I wish I hadn’t done but by the time I realized schedule length could be done first without changing the overall results, it would have taken way too much effort to change the order of operations). Adjustment totals are to career WAR (with negative seasons removed; more on that later).

Negro League players:

Player	WAR	Schedule Adj
Jud Wilson	32.7	+26.6
Hank Thompson	31.8	+7.4
John Beckwith	21.8	+16.5

Much more to come on NeL players as we continue in this series; for now, suffice it to say that the limited documentation of their individual stats results in some truly prodigious schedule adjustments.

19^th-century players:

Player	WAR	Schedule Adj
Lave Cross	46.6	+3.0
John McGraw	46.0	+4.7
Ed Williamson	36.0	+12
Denny Lyons	35.6	+4.0
Billy Nash	33.8	+3.3
Ezra Sutton	33.0	+12.2
Bill Joyce	31.1	+3.1
Davy Force	17.8	+13.5

Note the highly disparate adjustment totals here; the schedules were much shorter in the 1870s than in the 1890s, and you can see the results of that pretty clearly.

Players active in 2020:

Player	WAR	Schedule Adj
Evan Longoria	58.7	+0.8
Manny Machado	57.9	+3.7
Nolan Arenado	56.8	+1.7
Jose Ramirez	52.4	+2.9
Josh Donaldson	46.5	+0.6
Alex Bregman	39.6	+1.3
Justin Turner	38.4	+1.7
Matt Chapman	38.4	+1.4
Kyle Seager	37.0	+1.3
Anthony Rendon	34.0	+2.5
Matt Carpenter	28.8	+0.2
Kris Bryant	27.5	+0.6
Todd Frazier	25.2	+0.5

The main question here, of course, is how good the player’s 2020 season was. Machado was great that year; Carpenter, not so much. (Ryan Zimmerman, another top-100 3B who was active as of 2020, sat out the season and gets no adjustment.)

Players from the early 20^th century:

Not posting the entire subset of the top 100 here, as this would be nearly a quarter of the top 100 list; just adding enough to get a feel for the size of the adjustments.

Player	WAR	Schedule Adj
Eddie Mathews	96.2	+2.3
Ken Boyer	62.9	+1.5
Home Run Baker	62.7	+3.2
Stan Hack	55.6	+1.9
Jimmy Collins	53.3	+4.0
Bob Elliott	51.1	+1.6
Heinie Groh	48.4	+2.3
Larry Gardner	48.2	+2.1
Harlond Clift	42.0	+1.3
Gil McDougald	40.9	+1.6

 This is mostly the 154-game schedule adjustment, although players from the very beginning of the century (Collins) and players who were affected by shortened schedules due to World War I (Baker, Groh, Gardner) will get bigger bumps.

Players from notable strike seasons (1981, 1994-95)

Player	WAR	Schedule Adj
Mike Schmidt	106.9	+3.0
Wade Boggs	91.5	+1.9
George Brett	88.6	+1.5
Chipper Jones	85.1	+0.3
Graig Nettles	68.0	+1.4
Buddy Bell	66.4	+2.9
Sal Bando	61.7	+0.3
Darrell Evans	58.9	+1.2
Robin Ventura	56.0	+1.4
Ron Cey	53.8	+1.3
Toby Harrah	51.7	+1.1
Matt Williams	46.5	+2.0
Gary Gaetti	42.2	+1.1
Doug DeCinces	41.9	+1.2
Carney Lansford	40.5	+1.4
Tim Wallach	38.7	+1.0
Bill Madlock	38.4	+1.5
Jeff Cirillo	34.5	+0.5
Ken Caminiti	33.5	+1.5

Again, not the full list of affected top-100 players, but an extremely impressive group (even omitting Dick Allen and Ron Santo’s 1972 seasons; the impacts there are much smaller).

So, why start with third base? First, I feel like Mike Schmidt is one of the least controversial choices for all-time greatest at his position; he dominates in WAR total and is recent enough that it’s hard to timeline him away. Second, I wanted to start the methodological discussion with schedule length, and Schmidt’s best season (with the adjustment) came in a strike year. Add at least one (and often far more than one) top-30 player in each of the above categories, and the topical match seems obvious.

We move now from tables to… more tables! First, a list of active players (as of 2024) who are in or within reasonable striking distance of the weighted WAR top 100 at third base. We’ll include their overall WAR numbers in three categories: raw WAR total, adjusted WAR total (aWAR), which includes adjustments not discussed yet, and weighted WAR total (wWAR). We’ll also look at how much 2024 affected the standing of the players who ended the season inside the top 100.

Player	Rank	Years	WAR	aWAR	wWAR	2024 WAR	Rank Change
Manny Machado	13	2012-24	57.9	61.6	48.0	3.1	+2
Nolan Arenado	16	2013-24	56.8	58.5	46.3	2.5	+1
Jose Ramirez	18	2013-24	52.4	55.4	45.0	6.8	+5
Alex Bregman	31	2016-24	39.6	40.9	35.2	4.1	+4
Matt Chapman	33	2017-24	38.4	39.8	34.6	7.1	+20
Justin Turner	40	2009-24	38.4	40.7	32.2	1.6	+6
Anthony Rendon	42	2013-24	34.0	36.5	31.9	0.6	+3
Kris Bryant	65	2015-24	27.5	29.8	26.6	-0.7	0
Matt Carpenter	66	2011-24	28.8	30.7	26.4	-0.1	0
Eugenio Suarez	96	2014-24	23.6	24.8	20.9	3.1	+10
Rafael Devers	97	2017-24	22.6	23.4	20.8	3.7	+20
Austin Riley	99	2019-24	20.9	21.5	20.2	2.9	+22
Ke’Bryan Hayes	143	2020-24	13.4	15.6	14.5
Yoan Moncada	150	2016-24	14.4	15.5	14.1

At the beginning of 2024, there were three third basemen within reasonable striking distance of the top 100. All of them posted quality seasons, but none were especially spectacular – and yet, all of them moved up at least 10 positions and made the top 100. Moreover, look at Anthony Rendon, who barely plays baseball at this point. Even his lackluster 0.6 WAR effort in 2024 moved him up 3 positions. Which is to say, the top 100 list at any single position is likely to get very tightly packed once you get beyond the top 20 or so. If a 2.9-WAR season can move you from #121 to #99, there’s not much of a difference between the players in those spots.

That being said, with Suarez, Devers, and Riley having all cracked the top century this year, nobody new looks likely to join within the next couple of seasons.

You can also start to see how the peak weighting affects the rankings – for instance, Chapman and Turner have identical career totals, with Turner even leading in adjusted WAR, but Chapman’s higher peak (three seasons over 7 WAR) allows him to take a respectable lead over Turner (career high of 5.6).

Second, here are the top 25 third basemen by weighted WAR, followed by the players at each subsequent multiple of 10 from 30 to 100.

Player	Rank	Years	WAR	aWAR	wWAR
Mike Schmidt	1	1972-89	106.9	110.0	76.9
Wade Boggs	2	1982-99	91.5	93.3	66.3
Eddie Mathews	3	1952-68	96.2	94.6	66.2
George Brett	4	1973-93	88.6	90.2	63.3
Adrian Beltre	5	1998-2018	93.7	93.5	61.0
Chipper Jones	6	1993-2012	85.1	85.1	56.8
Ron Santo	7	1960-74	70.6	70.3	55.8
Brooks Robinson	8	1955-77	78.4	77.8	54.2
Scott Rolen	9	1996-2012	70.0	70.0	51.4
Buddy Bell	10	1972-89	66.4	69.5	50.5
Graig Nettles	11	1967-88	68.0	69.1	49.7
Ken Boyer	12	1955-69	62.9	61.7	48.3
Manny Machado	13	2012-24	57.9	61.6	48.0
Sal Bando	14	1966-81	61.7	61.2	47.3
Dick Allen	15	1963-77	58.6	58.2	47.2
Nolan Arenado	16	2013-24	56.8	58.5	46.3
Evan Longoria	17	2008-23	58.7	59.5	45.6
Jose Ramirez	18	2013-24	52.4	55.4	45.0
Darrell Evans	19	1969-89	58.9	61.6	44.6
Home Run Baker	20	1908-22	62.7	54.9	43.9
Robin Ventura	21	1989-2004	56.0	56.8	42.6
Ron Cey	22	1971-87	53.8	54.8	41.7
Jud Wilson	23	1923-45	32.7	51.3	41.1
David Wright	24	2004-18	49.4	49.4	40.4
Toby Harrah	25	1969-86	51.7	52.3	40.0
John McGraw	30	1891-1907	46.0	42.9	35.9
Justin Turner	40	2009-24	38.4	40.7	32.2
Aramis Ramirez	50	1998-2015	32.6	37.7	30.5
Ken McMullen	60	1962-77	33.9	33.6	27.9
Terry Pendleton	70	1984-98	28.4	30.5	25.6
Chase Headley	80	2007-18	26.1	26.7	23.0
Todd Frazier	90	2011-21	25.2	26.1	21.7
Billy Nash	100	1884-98	33.8	24.1	20.0

Since this is our first top-100 list, let’s make some general observations. As noted below the active players list, the gaps as you get toward the bottom half of the top 100 become vanishingly small. The distance between #80 Headley and #90 Frazier would be covered by one 2.5-WAR season, yet there are nine players between them. The paper-thin differences between players this low in the rankings render any pretense of precision rather absurd, and this will be a recurring theme as we continue. (This is why I’m not posting the full top-100 list; the return on effort invested diminishes significantly after a while. If you’re curious where a particular player ranks, let me know!)

Third base also serves as a useful introduction to these lists as its #100 score is the median among the non-catcher positions, so it gives a reasonable idea of where the cutoff points will usually end up.

On to position-specific conclusions. Third base skews very modern, especially at the top of the list; only two players in the top 25 debuted before 1950, and the higher-ranked of them is Home Run Baker at #19. Part of this is expansion bringing more players into the league, and part is the timeline adjustment, which we’ll cover later. But part of it is also the evolution of the position over time, as Mathews, Robinson, Boyer and Santo served to usher in the golden age of third base. Per weighted WAR, 10 of the top 15 third basemen of all time started their careers between 1952 and 1973.

One additional note: Most of the players listed in the above table spent all or nearly all of their careers at third base, but some (notably Allen, Evans, and Harrah) spent significant time at other positions. Allen won his 1972 AL MVP as a first baseman, while Harrah had his best year (1975) as a primary shortstop. Those seasons both contribute to their overall scores, and therefore to their rankings at third base – but should they? Up next, we’ll discuss the topic of positional classification in all-time rankings, while also exploring the top 100 left fielders by weighted WAR.

Introducing Weighted WAR

The last several decades have seen a veritable explosion of baseball statistical analysis, both by fans and by actual MLB teams. One of the primary products of this effort has been Wins Above Replacement (WAR), the general estimate of the number of wins a particular player provided in comparison to a freely available substitute. The merits of WAR have been debated ad nauseam since its introduction, but its appeal is obvious; it combines all of a player’s contributions (or at least, all that we have measurements for) into a single number that allows direct comparisons between players on different teams, at different positions, or from different time periods. For all its faults, WAR at least serves as a reasonable estimate and a good place to begin discussion.

It is not, however, the end of the discussion. I have been experimenting with a series of adjustments to baseline WAR totals which I feel lead to a more reasonable overall rating of baseball players throughout history, and will be presenting both those adjustments and some of the results over a series of upcoming posts. The adjustment I started with is a method of combining peak and career value into one number.

Career value is a straightforward concept – just add up the WAR totals from each of the player’s individual seasons and you get an estimate of the number of wins he added to his teams over the course of his career. Peak is more nebulous. It can be broadly defined as the player’s performance in his best seasons, but the follow up question is obvious: how many seasons constitute a peak?

Various analysts have used various answers to this question. In the New Historical Abstract, Bill James uses both best three seasons and best five consecutive seasons. The JAWS system, developed by Jay Jaffe and currently published by Baseball Reference, uses the best 7. One could reasonably argue that a true peak should just be the player’s best season; one could also make a case for a top 10 measure, trying to capture a player’s extended prime rather than their absolute pinnacle.

None of these answers is necessarily wrong, but each of them presents a different version of the same problem. Whatever cutoff you use when defining a player’s peak will arbitrarily benefit a subset of players who have exactly the right number of outstanding seasons.

If you focus only on a player’s single best season, Norm Cash does extremely well, with 9.2 Baseball Reference WAR (bWAR) in 1961. His second-best season was 1965, with a comparatively unimpressive 5.4. If you go for top two years, you’ll be fairly fond of Roger Maris, whose best years are a 7.5 and a 6.9 (which won him back-to-back MVPs), followed by a drop to 3.8. Top 3? Carl Yastrzemski looks spectacular (12.5, 10.5, 9.5); never mind the subsequent drop to 6.6.

The problem is mitigated somewhat if you increase the number of seasons, but it never goes away entirely. Ernie Banks has a terrific top four seasons (10.2, 9.3, 8.1, 7.9); his next four entries drop off by about one win each (6.7, 5.3, 4.6, 3.5). He looks gradually worse as your peak consideration set increases from four years to eight. Speaking of shortstops, Nomar Garciaparra has six years between 6.1 and 7.4 bWAR; his seventh-best season is a 2.5. Troy Tulowitzki is in the same boat, with a top six between 5.0 and 6.8, then a drop to 3.2. It’s not a new phenomenon either; George Sisler had a tremendous six-year peak from 1917-22 (all between 5.7 and 9.7 bWAR), plus a very solid 4.1 in 1916. After that, he developed double vision and was never the same, failing to exceed 2.7 bWAR in any of his remaining seven seasons.

How can we avoid the problem of arbitrarily favoring a particular subset of players that benefits from choosing a particular number of peak seasons? Simple: Don’t choose a particular number. Instead, take the average of (best season), (best two seasons), (best three seasons), and so on. You still have to end at some arbitrary point (I’m ending it after 19 years and adding total career rating as a twentieth term in the average), but the effect of the arbitrariness is greatly reduced. (A vast majority of players don’t play 19 seasons at all; even among those who do, very few of them have 19 GOOD seasons. By my count, only one position player in baseball history has over 20 seasons with noteworthy positive WAR values, and he’s been dead for over a century.)

Fortunately, it is not necessary to calculate totals for the top 2 and top 3 and top 11 seasons for each player. The average presented above is mathematically equivalent to the following, which is much easier to calculate:

Weighted WAR = Best season + 0.95*(season 2) + 0.9*(season 3) + … + 0.1*(season 19) + 0.05*(all seasons past 19).

This is the formula whose results we’ll be examining moving forward. It is intended to present a reasonable hybrid of peak, prime and career value; I think it works quite well for this purpose, but the results will be left to the reader’s evaluation.

The weighting formula itself, however, is just one of many adjustments that will be presented. We’ll be exploring the others while also presenting the overall results of the evaluation via the medium of top 100 lists at each non-pitching position.

A few additional notes on the project: First, which players are considered for the rankings? The ideal answer would be “all of them,” but sadly, the time and computing power available to me are both limited, so I had to pick a cutoff of some kind. I ended up entering every player who has at least one (schedule adjusted) 3-WAR season. That gives me a set of over 2400 players, including over 250 at every non-DH position. It also covers all but two of the top 1000 players by total bWAR (for position players); I entered both of them into the database just in case, and neither one made the top 100 at his position. It is still possible that I’ve missed someone who deserves a top 100 spot, but I think it's unlikely.

As a housekeeping note, one of the additional adjustments made in order to calculate weighted WAR is both obvious and (hopefully) uncontroversial. If a player is traded midseason, his WAR totals for each part of the season are added together before the weighting is done. This seems barely worthy of mention except for the fact that Baseball Reference has not always done this automatically in the WAR tables presented on its player pages. As of last year, if you simply scanned Mark Teixeira’s page, he appeared to have a stretch in the middle of his career including seasons of 2.6, 2.0, 4.1 and 3.7 WAR. In actuality, Teixeira was traded in both 2007 and 2008, which obscured his much better seasonal totals of 4.6 and 7.8 WAR.

With the exception of Negro League data (to be discussed in more depth later), all WAR values presented will be taken from Baseball Reference. This is not because bWAR is necessarily the best system; it is, however, readily available for all of recorded baseball history, from 1871 to the present. (This is also true of Fangraphs WAR; I have a slight preference for bWAR as its methodology is more clearly explained on the site, particularly for older players. Every other WAR source I am aware of has missing seasons.)

Finally, it should be noted that while the weighting formula and the other adjustments presented here are my work, the positional top 100 lists themselves are not my own personal top 100 lists. I would tend toward using a combination of WAR systems in order to mitigate the idiosyncrasies of any individual measure, and even with that in mind, all WAR systems will still omit factors that I would consider (some of which will also be discussed as we go through the rankings). So don’t sweat it if you disagree with the system’s outcomes; so do I. This is less an exercise intended to produce unassailable results than it is an exploration of what we can accomplish using a single WAR system as a starting point.

With all that in mind, next up will be our first ranking list (third base), combined with a discussion of how to adjust for schedule length.

Melog Rankings: Post-Indian Wells 2017

So, this is later than I thought it would be. Through a combination of happenstance and procrastination, I failed to run fully updated rankings before Indian Wells started - and I would have put them up after the tournament began, except that the draw promised extensive madness and then delivered in spades, and it seemed like a good idea to let that play out.

Everyone knows the big story: Roger Federer sat out the last six months of 2016, then came back to start this year and has won the two biggest events played to date. Which is marvelous theater, of course, but it's accompanied by so many other things that I'm reluctant to discuss them in too much detail before we get to the numbers. So let's do that.

Melog Rankings: Post-Australian Open 2017

At the end of last year, tennis appeared to be well on its way out of the Big Four era. Yes, Andy Murray and Novak Djokovic were still the top two players in the world (and with a wide gap between them and everyone else), but their counterparts in the group, Rafael Nadal and Roger Federer, had both struggled with various injuries throughout the year as age took its inevitable toll on them.

One month later, two of the members of the Big Four absorbed early, shocking upsets in Melbourne, while the other two faced off in the final of the season's first Grand Slam, with the winner overcoming a long-standing, well-known head-to-head disadvantage. If that description had been provided to anyone before the event began, it would have been expected that Murray had finally bested Djokovic and claimed his first Australian Open crown.

Instead, it was Federer and Nadal facing off in their first Grand Slam final in almost six years - a remarkable match that more than lived up to the hype (and easily justified my decision to wake up at 2:30 AM to watch it live). Federer's victory, rallying from a break deficit in the fifth set, gives him an 18th Grand Slam title. We'll cover the historical ramifications of said title when we update Grand Slam Scores in a later post, but for now, how does this affect his standing in the shorter-view Melog ratings?

Melog Rankings: Post-Finals 2016

One of the things I like most about tennis is the way it rewards long-haul performance. So many other sports seem to be all about the playoffs; you can win 116 games in a baseball season (like the 2001 Mariners), or 73 games in a basketball season (like last year's Warriors), and still have people scoff at you if you get upended in a short playoff series (or, like the 18-1 2007 Patriots, one game of football).

Tennis is different. While it has its own version of the playoffs, taking the top 8 players of the year and matching them up with each other in the World Tour Finals, it does not automatically designate the winner of that event to be the champion of the year. Instead, the World Number One is the player who performed best through the grind of the ten-month tennis season.

That designation is often secured very early - in 2015, for instance, Novak Djokovic clinched the top spot by winning the US Open in early September. Through the first half of the 2016 season, it looked like we'd have another blowout, as Djokovic won the first two Slams of the year, and three of the first five Masters. After he wrapped up the French Open title (completing the career Slam, and making him the first player in nearly 50 years to hold all four Slams at once), his lead in the 2016 points race was over 3000.

And after that... well, after that, Andy Murray had the run of his life, winning seven titles from nine events stretching from Queen's Club to Paris. Djokovic won only one title in that span, and entering the World Finals, Murray held the #1 spot by a margin of just over 400 points - a gap that could easily be closed using the points available in the Finals.

The event played out perfectly, as Djokovic and Murray both swept their round robin groups and won their semifinal matches, bringing up a first in tennis history: the last match of the Tour Finals would result in the year-end #1 ranking being awarded to the winner, whichever player that was.

Murray won the match, and the title, and the year-end #1 ranking. And given how the Melog ratings have looked during the last couple of updates, the top of the upcoming table should come as little surprise...

2016 MLB Postseason: The statistical view

THE CUBS WON THE WORLD SERIES!!!!!

It's been just over a week and it still hasn't fully sunk in. But I'll try to cut through the haze of post-title bliss and be as rational as possible in breaking down what the numbers have to say about the highly memorable 2016 postseason.