2000 Montreal Expos
Baseball Player Won-Loss Records
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2000 Montreal Expos*
Traditional Statistics and Splits (Retrosheet)
Montreal Expos - Franchise Page

(Regular Season)

(2000 National League)
Positional Average (Weighted Avg of Options): 0.500 (1-Year) (9-Year) (Long-Run)

Positional Averages for Pitchers (Wgted Avg of Options): "0.500 "vs. Different Values for Starting vs. Relief Pitchers: (Empirical) (Based on Pitchers who Do Both)

Basic Player Won-Lost Records
Position Players
(Sorted by pWORL)
Player pWins pLosses pWOPA pWORL eWins eLosses eWOPA eWORL
24.320.62.4
4.5
25.119.84.0
6.2
20.318.02.5
4.4
20.617.73.1
5.0
10.59.80.1
1.1
10.69.70.3
1.3
12.914.3-0.8
0.5
12.714.6-1.4
-0.1
2.22.00.2
0.4
2.21.90.4
0.6
6.87.6-0.5
0.2
6.97.5-0.3
0.4
0.30.10.2
0.2
0.20.10.1
0.2
4.75.1-0.4
0.1
4.84.9-0.2
0.3
0.10.1-0.0
-0.0
0.10.1-0.1
-0.1
0.20.5-0.3
-0.2
0.30.4-0.1
-0.0
1.01.3-0.4
-0.2
1.01.3-0.3
-0.2
4.75.5-1.1
-0.6
5.25.00.0
0.6
4.45.3-1.1
-0.6
4.45.3-1.0
-0.5
2.13.1-0.9
-0.6
2.23.0-0.8
-0.5
1.01.9-0.8
-0.7
1.11.7-0.5
-0.4
9.111.1-1.9
-0.9
10.110.10.0
1.0
14.616.5-2.7
-1.1
15.016.1-1.7
-0.2
7.08.6-2.0
-1.1
7.18.4-1.6
-0.8
4.15.8-1.9
-1.4
4.35.6-1.4
-0.9
11.212.4-2.7
-1.6
12.211.5-0.8
0.4
3.75.5-2.1
-1.7
4.54.8-0.6
-0.1
6.49.3-2.9
-2.1
6.49.3-3.0
-2.2
------ ------ ------ ------ ------ ------ ------ ------
SUB-TOTAL
151.7164.3-16.9
-1.3
157.1158.9-5.8
9.8


Basic Player Won-Lost Records
Pitchers
(Sorted by pWORL)
Player pWins pLosses pWOPA pWORL eWins eLosses eWOPA eWORL
13.512.81.6
3.3
13.612.71.9
3.6
5.14.10.9
1.7
4.84.40.3
1.2
6.05.50.9
1.6
5.95.70.7
1.5
3.12.01.0
1.5
3.02.10.9
1.4
3.83.40.3
1.0
3.73.40.3
0.9
1.41.20.2
0.5
1.51.10.3
0.6
6.47.4-0.6
0.3
7.06.80.6
1.5
12.014.3-1.5
0.3
12.014.3-1.4
0.4
1.21.2-0.0
0.2
1.21.20.0
0.3
0.00.1-0.1
-0.0
0.10.1-0.0
-0.0
0.10.2-0.1
-0.1
0.10.3-0.2
-0.2
0.00.2-0.1
-0.1
0.10.1-0.0
-0.0
0.10.2-0.1
-0.1
0.10.2-0.1
-0.0
0.20.3-0.1
-0.1
0.20.3-0.1
-0.1
0.00.2-0.2
-0.2
0.10.2-0.1
-0.1
0.10.3-0.2
-0.2
0.10.2-0.2
-0.1
3.34.2-0.7
-0.2
3.44.0-0.4
0.1
0.30.7-0.4
-0.3
0.50.5-0.1
0.0
4.76.1-1.1
-0.3
4.76.1-1.1
-0.4
0.41.1-0.6
-0.5
0.41.1-0.7
-0.6
0.61.5-0.9
-0.7
0.91.1-0.2
0.0
4.36.1-1.6
-0.8
5.15.3-0.1
0.7
2.03.2-1.2
-0.8
2.42.8-0.3
0.1
0.21.2-1.0
-0.9
0.40.9-0.5
-0.4
1.02.2-1.2
-1.0
1.51.7-0.2
0.1
2.14.4-2.0
-1.6
2.34.2-1.8
-1.4
5.38.6-3.0
-2.0
5.88.0-1.7
-0.8
------ ------ ------ ------ ------ ------ ------ ------
SUB-TOTAL
77.392.7-11.9
0.4
81.089.0-4.0
8.3


pWins pLosses pWOPA pWORL eWins eLosses eWOPA eWORL
TEAM TOTALS
229.0257.0-28.8
-0.9
238.1247.9-9.8
18.1


Basic Player Won-Lost Records
Position Players
(Sorted by pWORL)
Batting Baserunning Pitching Fielding
Player eWins eLoss Win Pct. eWins eLoss Win Pct. eWins eLoss Win Pct. eWins eLoss Win Pct.
16.711.1
0.601
1.41.4
0.500
0.00.0
0.000
6.97.4
0.481
14.211.5
0.554
0.91.0
0.451
0.00.0
0.000
5.45.3
0.502
6.55.3
0.551
0.40.5
0.451
0.00.0
0.000
3.73.9
0.484
7.89.6
0.449
0.60.7
0.467
0.00.0
0.000
4.34.2
0.501
1.31.1
0.534
0.20.1
0.728
0.00.0
0.000
0.80.8
0.498
5.76.0
0.491
0.30.5
0.410
0.00.0
0.000
0.91.1
0.446
0.10.1
0.682
0.00.0
0.598
0.00.0
0.000
0.10.0
0.769
3.23.1
0.510
0.20.4
0.384
0.00.0
0.000
1.31.5
0.476
0.00.1
0.225
0.00.0
0.493
0.00.0
0.000
0.00.0
0.527
0.30.3
0.472
0.00.0
0.217
0.00.0
0.000
0.00.1
0.313
0.71.0
0.408
0.00.1
0.431
0.00.0
0.000
0.20.2
0.524
4.23.6
0.538
0.30.4
0.421
0.00.0
0.000
0.71.0
0.412
2.33.2
0.424
0.50.4
0.556
0.00.0
0.000
1.61.7
0.488
1.62.5
0.394
0.10.1
0.466
0.00.0
0.000
0.40.4
0.523
0.81.4
0.377
0.10.1
0.514
0.00.0
0.000
0.20.2
0.444
6.66.6
0.499
0.40.6
0.417
0.00.0
0.000
3.12.9
0.516
8.610.0
0.463
1.11.1
0.514
0.00.0
0.000
5.35.0
0.518
4.75.7
0.450
0.40.6
0.400
0.00.0
0.000
2.12.1
0.502
2.53.2
0.441
0.20.3
0.409
0.00.0
0.000
1.72.1
0.450
9.68.9
0.517
0.50.6
0.453
0.00.0
0.000
2.12.0
0.520
3.63.4
0.509
0.10.3
0.313
0.00.0
0.000
0.81.0
0.440
4.46.9
0.392
0.30.3
0.564
0.00.0
0.000
1.72.1
0.438
------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
SUB-TOTAL
105.7104.6
0.503
8.19.2
0.469
0.00.0
0.000
43.345.1
0.490


Basic Player Won-Lost Records
Pitchers
(Sorted by pWORL)
Batting Baserunning Pitching Fielding
Player eWins eLoss Win Pct. eWins eLoss Win Pct. eWins eLoss Win Pct. eWins eLoss Win Pct.
0.71.2
0.372
0.10.1
0.450
12.311.2
0.522
0.40.3
0.603
0.00.0
0.000
0.00.0
0.000
4.84.3
0.523
0.00.1
0.403
0.20.5
0.272
0.00.0
0.467
5.64.9
0.531
0.10.2
0.344
0.00.0
0.000
0.00.0
0.000
3.02.0
0.600
0.00.1
0.450
0.00.0
0.259
-0.00.0
-1.292
3.73.3
0.523
0.00.0
0.782
0.00.0
0.000
0.00.0
0.000
1.51.1
0.572
0.00.0
0.000
0.30.7
0.305
0.00.0
0.562
6.56.1
0.516
0.10.0
0.730
0.41.2
0.269
0.00.1
0.315
11.712.7
0.478
0.10.1
0.484
0.00.0
0.000
0.00.0
0.000
1.21.1
0.514
0.00.0
0.830
0.00.0
0.000
0.00.0
0.000
0.10.1
0.443
0.00.0
0.000
0.00.0
0.000
0.00.0
0.000
0.10.3
0.256
0.00.0
0.000
0.00.0
0.000
0.00.0
0.000
0.10.1
0.459
0.00.0
0.000
0.00.0
0.000
0.00.0
0.000
0.10.2
0.414
0.00.0
0.000
0.00.0
0.006
0.00.0
0.000
0.20.3
0.408
0.00.0
0.030
0.00.0
0.000
0.00.0
0.000
0.10.2
0.259
0.00.0
0.000
0.00.0
0.000
0.00.0
0.000
0.10.2
0.368
0.00.1
0.054
0.20.3
0.378
0.00.0
0.235
3.23.6
0.470
0.00.0
0.538
0.00.0
0.000
0.00.0
0.000
0.50.4
0.528
0.00.1
0.169
0.20.5
0.323
0.00.0
0.250
4.55.5
0.450
0.10.1
0.395
0.00.0
0.918
0.00.0
0.876
0.41.0
0.293
0.00.0
0.907
0.00.0
0.000
0.00.0
0.000
0.91.1
0.455
0.00.0
0.668
0.30.3
0.492
0.00.0
0.561
4.74.9
0.487
0.10.1
0.518
0.00.1
0.088
0.00.0
0.091
2.42.6
0.473
0.10.0
0.851
0.00.0
0.528
0.00.0
0.175
0.40.9
0.338
0.00.0
0.019
0.10.0
0.725
0.00.0
0.119
1.41.7
0.460
0.00.0
0.700
0.10.3
0.313
0.00.0
0.459
2.23.6
0.379
0.00.1
0.184
0.10.6
0.149
0.00.0
0.426
5.86.9
0.459
0.10.3
0.249
------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
SUB-TOTAL
2.76.0
0.316
0.20.3
0.410
77.480.5
0.490
1.31.6
0.448


Batting Baserunning Pitching Fielding
eWins eLoss Win Pct. eWins eLoss Win Pct. eWins eLoss Win Pct. eWins eLoss Win Pct.
TEAM TOTALS
108.4110.5
0.495
8.49.5
0.467
77.480.5
0.490
44.646.6
0.489


eWin, eLoss, eWOPA by Position
Catcher
Player eWins eLosses eWOPA
Chris Widger7.07.5-0.3
Brian Schneider2.12.9-0.6
Michael Barrett1.92.7-0.7
Lenny Webster1.11.7-0.5
Charlie O'Brien0.30.4-0.1
Yohanny Valera0.10.1-0.1
------ ------ ------
TOTAL12.415.2-2.3
  
First Base
Player eWins eLosses eWOPA
Lee Stevens12.111.4-0.7
Fernando Seguignol1.92.3-0.7
Andy Tracy2.01.60.1
Geoff Blum0.60.40.2
Talmadge Nunnari0.20.10.1
Mike Mordecai0.00.00.0
------ ------ ------
TOTAL16.715.8-1.0
  
Second Base
Player eWins eLosses eWOPA
Jose Vidro20.517.83.0
Geoff Blum0.81.0-0.2
Mike Mordecai0.00.1-0.0
Trace Coquillette0.00.0-0.0
Wilton Guerrero0.00.0-0.0
Orlando Cabrera0.00.0-0.0
------ ------ ------
TOTAL21.419.02.7
  
Third Base
Player eWins eLosses eWOPA
Michael Barrett4.56.4-2.0
Geoff Blum5.75.20.3
Mike Mordecai3.83.9-0.2
Andy Tracy2.42.5-0.1
Trace Coquillette0.91.2-0.2
------ ------ ------
TOTAL17.319.2-2.3
  
Shortstop
Player eWins eLosses eWOPA
Orlando Cabrera12.614.4-1.3
Geoff Blum2.72.80.0
Tomas de la Rosa2.11.90.2
Mike Mordecai0.70.50.2
------ ------ ------
TOTAL18.119.6-0.8
  
Left Field
Player eWins eLosses eWOPA
Rondell White10.59.60.3
Wilton Guerrero3.84.9-1.5
Peter Bergeron2.83.2-0.7
Terry Jones2.52.8-0.5
Fernando Seguignol0.91.3-0.5
Trace Coquillette0.00.0-0.0
------ ------ ------
TOTAL20.521.9-2.8
  
Center Field
Player eWins eLosses eWOPA
Peter Bergeron12.212.7-1.0
Milton Bradley4.45.5-1.4
Terry Jones1.41.8-0.5
Wilton Guerrero0.70.8-0.0
------ ------ ------
TOTAL18.620.7-2.8
  
Right Field
Player eWins eLosses eWOPA
Vladimir Guerrero Sr.24.919.63.9
Wilton Guerrero0.91.0-0.2
Fernando Seguignol0.90.60.3
Terry Jones0.10.1-0.1
Trace Coquillette0.00.0-0.0
------ ------ ------
TOTAL26.721.34.0
  
Designated Hitter
Player eWins eLosses eWOPA
Wilton Guerrero0.50.6-0.1
Vladimir Guerrero Sr.0.10.2-0.1
Fernando Seguignol0.00.1-0.1
------ ------ ------
TOTAL0.60.9-0.4
  
Pinch Hitter
Player eWins eLosses eWOPA
Wilton Guerrero1.11.10.1
Andy Tracy0.80.90.0
Fernando Seguignol0.80.50.4
Geoff Blum0.40.7-0.2
Terry Jones0.40.5-0.1
Mike Mordecai0.20.5-0.2
Michael Barrett0.10.1-0.0
Peter Bergeron0.10.2-0.1
Orlando Cabrera0.10.1-0.0
Brian Schneider0.00.2-0.2
Rondell White0.10.1-0.0
Trace Coquillette0.10.10.0
Lee Stevens0.10.1-0.0
Tomas de la Rosa0.10.00.1
Talmadge Nunnari0.10.00.1
Lenny Webster0.00.1-0.0
Vladimir Guerrero Sr.0.00.1-0.0
Milton Bradley0.00.00.0
Chris Widger0.00.0-0.0
Felipe Lira0.00.0-0.0
------ ------ ------
TOTAL4.45.2-0.3
  
Pinch Runner
Player eWins eLosses eWOPA
Terry Jones0.10.00.1
Wilton Guerrero0.00.0-0.0
Tomas de la Rosa0.00.00.0
Peter Bergeron0.00.00.0
Talmadge Nunnari0.00.00.0
Mike Mordecai0.00.00.0
------ ------ ------
TOTAL0.20.10.1
  
Pitcher Offense
Player eWins eLosses eWOPA
Javier Vazquez0.81.30.2
Dustin Hermanson0.51.2-0.1
Tony Armas Jr.0.30.8-0.0
Mike Thurman0.10.7-0.2
Carl Pavano0.20.5-0.1
Mike K. Johnson0.20.5-0.0
Felipe Lira0.30.30.2
Hideki Irabu0.20.40.0
Trey Moore0.10.30.0
Julio Santana0.00.1-0.1
Jeremy Powell0.10.00.1
Anthony Telford0.00.0-0.0
T.J. Tucker0.00.00.0
Scott Downs0.00.0-0.0
Matt Blank0.00.0-0.0
Miguel Batista0.00.00.0
Scott Strickland0.00.0-0.0
Brad Rigby0.00.0-0.0
Guillermo Mota0.00.0-0.0
Ugueth Urbina0.00.0-0.0
Steve Kline (1997)0.00.0-0.0
------ ------ ------
TOTAL3.06.3-0.1
  
Starting Pitching
Player eWins eLosses eWOPA
Javier Vazquez12.711.51.4
Dustin Hermanson10.811.5-0.6
Mike Thurman5.97.2-1.1
Tony Armas Jr.6.66.10.6
Carl Pavano5.75.10.7
Mike K. Johnson3.74.8-1.1
Hideki Irabu3.23.6-0.4
Trey Moore2.33.8-1.5
Felipe Lira2.02.7-0.7
Jeremy Powell1.31.5-0.2
Julio Santana0.91.3-0.3
T.J. Tucker0.41.0-0.6
Scott Downs0.20.3-0.1
------ ------ ------
TOTAL55.660.4-4.0
  
Relief Pitching
Player eWins eLosses eWOPA
Steve Kline (1997)4.84.40.3
Anthony Telford3.73.40.3
Felipe Lira2.62.50.1
Scott Strickland3.02.10.9
Julio Santana1.41.4-0.0
Ugueth Urbina1.51.10.3
Dustin Hermanson1.01.5-0.5
Guillermo Mota1.21.10.0
Scott Forster0.91.1-0.2
Mike K. Johnson0.80.9-0.1
Miguel Batista0.40.9-0.4
Matt Blank0.50.5-0.0
Jim Richard Poole0.10.3-0.2
Matt Skrmetta0.10.3-0.2
Jeremy Powell0.20.2-0.0
Sean Spencer0.10.2-0.1
David Moraga0.10.2-0.1
Brad Rigby0.10.1-0.0
Yovanny Lara0.10.1-0.0
------ ------ ------
TOTAL22.622.00.1


Team eWins over Positional Average by Position
PC1B2B3BSSLFCFRFOth
2000 Montreal Expos-4.0-2.3-1.02.7-2.3-0.8-2.8-2.84.0-0.5


pWin, pLoss, pWOPA by Position
Catcher
Player pWins pLosses pWOPA
Chris Widger6.87.6-0.5
Brian Schneider2.12.9-0.7
Michael Barrett1.82.7-0.8
Lenny Webster0.91.8-0.8
Charlie O'Brien0.20.5-0.3
Yohanny Valera0.10.1-0.0
------ ------ ------
TOTAL12.115.6-3.0
  
First Base
Player pWins pLosses pWOPA
Lee Stevens11.212.4-2.7
Fernando Seguignol1.82.4-0.9
Andy Tracy1.81.7-0.1
Geoff Blum0.70.30.3
Talmadge Nunnari0.20.00.2
Mike Mordecai0.00.00.0
------ ------ ------
TOTAL15.716.8-3.3
  
Second Base
Player pWins pLosses pWOPA
Jose Vidro20.318.02.5
Geoff Blum0.90.9-0.0
Mike Mordecai0.00.1-0.1
Trace Coquillette0.00.0-0.0
Wilton Guerrero0.00.0-0.0
Orlando Cabrera0.00.0-0.0
------ ------ ------
TOTAL21.219.22.4
  
Third Base
Player pWins pLosses pWOPA
Michael Barrett4.56.4-2.1
Geoff Blum4.96.0-1.2
Mike Mordecai3.73.9-0.2
Andy Tracy2.22.8-0.7
Trace Coquillette0.91.2-0.3
------ ------ ------
TOTAL16.120.3-4.6
  
Shortstop
Player pWins pLosses pWOPA
Orlando Cabrera12.914.1-0.7
Geoff Blum2.33.2-0.8
Tomas de la Rosa2.12.00.1
Mike Mordecai0.70.60.1
------ ------ ------
TOTAL17.919.8-1.3
  
Left Field
Player pWins pLosses pWOPA
Rondell White10.49.8-0.0
Wilton Guerrero3.65.2-1.9
Peter Bergeron2.33.7-1.6
Terry Jones2.52.8-0.5
Fernando Seguignol0.71.5-0.9
Trace Coquillette0.00.0-0.0
------ ------ ------
TOTAL19.522.9-4.9
  
Center Field
Player pWins pLosses pWOPA
Peter Bergeron12.112.8-1.1
Milton Bradley4.15.8-1.9
Terry Jones1.61.6-0.1
Wilton Guerrero0.70.8-0.1
------ ------ ------
TOTAL18.520.9-3.2
  
Right Field
Player pWins pLosses pWOPA
Vladimir Guerrero Sr.24.220.32.6
Wilton Guerrero0.91.0-0.1
Fernando Seguignol0.70.7-0.1
Terry Jones0.00.2-0.2
Trace Coquillette0.00.0-0.0
------ ------ ------
TOTAL25.922.22.1
  
Designated Hitter
Player pWins pLosses pWOPA
Wilton Guerrero0.50.6-0.1
Vladimir Guerrero Sr.0.10.2-0.1
Fernando Seguignol0.00.1-0.1
------ ------ ------
TOTAL0.61.0-0.4
  
Pinch Hitter
Player pWins pLosses pWOPA
Wilton Guerrero1.21.00.3
Andy Tracy0.71.0-0.3
Fernando Seguignol0.50.8-0.2
Geoff Blum0.40.7-0.2
Terry Jones0.20.7-0.4
Mike Mordecai0.20.5-0.2
Michael Barrett0.10.1-0.0
Peter Bergeron0.10.10.0
Orlando Cabrera0.10.1-0.1
Brian Schneider0.00.2-0.2
Rondell White0.10.00.1
Trace Coquillette0.10.10.0
Lee Stevens0.10.10.0
Tomas de la Rosa0.10.00.1
Talmadge Nunnari0.10.00.0
Lenny Webster0.00.1-0.0
Vladimir Guerrero Sr.0.00.1-0.0
Milton Bradley0.00.00.0
Chris Widger0.00.0-0.0
Felipe Lira0.00.0-0.0
------ ------ ------
TOTAL4.15.5-0.9
  
Pinch Runner
Player pWins pLosses pWOPA
Terry Jones0.10.10.1
Wilton Guerrero0.00.0-0.0
Tomas de la Rosa0.00.00.0
Peter Bergeron0.00.00.0
Talmadge Nunnari0.00.00.0
Mike Mordecai0.00.0-0.0
------ ------ ------
TOTAL0.20.10.1
  
Pitcher Offense
Player pWins pLosses pWOPA
Javier Vazquez0.81.30.2
Dustin Hermanson0.41.3-0.2
Tony Armas Jr.0.40.70.1
Mike Thurman0.20.6-0.1
Carl Pavano0.20.6-0.1
Mike K. Johnson0.20.6-0.1
Felipe Lira0.20.40.1
Hideki Irabu0.20.40.1
Trey Moore0.10.3-0.0
Julio Santana0.00.1-0.1
Jeremy Powell0.10.10.0
Anthony Telford0.00.1-0.0
T.J. Tucker0.00.00.0
Scott Downs0.00.0-0.0
Matt Blank0.00.0-0.0
Miguel Batista0.00.00.0
Scott Strickland0.00.0-0.0
Brad Rigby0.00.0-0.0
Guillermo Mota0.00.0-0.0
Ugueth Urbina0.00.0-0.0
Steve Kline (1997)0.00.0-0.0
------ ------ ------
TOTAL2.86.4-0.3
  
Starting Pitching
Player pWins pLosses pWOPA
Javier Vazquez12.711.51.4
Dustin Hermanson10.611.7-1.0
Mike Thurman5.18.0-2.8
Tony Armas Jr.6.06.7-0.7
Carl Pavano5.94.91.0
Mike K. Johnson3.84.7-0.9
Hideki Irabu3.13.8-0.7
Trey Moore2.04.0-2.0
Felipe Lira1.63.1-1.4
Jeremy Powell0.81.9-1.1
Julio Santana0.81.3-0.5
T.J. Tucker0.41.1-0.7
Scott Downs0.20.3-0.1
------ ------ ------
TOTAL53.063.1-9.4
  
Relief Pitching
Player pWins pLosses pWOPA
Steve Kline (1997)5.14.10.9
Anthony Telford3.73.30.3
Felipe Lira2.52.6-0.2
Scott Strickland3.12.01.0
Julio Santana1.11.8-0.7
Ugueth Urbina1.41.20.2
Dustin Hermanson1.11.4-0.3
Guillermo Mota1.21.2-0.0
Scott Forster0.61.5-0.9
Mike K. Johnson0.80.8-0.0
Miguel Batista0.11.2-1.1
Matt Blank0.30.7-0.3
Jim Richard Poole0.10.2-0.1
Matt Skrmetta0.10.3-0.2
Jeremy Powell0.10.2-0.1
Sean Spencer0.10.2-0.1
David Moraga0.00.2-0.2
Brad Rigby0.00.1-0.1
Yovanny Lara0.00.1-0.1
------ ------ ------
TOTAL21.523.1-2.1


Team pWins over Positional Average by Position
PC1B2B3BSSLFCFRFOth
2000 Montreal Expos-11.9-3.0-3.32.4-4.6-1.3-4.9-3.22.1-1.3


*Player won-loss records are calculated using Retrosheet play-by-play data. For some seasons, Retrosheet does not have full play-by-play data for all games. In these cases, some games may be missing.
The number of Games shown here are the number of games for which Retrosheet play-by-play data exist and were used in these calculations.



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