Daniel P. Glatt

David U. Cha

Paul M. Sommers

English

Research Papers in Economics

 
 
 
 
 
“AN EMPIRICAL TEST OF BILL JAMES’S 
PYTHAGOREAN FORMULA” 
 
by 
 
Paul M. Sommers 
David U. Cha 
And 
Daniel P. Glatt 
 
 
 
 
March  2010 
 
 
 
 
 
MIDDLEBURY COLLEGE ECONOMICS DISCUSSION PAPER NO. 10-06 
 
 
 
 
 
 
 
 
DEPARTMENT OF ECONOMICS 
MIDDLEBURY COLLEGE 
MIDDLEBURY, VERMONT 05753 
 
http://www.middlebury.edu/~econ                                                                                                                                             2 
 
 
 
 
 
 
 
 
 
 
 
AN EMPIRICAL TEST OF BILL JAMES’S 
PYTHAGOREAN FORMULA 
 
 
 
 
 
 
 
by 
 
 
 
 
 
 
David U. Cha 
Daniel P. Glatt 
Paul M. Sommers 
 
 
 
 
 
 
 
Department of Economics 
Middlebury College 
Middlebury, Vermont 05753 
psommers@middlebury.edu                                                                                                                                             3 
 
 
AN EMPIRICAL TEST OF BILL JAMES’S 
PYTHAGOREAN FORMULA 
 
 
 
 
  The Giants do not usually need to score many runs. 
  All that we must do is score more than the other fellow. 
                                                             
                                                                                              — Bill Terry, manager of the 
                                                                                                   1932 New York Giants  
                                                                                                   [1, p.136] 
 
 
 
  Bill James, baseball writer and statistician, in 1980 developed a formula that related a 
team’s won-lost percentage to the number of runs they scored and allowed, as follows: 
 
(1)                             Won-Lost Percentage  =   ( )
() ( )
2 2
2
d RunsAllowe RunsScored
RunsScored
+
  
 
 
 
Since the Won-Lost Percentage is the ratio of games won to the total number of games played 
(games won plus games lost), equation (1) can be re-written as follows: 
 
(2)                                     ( )
()
2
2
2
⎟
⎠
⎞
⎜
⎝
⎛ = =
d RunsAllowe
RunsScored
d RunsAllowe
RunsScored
Losses
Wins
 
 
 
If, for example, the Boston Red Sox score 867 runs and allow 657 runs (as they did in 2007), Bill 
James’s “Pythagorean” method [so dubbed because of the presence of three squared terms in 
equation (1)]
1 projects that the team would have a won-lost percentage of (867)
2/[(867)
2 + (657)
2] 
or .635 (and hence win about .635 × 162 or 103 games).  In fact, the Red Sox (world champions 
in 2007) won 96 regular season games or about 6.8 percent fewer games than the Pythagorean 
method would predict.  In this instance, the exponent of “2” on the right-hand side of equation (2)                                                                                                                                             4 
 
is too high.  Or, one could argue that “2” is accurate, but the Boston Red Sox should have won 
more regular season games in 2007 than they actually did.   
  The purpose of this brief note is to empirically test Bill James’s Pythagorean method for 
all teams in both leagues, by decade, from 1950 to 2007.  Does the method work as well since 
1980 as it did before 1980?  Does the method work better for one league (American or National) 
than the other?  Has the exponent in equation (2) changed in recent decades? 
 
The Models 
  Equation (2) can be written in log-linear form as follows: 
 
(3)                                                      ⎟
⎠
⎞
⎜
⎝
⎛ = ⎟
⎠
⎞
⎜
⎝
⎛
RA
RS
Losses
Wins
ln 2 ln  
 
 
where “ln” is the natural logarithm, RS denotes runs scored, and RA denotes runs allowed.  That is, 
if one first takes logs of both sides of equation (2) and then if we define 
t i
t i Losses
Wins
y
,
, ln ⎟
⎠
⎞
⎜
⎝
⎛ =  and  
t i
t i RA
RS
x
,
, ln ⎟
⎠
⎞
⎜
⎝
⎛ = for each team i in year t, we can estimate the coefficients β0 and β1 by applying 
least squares to y and x in the following regression: 
 
(4)                                                           t i t i t i x y , , 1 0 , ε β β + + =  
 
 
where  t i, ε  is a disturbance term.  According to Bill James, β0 should be indistinguishable from 
zero and β1 should be close to “2”.  To test the null hypothesis H0: β1 = 2, we employ a t-test.  The 
test statistic is  () 1
1 2
b SE
b
tcalc
−
= , where b1 is the estimated slope coefficient and SE(b1) is the                                                                                                                                             5 
 
standard error of the estimated slope coefficient.
2  Hereafter, equation (4) where 
⎟
⎠
⎞
⎜
⎝
⎛ =
Losses
Wins
y ln  and  ⎟
⎠
⎞
⎜
⎝
⎛ =
RA
RS
x ln  will be called Model (1). 
  Model (1) assumes that one more run scored has the same impact on a team’s win 
percentage as does one less run allowed.  But what if scoring runs was more (or less) important to 
winning games than allowing runs?  Model (1) might then be revised as follows: 
 
(5)                             t i t i t i
t i
RA RS
Losses
Wins
, , 2 , 1 0
,
) ln( ) ln( ln ε β β β + + + = ⎟
⎠
⎞
⎜
⎝
⎛
 
 
 
If we relax the assumption that the exponent on the ratio  ⎟
⎠
⎞
⎜
⎝
⎛
RA
RS
 is the same (and, according to 
James, equal to “2”), then the revised model would be described by equation (5), hereafter,  
Model (2). 
 
The Data 
  Data on regular season wins, losses, runs scored, and runs allowed for all teams were 
gleaned from two primary sources: Total Baseball [3] for the years 1950 through 2003 and  
http://sports.espn.go.com/mlb/standings for the years 2004 through 2007. 
 
The Results 
  Table 1 shows the regression results for each league (as well as for both leagues 
combined) for each decade since the 1950s.  The estimated intercept (b0) in all regressions is 
not discernible from zero, as Bill James would expect.  Since the year 2000, the exponent in the 
ratio of runs scored to runs allowed in James’s Pythagorean formula has been indistinguishable 
from “2”.  But, in decades before the turn of the millennium the exponent was not equal to “2”.  
And, in all cases when we could reject H0: β1 = 2 (in favor the alternative hypothesis HA: β1 ≠ 2),                                                                                                                                             6 
 
our estimate b1 was invariably less than “2”.  A comparison of the 30-year period 1950-1979 to 
the 28-year period 1980-2007 shows that b1 was in most cases (with the exception of the 
American League (AL) from 1980 to 2007) significantly less than “2”.  The impact of RS/RA on 
winning is marginally higher now (1980-2007) than it was in the earlier period (1950-1979).  
Compare the value of b1 (1.9202) estimated for both leagues combined in the period 1980-2007 to 
the corresponding estimate for b1 (1.8099) in the period 1950-1979.  Moreover, it is worth noting 
that the average number of runs scored is also higher in the National (American) League in the 
period 1980-2007 than it was in the period 1950-1979 [ NL RS , 2007 1980−   =  699,  NL RS , 1979 1950−  = 
667.3,  p-value on the difference between means is less than .001;  AL RS , 2007 1980−  = 747, 
AL RS , 1979 1950−  = 669.6,  p-value on the difference is again less than .001]. 
  Figures 1 and 2 show scatterplots of ln(W/L) against ln(RS/RA) for each subperiod (1950-
1979 and 1980-2007, respectively) for each league.  Each point represents an observation on one 
team in one year.  The points more closely fall on a straight line for the National League, 1950-
1979 than they do for the National League, 1980-2007 (compare R
2 = .878 for 1950-1979 with  
R
2 = .847 for 1980-2007 in Table 1).  Still, the differences between the two periods by league are 
admittedly very small.   
  Table 2 shows the regression results for Model (2), which isolates the impact of runs 
scored from the impact of runs allowed on the win-loss ratio.  The right-hand column reports the 
coefficient of determination (R
2) for each regression each decade, by league.  A look down this 
column and the corresponding column in Table 1 clearly shows that the explanatory power (that 
is, how well the regressors as a group explain variation in the dependent variable, namely,  
ln(Wins/Losses) ) of Model (2) is not an improvement over Model (1).  In other words, runs 
scored and runs allowed seemingly have an equal (and opposite) effect on the win-loss ratio.                                                                                                                                             7 
 
Concluding Remarks 
  
  Early in the 1980s, Bill James developed a formula in response to the question: Can you 
tell how many games a team will win, based on its runs scored and runs allowed?  A regression 
analysis of data on regular season runs scored, runs allowed, and wins (and losses) for each team 
each season in Major League Baseball since 1950 reveals that Bill James’s Pythagorean formula 
has stood the test of time very well indeed.  Runs scored and runs allowed have equal (and 
opposite) effects on team winning, in both leagues and in years before and since 1980.  If any 
modification should be made to the formula, the exponent on runs scored and runs allowed should 
be reduced to a power slightly below “2” [“1.92” for both leagues since the year 1980].                                                                                                                                              8 
 
Table 1. Regression Results for Model (1) 
ln(WINS/LOSSES)  = b0 + b1 ln(RS/RA) 
 
 
   Slope  coefficient  on 
 Intercept  ln(RS/RA) R
2 
  b0 b 1 
 
 
1950-1959 
 AL  -.0059    [.0131]
a   1.7543  [.0598] .917 
 NL  .00003  [.0122]  1.8758  [.0737]  .893 
Both leagues  -.0030  [.0089]  1.7985  [.0461] .906 
 
1960-1969 
  AL -.0017  [.0094]  1.8757  [.0593] .911 
 NL  -.0013  [.0111]  1.9323  [.0655]  .901 
Both leagues  -.0016  [.0072]  1.9055  [.0441] .905 
 
1970-1979  
  AL .00001  [.0091]  1.8139  [.0560] .894 
 NL  .0012  [.0101]  1.6576  [.0642] .850 
Both leagues  .0006  [.0068]  1.7398 [.0425]  .873 
 
1980-1989 
 AL  .0005  [.0078]  1.8849 [.0577]  .885 
 NL  -.0017  [.0100]  2.0195  [.0848]  .828 
Both leagues  -.0005  [.0063]  1.9381  [.0489]  .859 
 
1990-1999 
  AL .00003  [.0078]  1.9324  [.0599]  .883 
 NL  -.0021  [.0090]  1.8370 [.0645]  .856 
Both leagues  -.0012  [.0060]  1.8814  [.0441] .869 
 
2000-2007  
 AL  -.0055  [.0099]  2.0026  [.0624]  .904 
 NL  -.0004  [.0089]  1.8720  [.0682]  .857 
Both leagues  -.0023  [.0066]  1.9445  [.0458]  .883 
 
1950-1979 
 AL  -.0021  [.0059]  1.8062  [.0333] .907 
 NL  .00005  [.0064]  1.8146  [.0393] .878 
Both leagues  -.0011  [.0044]  1.8099  [.0255] .893 
 
1980-2007 
  AL -.0012  [.0048]  1.9415  [.0344]  .891 
 NL  -.0014  [.0054]  1.8951  [.0411] .847 
Both leagues  -.0013  [.0036]  1.9202 [.0266]  .870 
 
aNumbers in brackets are standard errors and numbers in boldface (italics) are significant at better than 
the .01 (.05) level.  The null hypothesis for the intercept is H0:β0 = 0 and the null hypothesis for the slope 
coefficient on ln(RS/RA) is H0: β1 = 2.  In both cases, the alternative hypothesis is two-tailed.                                                                                                                                              9 
 
     
Table 2. Regression Results for Model (2) 
ln(WINS/LOSSES)  = b0 + b1 ln(RS) + b2 ln(RA) 
 
 
 
                                                                         Slope coefficient on: 
                             Intercept                        ln(RS)                     ln(RA)   R
2 
    b0       b1  b2 
 
 
1950-1959 
  AL .3888  [.9790]  1.7224 [.0995]  -1.7829 [.0930]  .917 
  NL  -1.0380  [1.1370]  1.9526 [.1119]  -1.7937 [.1164]  .894 
Both leagues -.2284  [.7351]  1.8162  [.0739]  -1.7816  [.0718] .906 
 
1960-1969 
  AL  .3171 [.5948]  1.8494 [.0771]  -1.8986 [.0733]  .911 
  NL  .8684 [.7091]  1.8708 [.0823]  -2.0052 [.0883]  .902 
Both leagues .5648  [.4578]  1.8624  [.0562] -1.9499  [.0567]  .906 
 
1970-1979 
 AL  -.3023  [.5620]  1.8365 [.0702]  -1.7900  [.0715] .895 
 NL  -.6484  [.7772]  1.7046  [.0854]  -1.6046  [.0903] .850 
Both leagues  -.4204 [.4613]  1.7060  [.0545]  -1.7060 [.0564]  .873 
 
1980-1989 
  AL  .0077 [.3052]  1.8844 [.0619]  -1.8855 [.0630]  .885 
  NL  -.3260 [.4232]  2.0462 [.0919]  -1.9959 [.0904]  .829 
Both leagues  -.1265 [.2429]  1.9477 [.0524]  -1.9283 [.0525]  .859 
 
1990-1999 
  AL  .0794 [.4242]  1.9265 [.0680]  -1.9385 [.0683]  .883 
 NL  -.1865  [.4482]  1.8535  [.0762]  -1.8253 [.0707]  .856 
Both leagues  -.0895 [.2951]  1.8887  [.0505]  -1.8753 [.0486]  .869 
 
2000-2007 
  AL  -.8735 [.9428]  2.0721 [.0980]  -1.9421 [.0907]  .904 
 NL  1.6195  [.7686]  1.7343  [.0938] -1.9788  [.0842]  .862 
Both leagues  .5502 [.5714]  1.8998 [.0652]  -1.9829 [.0666]  .884 
 
aNumbers in brackets are standard errors and numbers in boldface (italics) are significant at better than 
the .01 (.05) level.  The null hypothesis for the intercept is H0:β0 = 0 and the null hypotheses for the slope 
coefficients on ln(RS) and ln(RA) are H0: β1 = 2 and H0: β2 = -2, respectively.  In all three cases, the 
alternative hypothesis is two-tailed.  
                 
 
                                                                                                                                             10 
 
 
Figure 1 
 
 
ln(RS/RA)
l
n
(
W
/
L
)
0.50 0.25 0.00 -0.25 -0.50
1.0
0.5
0.0
-0.5
Scatterplot of ln(W/L) against ln(RS/RA), 1950 - 1979
American League
 
 
ln(RS/RA)
l
n
(
W
/
L
)
0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3 -0.4 -0.5
1.0
0.5
0.0
-0.5
-1.0
Scatterplot of ln(W/L) against ln(RS/RA), 1950 - 1979
National League
                                                                                                                                             11 
 
 
Figure 2 
 
 
ln(RS/RA)
l
n
(
W
/
L
)
0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3 -0.4 -0.5
1.0
0.5
0.0
-0.5
-1.0
Scatterplot of ln(W/L) against ln(RS/RA), 1980 - 2007
American League
 
 
ln(RS/RA)
l
n
(
W
/
L
)
0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3 -0.4
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
Scatterplot of ln(W/L) against ln(RS/RA), 1980 - 2007
National League
                                                                                                                                             12 
 
  
 
References 
 
 
 
1. P.  Williams,  When the Giants Were Giants, Chapel Hill, NC: Algonquin Books, 1994. 
 
2. B.  James,  The Bill James Baseball Abstract 1983, New York: Ballantine Books, 1983. 
 
 
3.  Total Baseball: The Ultimate Baseball Encyclopedia (edited by H. Thorn, P. Birnbaum,  
 
  and B. Deane), Wilmington, DE: Sports Media Publishing, 2004.                                                                                                                                             13 
 
 
 
Footnotes 
 
 
 
1.  See, for example, the reference to “The Pythagorean Formula” in [2, p. 10]. 
 
2. The  b1 estimate also interprets as an elasticity of (Wins/Losses) to (RS/RA), where (in 
 
  general) the elasticity of Y with respect to X is defined as 
dX
dY
Y
X
⋅  .  In other words, a  
  1 percent increase in (RS/RA) will lead to a “b1” percent increase in (Wins/Losses).  
  Moreover, since “Wins + Losses” is equal to a constant (162 games since the year 1962 
  and 154 games in years before 1962), it should also be noted that a given percentage  
 change  in  Wins is equal to the percentage change in the winning percentage, 
 [ Wins/(Wins + Losses)].    
                          

An Empirical Test of Bill James’s Pythagorean Formula

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