A Note on the Pythagorean Theorem of Baseball Production

MANAGERIAL AND DECISION ECONOMICS, VOL. 18, 335±342 (1997)

A Note on the Pythagorean Theorem ofBaseball Production

John Ruggiero, Lawrence Hadley,* Gerry Ruggiero and Scott Knowles

University of Dayton, OH, USA

Recent analyses of baseball managers' performances have employed the so-called`Pythagorean Theorem' of baseball. This `theorem' states that the ratio of wins to lossescan be approximated by the square of the ratio of team runs scored to opposition runsscored. Recent publications assume this approximate relationship can be used to evaluatemanagers; implicit is the additional assumption that the Pythagorean relationshipconstitutes a production process. It does not. This paper derives the exact relationshipbetween the wins, losses, runs scored and runs allowed. The result is an identity. Weconclude that application of the `Pythagorean Theorem' for manager evaluation isinappropriate. # 1997 by John Wiley & Sons, Ltd.

Manage. Decis. Econ. 18: 335±342 (1997)No. of Figures: 0 No. of Tables: 4 No. of References: 7

INTRODUCTION

There are various models in the baseball economics

literature for the evaluation of managerial perfor-

mance. Porter and Scully (1982) provided the earliest

approach based on an economic model of production.

Speci®cally, they employ linear programming to

measure the production frontier and evaluate team

ef®ciency. Kahn (1993) integrated a salary equation

with a production function to analyze managerial

quality. Ruggiero, Hadley and Gustafson (1996) use

a production function to measure and compare deter-

ministic and stochastic production frontiers. Finally,

Horowitz (1994a,b) applies the Pythagorean Theorem

of baseball (James, 1986) as a tool to evaluate major

league baseball (MLB) managers.

The Pythagorean Theorem states that the ratio of

team wins to losses can be approximated by the

square of the ratio of runs scored to runs allowed.1

Horowitz's application of this theorem views MLB as

a production process. The ratio of runs scored to runs

allowed is a team's input, and the ratio of wins to

losses is its output. Horowitz justi®es the implemen-

tation of this theorem based on its empirical validity,

and he uses the theorem to derive a measure of

managerial performance. This measure is based on

regression coef®cients estimated with career perfor-

mance data for each manager separately.

It is conceptually correct to measure the ef®ciency

of a manager in terms of the games won and lost by

his team while controlling for the quality of his

team's players. The variables speci®ed by

Horowitz, however, do not model a production pro-

cess. Rather, the `Pythagorean Theorem' is only an

identity that links wins and losses with runs scored

and runs allowed. This identity is derived in this

paper.

In a given game, if the ratio of team runs to

opposition runs is greater than one then, by de®nition,

the game is won. If this ratio is less than one, the

game is lost. The number of wins in a given season is

determined by the number of games for which a team

scores more runs than the opposition. The season's

ratio of team runs to opposition runs contains the

information concerning the number of games won.

The relationship, however, is not as obvious as it is

CCC 0143±6570/97/040335±08 $17.50

# 1997 by John Wiley & Sons, Ltd.

* Correspondence to: Lawrence Hadley, Department of Economicsand Finance, University of Dayton, Dayton, OH 45469-2240, USA.E-mail [email protected]

for an individual game due to aggregation. Still, the

exact relationship between team and opposition runs

and team wins and losses can be derived from

de®nitions, and the resulting equation is therefore

an identity. Consequently, the ratio of runs scored to

runs allowed is a proxy for winning percent rather

than an input into a production process. The

Pythagorean Theorem is nothing more than an

approximation of an ex post identity and should not

be used to model a production process nor to evaluate

managers.

The next section derives the identity that links wins

and losses to the aggregate runs scored and allowed

for a particular team in a given season. The

Pythagorean Theorem and the estimating equations

used in Scully (1994) and Horowitz (1994a,b) are

presented and critiqued in the third section.

The fourth section presents an empirical analysis

of the managers considered by Horowitz (1994a).

This analysis will be used to highlight the major

points of this paper. The ®nal section presents con-

clusions.

WINS, LOSSES AND RUNS IN BASEBALL

Before evaluating the Pythagorean approach, it is

useful to derive the exact relationship between

wins, losses and the aggregate runs scored by a

team and by their opposition. For any team playing

G games, let

Rg � the runs scored by the team in game g

Og � the runs scored by the opponent in game g

DRg � Rg ÿ Og � the number of runs by which

game g is won or lost

Eg � the runs in excess of the deciding run2,

TR� the total runs scored by the team in the

season

OR� the total runs allowed by the team in the

season

W� the number of team wins

L� the number of team losses (where

L � G ÿW ).

Total runs scored in a season can be expressed as:

TR � PGg�1

Rg �PGg�1

�DRg � Og� � OR� PGg�1

DRg

� OR�PWi�1

DRi �PLi�1

DRi �1�

For any particular game i won, DRi 5 1 and

DRi � 1� Ei �2�

Also, for any particular game j lost, DRj 4ÿ 1 and

DRj � ÿ1� Ej �3�

Substituting Eqns (2) and (3) into Eqn (1) results in

the following equation:

TR � OR�PWi�1

�1� Ei� ÿPLj�1

�1ÿ Ej�

� OR�W ÿ L� PGg�1

Eg �4�

Rewriting Eqn (4) leads to the following identity:

W ÿ L � TRÿ ORÿ E �5�

where E is the aggregate number of excess runs for a

team in a season. If E� 0, then the team won and lost

each game by one run, on average.

This relationship was derived using only de®ni-

tions; therefore, Eqn (5) is an identity. Estimation of

Eqn (5) omitting E will lead to biased estimates.

Moreover, E is not an error term representing mea-

surement error of any other statistical noise. The

value of E is obtained from the addition of each

individual game's excess runs; it is deterministic (not

stochastic) just as TR and OR. Further, the identity

suggests that the runs ratio is a proxy for the stated

output, ratio of wins to losses.

THE PYTHAGOREAN THEOREM OFBASEBALL

Now consider the Pythagorean Theorem. With a

given level of player skills, the theory maintains

that the ratio of wins to losses in a season can be

approximated by the ratio of runs scored to runs

allowed:

W

L� a

TR

OR

� �b

E �6�

The theory suggests that a � 1 and b � 2. Thus, a

team's win-to-loss ratio is approximated by the

square of the runs scored to runs allowed ratio.

336 J. RUGGIERO ET AL.

Managerial and Decision Economics, 18: 335±342 # 1997 by John Wiley & Sons, Ltd.

Based on identity (5), there are two problems with

the empirical implementation of (6): the functional

form of the equation is misspeci®ed, and E is omitted.

The functional relationship between wins, losses and

runs is derived from identity (5). Also, the omission

of total excess runs scored (E) is expected to generate

biased parameter estimates due to the expected cor-

relation of E and (TR-OR).

Despite these problems, Horowitz (1994a,b) esti-

mates a constrained regression for a sample of

managers based on career data. He estimates a

regression equation for each individual manager

using a related but alternative functional form:

W

L� b1

TR

OR

� �� b2

TR

OR

� �2

� E �7�

where E is a normally distributed error term.3

Horowitz proposes using the parameter estimates b1

and b2 as an index (PH ) measuring managerial

performance, de®ned as:

PH � b1 � b2 �8�PH is the predicted winning percent of a manager's

team assuming that his team scored the same number

of runs that it allowed. Horowitz (1994a, p. 189)

argues that each manager's historic PH is a basis for

evaluation because, `over a full season, all major-

league-quality managers will generate and permit just

about the same number of runs from a given roster,

but the better managers see to it that these runs are

distributed more effectively'.4 In light of identity (5),

however, it is possible that any pair of managers with

equal ability will have different PH values due to

improper functional form speci®cation of Eqn (7) and

the omitted variable (E ).

According to the Pythagorean theorem, better

managers are able to win more games due to an

ability to distribute runs more ef®ciently. Apparently,

the belief is that an ef®cient manager will forgo runs

in a ball game once a lead is obtained, and then use

these forgone runs during future games when his

team is behind. In the course of a given game,

however, the goal of the manager is to score as

many runs as possible because of the uncertainty of

the number of runs that will be scored by the

opposition. For example, if a team scores 4 runs in

the ®rst inning, a manager does not attempt to limit

run production in all other innings. There may be

alternative strategies, such as resting players, depend-

ing on the score during a given game, but a manager

desires more runs, not less. Importantly, resting key

players is not inconsistent with the goal of maximiz-

ing the number of wins during a given season.

The amount of excess runs (E) for a given team

depends not only on the number of runs scored but

also on the number of runs scored by the opposition.

Consider the following hypothetical case where two

teams that are similar, except for the quality of the

manager, play a game. Given the available player

talent, suppose that each team is expected to score 5

runs, but team A has a better manager than team B.

Suppose that team A wins this hypothetical game 5 to

1. Team A scored the expected number of runs;

hence, manager A was ef®cient. Team B, however,

scored only 1 run, suggesting that manager B was

inef®cient. The maximum possible runs given team

B's available talent were not scored. The fact that

team B did not produce the maximum runs should not

implicate manager A. However, the number of runs

scored by team B will impact the amount of excess

runs of team A, and therefore will affect the PH of

team A's manager.

Baseball managers do not have equal player inputs.

But predictions of a team's expected winning percent

can be made on the basis of the quality of the team's

player inputs.5 The best managers are able to win

more ball games than predicted given the quality of

their teams. Therefore, managers should be evaluated

on the basis of wins achieved relative to their pre-

dicted wins, given player quality. The Pythagorean

approach does not consider differences in player

quality.

EMPIRICAL ANALYSIS OF MAJORLEAGUE MANAGERS

Horowitz (1994a) calculated his proposed measure

PH for the 18 individual managers who had at least

10 full seasons of MLB experience. In this section,

we analyze the results for these managers.6 Table 1

reports descriptive statistics for these 18 managers.

Included in the table are the number of seasons

managed, the estimated coef®cients from the con-

strained regressions, the calculated PH (which equals

the sum of the reported parameter estimates), each

managers average win-loss ratio (W=L) and average

runs ratio (TR=OR). Following Horowitz, the

win=loss ratio (W=L) is the dependent variable in

the constrained regression, and the runs ratio and its

square are the independent variables.7 The results

discussed in this section are only reported to demon-

strate the empirical problems with the Pythagorean

approach.

PYTHAGOREAN THEOREM OF BASEBALL PRODUCTION 337

# 1997 by John Wiley & Sons, Ltd. Managerial and Decision Economics, 18: 335±342

Interesting results emerge from the descriptive

statistics reported in Table 1. First, of the managers

analyzed, Al Lopez achieved the highest win±loss

ratio (W=L� 1.463), but he was ranked the lowest by

Horowitz's PH index. Thus, the Pythagorean

approach would have us believe that Lopez is the

worst manager from this group despite the fact that he

has the best win±loss ratio. On average, Lopez had a

similar runs ratio to Earl Weaver (the best manager

according to Horowitz's PH index) and a higher win±

loss ratio. This peculiar result suggests that the

Pythagorean approach does not properly evaluate

managerial performance.

A closer examination of the arithmetic properties

of Horowitz's PH index reveals its inherent ¯aw as a

measure of managerial performance. For any pair of

managers A and B, there is a unique runs ratio that

will equalize the predicted win±loss ratios of both

managers. Let (TR=OR)E be the ratio that equates the

predicted win±loss ratio of managers A and B. This

ratio can be identi®ed as follows:

b1A

TR

OR

� �E

�b2A

TR

OR

� �2

E

� b1B

TR

OR

� �E

�b2B

TR

OR

� �2

E

�9�

which implies

TR

OR

� �E

� b1A ÿ b1B

b2B ÿ b2A

�10�

Once b1 and b2 have been estimated for each manager

(e.g. for each of the 18 managers analyzed by

Horowitz and identi®ed in Table 1), a unique solution

for any pair of managers can be obtained from Eqn

(10).

As an example, consider Al Lopez and Earl

Weaver. A runs ratio of 1.2 approximately equalizes

these two managers' win±loss ratios at 1.45 given the

parameters reported in Table 1. The results indicate

that Lopez has a higher predicted win±loss ratio than

Weaver for any runs ratio greater than 1.2 while

Weaver has a higher predicted win±loss ratio for any

ratio less than 1.2. In other words, Horowitz's index

indicates that Lopez is the better manager as long as a

team is expected to outscore their opponents by 20%;

otherwise Weaver is the better manager. We believe

that this is illogical. If one manager is better than

another, he should be better regardless of the strength

of the team. An index that claims to rank managerial

performance should not systematically generate

opposite rankings depending on the strength of the

team.8

To extend this analysis, we have computed the

predicted win±loss ratios for Horowitz's 18 managers

for each of the 15 full seasons in which Earl Weaver

managed. Each of these managers is `given Weaver's

team' in the sense that Weaver's runs ratios are used

to predict each manager's win±loss ratios for these 15

seasons. In other words, we pretend that all the

managers were managing in the same 15 seasons as

Weaver did, and we pretend that their runs ratios are

Table 1. Descriptive Statistics and Pythagorean Results

Average

Manager Seasons b1 b2 PH W=L TR=OR

Al Lopez 15 ÿ0.281 1.243 0.962 1.463 1.198

Earl Weaver 15 0.714 0.415 1.129 1.457 1.195

Sparky Anderson 22 0.064 0.957 1.021 1.295 1.122

Walter Alston 23 0.897 0.207 1.103 1.295 1.142

Billy Martin 10 0.399 0.662 1.062 1.276 1.113

Tony LaRussa 12 0.007 1.027 1.034 1.261 1.095

Leo Durocher 22 ÿ0.002 0.989 0.986 1.247 1.114

Whitey Herzog 13 0.168 0.871 1.040 1.241 1.096

Danny Murtaugh 12 0.418 0.593 1.011 1.210 1.115

Dick Williams 15 0.396 0.639 1.034 1.191 1.086

Al Dark 10 0.325 0.673 0.998 1.163 1.086

Ralph Houk 19 0.052 1.090 1.038 1.117 1.024

Red Schoendienst 12 ÿ0.133 1.123 0.990 1.117 1.053

Tommy Lasorda 16 0.221 0.751 0.972 1.095 1.095

John McNamara 13 0.388 0.650 1.038 1.039 0.992

Chuck Tanner 17 ÿ0.058 1.057 0.999 1.023 1.006

Gene Mauch 22 0.265 0.737 1.001 0.987 0.985

Bill Rigney 15 0.236 0.771 1.008 0.960 0.967

Managers are ranked according to their average win=loss ratio.



the same as Weaver's actual runs ratio for each of

these seasons.

Table 2 presents the results for Weaver and the

other seven managers who have predicted win±loss

ratios greater than Weaver's ratios in a majority of

the 15 seasons. These managers have predicted win±

loss ratios greater than Weaver's for the seasons in

which his teams had relatively high values for

TR=OR. Like Lopez, all these managers are expected

to be superior to Weaver when managing teams with

high runs ratios and inferior to Weaver when merging

teams with low runs ratios.9

These results illustrate the illogical arithmetic

property of Horowitz's PH index. An index that

claims to measure managerial ef®ciency should con-

sistently rank managers with respect to their pre-

dicted performance. On the surface, the PH index

may appear to exhibit this property. But our analysis

reveals that these rankings are systematically depen-

dent on the relative values of b1; b2, and the team's

runs ratio. The primary reason for this problem is the

mis-speci®cation in the Pythagorean approach. The

replacement of Eqn (6) with Eqn (7) is the source of

the arithmetic ¯aw discussed above. This ¯aw com-

pounds the fact that the Pythagorean Theorem is

based on an identity.

CONCLUSIONS

This note has critiqued recent papers that have

applied the Pythagorean Theorem as a tool for the

evaluation of managerial performance in MLB. Any

attempt to use this so-called theorem for explanation

or evaluation is inherently ¯awed because it is an

identity rather than a theory. Horowitz's version of

the Pythagorean Theorem is twice ¯awed because he

also mis-speci®es the identity. The correct approach

for the measurement of managerial ef®ciency in any

professional sport is the estimation of a production

function with team winning percent as the dependent

(output) variable and team performance variables as

the independent (input) variables. This production

approach is illustrated by Porter and Scully (1982),

Kahn (1993) and Ruggiero, Hadley and Gustafson

(1996).

APPENDIX

In response to an earlier version of this paper, a

referee raised a number of issues. This appendix

discusses these.

The Pythagorean Approach and Winning Close

Ballgames

Following Horowitz, a referee has suggested that the

Pythagorean approach is useful because it correctly

evaluates managers according to the ability to win

close ballgames. In particular, it is implied that a runs

ratio of unity provides evidence of the number of

close ballgames in a given season. In this section, we

will show that a runs ratio of 1 (i.e. a team allows the

Table 2. Predicted Winning Percents of Selected Managers Using Earl Weaver's Seasonal Data

Earl Weaver's Predicted W=L

Year TR=OR W=L Weaver Houk LaRussa Anderson Herzog Schoendienst Lopez Tanner

1969 1.507 2.057 2.017 2.396 2.342 2.270 2.232 2.350 2.399 2.313

1970 1.380 2.000 1.775 2.003 1.964 1.911 1.891 1.955 1.979 1.933

1971 1.400 1.772 1.812 2.063 2.022 1.966 1.943 2.015 2.043 1.991

1972 1.207 1.081 1.466 1.525 1.504 1.472 1.472 1.476 1.472 1.470

1973 1.344 1.492 1.709 1.898 1.864 1.815 1.800 1.850 1.868 1.832

1974 1.077 1.282 1.250 1.207 1.198 1.179 1.191 1.159 1.139 1.163

1975 1.233 1.304 1.511 1.593 1.570 1.535 1.533 1.544 1.544 1.537

1976 1.035 1.189 1.183 1.114 1.107 1.092 1.108 1.066 1.041 1.073

1977 1.101 1.516 1.289 1.264 1.252 1.231 1.242 1.215 1.197 1.218

1978 1.041 1.268 1,193 1.127 1.120 1.104 1.119 1.079 1.055 1.085

1979 1.301 1.789 1.630 1.776 1.746 1.703 1.693 1.727 1.737 1.713

1980 1.258 1.613 1.554 1.658 1.633 1.595 1.590 1.610 1.613 1.600

1981 0.982 1.283 1.101 0.999 0.996 0.985 1.005 0.952 0.922 0.962

1982 1.127 1.382 1.331 1.325 1.311 1.287 1.295 1.276 1.261 1.277

1986 0.932 0.820 1.025 0.897 0.897 0.890 0.913 0.851 0.817 0.863

Data are taken from Earl Weaver's season data. W=L reported in column 2 is the actual W=L achieved by Weaver.



same number of runs as it scores) does not necessa-

rily correspond to a manager's ability to win close

ballgames. Assume that three different managers

have a season in which TR=OR � W=L � 1. Thus,

each manager wins and loses 81 games in the season.

Also, de®ne a close game (perhaps arbitrarily) as a

game decided by only one run. We now consider

three different cases in terms of the number of close

games that are won.

In case 1, the manager wins and loses 81 games by

a score of 6-1. In this case, the manager has not won

or lost any close games but was still able to win half

of the games. In case 2, suppose the manager wins 40

close games by a score of 3-2. Also, he wins 41

games by a score of 6-1. The team loses 80 games by

a score of 5-2 and one game by a score of 5-0. In this

case, the manager was able to win all close games

that he was involved in. Finally, in case 3, the manger

wins 80 games by a score of 5-2 and one game by a

score of 5-0. The team loses 40 close games by a

score of 3-2 and 41 games by a score of 6-1. The

manager in case 3 has lost all close games.

In all three hypothetical cases a seasonal data point

of TR=OR � W=L � 1 is recorded for the manager.

But this result has nothing to do with the number of

close games that are won. Extending this example to

many seasons, we conclude that the Pythagorean

approach does not evaluate a manager according to

the number of close games that are won. If close

games is the appropriate criterion for managerial

ef®ciency (which it is not, in our view), then a

manager's winning percent in only close games can

be observed directly.

APPLYING THE IDENTITY TOPROFESSIONAL FOOTBALL

In response to an earlier version of this paper, a

referee has suggested that the Pythagorean approach

is a `mysterious fact of baseball life', suggesting that

the approach does not work for other sports. The

reason the Pythagorean approach does not appear to

work in other sports may be attributed (perhaps) to

the number of games played and the number of points

scored. Given the similarities in scoring and the

number of games played, we would expect to see

results for hockey similar to MLB. Regardless, we

will show that the identity developed in the paper

applies to professional football. Data for the

Clevelent Browns in 1992 is reported in Table 2.

Using the data in the table, we see that

TRÿ ORÿ E � 272ÿ 275� 1 � ÿ2. Note that

Cleveland won seven games and lost nine games in

1992. This shows that the identity in the paper

extends to all other sports where ties are ignored.

Why, then, would the Pythagorean approach fail?

The reason perhaps is the small number of games

played and the scoring differences between baseball

and football. Whatever the reason, this example

highlights the proof of the identity in the paper.

Importantly, excess runs (points) are calculated in

Table A1. Game Results for the 1992 Cleveland Browns

Game Cleveland's score Opponent's score Excess points

1 (Indianapolis) 3 14 ÿ10

2 (Miami) 23 27 ÿ3

3 (L.A. Raiders) 28 16 11

4 (Denver) 0 12 ÿ11

5 (Pittsburgh) 17 9 7

6 (Green Bay) 17 6 10

7 (New England) 19 17 1

8 (Cincinnati) 10 30 ÿ19

9 (Houston) 24 14 9

10 (San Diego) 13 14 0

11 (Minnesota) 13 17 ÿ3

12 (Chicago) 27 14 12

13 (Cincinnati) 37 21 15

14 (Detroit) 14 24 ÿ9

15 (Houston) 14 17 ÿ2

16 (Pittsburgh) 13 23 ÿ9

Sum TR� 272 OR� 275 E�ÿ1



the same manner as total runs and opposition runs

(points) for a given ballgame.

The Runs Ratio as a Proxy for the Win±loss ratio

In this section we show that the runs ratio can be

considered to be an alternative measure of output. As

such, it is a proxy to the win±loss ratio. Thus, the

Pythagorean approach considers a regression of out-

put on a proxy for output. This is empirically shown

with the following example.

Using team data (variables are measured in loga-

rithmic form) from the 1982 to 1993 MLB seasons,

we consider two models of production using the

inputs selected by Porter and Scully (slugging percent

and the ratio of walks to strikeouts). The models

differ with the selection of output (i.e. the dependent

variable): Winning percent (a variant to W=L) and the

runs ratio. Regression estimates of these two models

are presented in Table A2. The only statistical dif-

ference between the two models is the intercept. This

con®rms that the runs ratio is actually a proxy for

winning percent.

The Pythagorean Approach and Production

Analyses

Finally, the referee has suggested that the

Pythagorean approach provides similar rankings as

the production function approach used by Porter and

Scully (1982). Importantly, there is a main difference

between these approaches. The production function

approach uses seasonal data to compare all managers

in a given season. Thus, even though Porter and

Scully use data across time, the production function

approach unlike the Pythagorean approach can be

applied to data in a season to measure relative

performance. Hence, the approaches are not compar-

able. Also, the Pythagorean approach does not even

allow differential ef®ciency for a given manager

across time. At best, it provides an average perfor-

mance ranking.

Acknowledgement

We would like to thank Ira Horowitz for his thoughtful commentson an earlier version of this paper.

NOTES

1. Note that Scully (1994) used a variant of this rule toanalyze managerial performance. He estimated analternative functional form using the same variables.The criticism of the Pythagorean approach, therefore,applies also to Scully (1994).

2. For example, if a team wins a game 6-4, the ®fth run isthe deciding run ex post, resulting in one excess run.Likewise, if a team loses 5-2, there are ÿ2 excess runs:the team would have lost even if it had allowed onlythree runs.

3. The regression is constrained with the restriction that theintercept equals zero.

4. A manager's PH is derived by estimating Eqn (7) usingonly the seasonal data of that manager. Thus, a separateregression equation is estimated for each manager.

5. See Ruggiero, Hadley and Gustafson (1996) for a furtherdiscussion.

6. Horowitz (1994b) analyzed additional mangers. In thispaper we analyze the smaller set of managers tohighlight the problems of the Pythagorean approachdiscussed in the paper. These problems also exist forHorowitz's complete list of managers.

7. A few minor discrepancies exist between Horowitz'sreported PH values and those reported in Table 1. Thesecan be attributed to rounding errors. Data problems wereruled out as the source of these discrepancies. Note thatthe discrepancies do not change the critique of thispaper.

8. A referee argued that the Pythagorean approachmeasures the ability of a manager to win close games.In particular, Weaver appears to be a better managerwhen the runs ratio is approximately one. However, aruns ratio equal to one does not provide suf®cientinformation on the number of close games in a givenseason. For a further discussion, see Appendix A.

9. Algebraically, this is explained by the fact that Weaver'sb2 value is lower than many of the other managers inTable 2. In these cases, the square term becomesdominant as TR=OR increases. If the same analysis wereperformed for a manager like Walter Alston, we would®nd that Weaver is the better manager when the runsratio is high, and Alston is better when the runs ratio islow. This is true because Alston's b2 is only half thevalue of Weaver's. Again, we believe this propertymakes Horowitz's PH value an illogical basis forranking managers.

Table A2. Selection of Output in Baseball

Production (OLS Regression Co-

ef®cients, m� 314)

Dependent variable

Variable Winning percent Runs ratio

Intercept ÿ0.069 0.626

(ÿ0.75) (7.58)

Slugging percent 0.863 0.868

(9.09) (10.23)

Base on balls per

strike out (pitchers)

ÿ0.345 ÿ0.365

(ÿ0.67) (ÿ10.27)

Adjusted R-square 0.33 0.40



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Documents

A Note on the Pythagorean Theorem of Baseball Production