Estimated Age Effects in Baseball

Volume 4, Issue 1 2008 Article 1

Journal of Quantitative Analysis in

Sports

Ray C. Fair, Yale University

Recommended Citation:

Fair, Ray C. (2008) "Estimated Age Effects in Baseball," Journal of Quantitative Analysis in

Sports: Vol. 4: Iss. 1, Article 1.

DOI: 10.2202/1559-0410.1074

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Estimated Age Effects in Baseball

Ray C. Fair

Abstract

Age effects in baseball are estimated in this paper using a nonlinear fixed-effects regression.

The sample consists of all players who have played 10 or more "full-time" years in the major

leagues between 1921 and 2004. Quadratic improvement is assumed up to a peak-performance

age, which is estimated, and then quadratic decline after that, where the two quadratics need not be

the same. Each player has his own constant term. The results show that aging effects are larger for

pitchers than for batters and larger for baseball than for track and field, running, and swimming

events and for chess. There is some evidence that decline rates in baseball have decreased slightly

in the more recent period, but they are still generally larger than those for the other events. There

are 18 batters out of the sample of 441 whose performances in the second half of their careers

noticeably exceed what the model predicts they should have been. All but 3 of these players

played from 1990 on. The estimates from the fixed-effects regressions can also be used to rank

players. This ranking differs from the ranking using lifetime averages because it adjusts for the

different ages at which players played. It is in effect an age-adjusted ranking.

KEYWORDS: aging, baseball

Author Notes: Cowles Foundation and International Center for Finance, Yale University, New

Haven, CT 06520-8281. Voice: 203-432-3715; Fax: 203-432-6167; email: [email protected];

website: http://fairmodel.econ.yale.edu. I am indebted to Danielle Catambay for research

assistance and to John Oster and Sharon Oster for their helpful comments.

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1 Introduction

This paper estimates the effects of aging on the performance of major league

baseball players. The performance measures used are on-base percentage (OBP)

and on-base percentage plus slugging percentage (OPS) for batters and earned run

average (ERA) for pitchers. The paper estimates 1) the rate of improvement up to

the peak-performance age, 2) the peak-performance age itself, and 3) the rate of

decline after this age. The improving and then declining age proﬁle is assumed

to be the same for each player, including the peak-performance age. Each player

has his own constant term, however, and so there are n dummy variables in the

regression (a ﬁxed-effects regression), where n is the number of players. Both

the improving and declining proﬁles are assumed to follow quadratic processes,

where the two processes need not be the same. The restrictions imposed are that

the two quadratic processes touch and have zero slopes at the peak-performance

age. The model is presented in Section 2; the data are discussed in Section 3; and

the estimates are presented in Section 4.

The sample is for the period 1921–2004 (1921 is the ﬁrst year of the “live”

ball). Only players who have played at least 10 “full-time” years in this period are

included in the sample, where a full-time year is a year in which a batter played in

at least 100 games and a pitcher pitched at least 450 outs. The aim of this paper

is to estimate aging effects for injury-free, career baseball players, and the sample

was chosen with this in mind. If a batter played fewer than 100 games or a pitcher

pitched fewer than 450 outs in a year, it is possible that the player was injured, and

so these “part-time” years were excluded. If a player played at least 10 full-time

years, he is clearly a career player. The estimated aging effects in this paper are

thus conditional on the player being a career player and not affected by injuries.

The biological decline rate is being estimated for injury-free players. No attempt

is made to estimate the effect of aging on injuries.

There is much work in sabermetrics on developing measures of performance

that might be improvements on OBP, OPS, ERA, and the like.

The standard

measures (like OPS and ERA) are adjusted for issues like 1) the introduction of

the designed hitter rule in the American League in 1973, 2) different ball parks

that players play in, and 3) different league yearly averages. These kinds of ad-

For example, OPS+ and ERA+ are featured on the website www.baseball-reference.com. OPS+

is OPS adjusted for ballparks, the league, and league yearly averages. ERA+ is ERA adjusted for

the same things. Another well known measure is Bill James’ (2001) Win Shares. Another is LW

(linear weights), developed by Thorn and Palmer (1984). Another is EqR (equivalent runs), used,

for example, by Silver (2006).

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justments, however, are problematic from the point of view of this paper. First,

the adjustments tend to be subjective. They are based on particular views about

what is and is not important in measuring players’ performances, and there are no

rigorous ways of testing whether one measure is better than another. Second, and

perhaps more important, adjusting for league averages is likely to over adjust a

player’s performance. If there are ﬂuctuations in league averages over time that

have no effect on a player’s performance, which seems likely, then it is not appro-

priate to divide, say, a player’s OPS for the year by the league-average OPS for

the year to get an “adjusted” OPS for the player. To take an obvious case, say that

the league-average OPS increased for the year because a number of players began

using steroids, but that player A did not use steroids. If player A’s actual OPS were

unchanged for the year, then his adjusted OPS would fall because of the higher

league average, and this would be an incorrect adjustment. Because of these prob-

lems, no adjustments to the standard OBP, OPS, and ERA measures were made for

the work in this paper. This work is based on the assumption that the 15-year-or-so

period that a player plays is stable for that player. This assumption is obviously

only an approximation, since some changes clearly take place within any 15-year

period, but it may not be a bad approximation. In future work, however, it may be

interesting to experiment with alternative measures.

Once the aging estimates have been obtained, they can be used in a variety

of ways. One way, as discussed below, is to compare them to estimates obtained

using the “delta approach.” This is done in Section 5, where it is argued that the

delta approach likely leads to estimated decline rates that are too large. Another

way is to search for players who have unusual age-performance proﬁles. It will be

seen that there are 18 batters out of the sample of 441 whose actual OPS values

late in their careers are noticeably larger than predicted by the equation. All but 3

of these players played from 1990 on. These results are presented in Section 6.

The estimates can also be compared to those for other events. In previous

work—Fair (1994, 2007)—I have estimated decline rates for various track and

ﬁeld, running, and swimming events and for chess. The methodology used in the

present paper is quite different from that used in this earlier work, which is based on

the use of world records by age, and it is of interest to see how the results compare.

It will be seen that the estimated rates of decline in baseball are somewhat larger

than those in the other events. These comparisons are discussed in Section 7, where

possible reasons for the larger rates in baseball are also discussed.

The stability of the estimates over time is examined in Section 8. There is some

evidence that decline rates in baseball are slightly smaller now than they were 40

years ago, although the evidence in general is mixed.

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Finally, the estimates provide a way of ranking players that adjusts for the ages

at which they played. Take two players, both of whom started at age 23. Say that

one played until age 32 and the other played until age 38. Given, as will be seen,

that the peak-performance age is about 28, the second player should be expected,

other things being equal, to have a worse lifetime performance record because he

played a larger fraction of his years below the peak. Ranking players by lifetime

OBP, OPS, or ERA does not correct for possible different ages played. One can

correct for this, however, by ranking players by the size of the coefﬁcient estimates

of the player dummy variables in the regression, i.e., by the players’ estimated

constant terms. This ranking is discussed in Section 9 and presented in Tables A.1

and A.2 for the sample of 441 batters and 144 pitchers.

Regarding previous work in this area, Bill James is the pioneer in using baseball

statistics. In his 1982 Baseball Abstract he evaluated thousands of ballplayers and

concluded that the majority of players peaked at age 27, with most others peaking

at age 26 or 28. The results below are consistent with this conclusion. For example,

the estimated peak age for batters using the OPS measure is 27.59 years, with an

estimated standard error of 0.23 years. One way of estimating aging effects (not

just peak ages) in the baseball literature is to use what is sometimes called the "delta

approach" (see www.tangotiger.net/aging.html). Silver (2006), for example, uses

this approach using equivalent runs (EqR) as his measure of performance. The

approach is to take, say, all 31 year olds in one’s sample who also played when

they were 32, compute the average of the measure across these players for age 31

and for age 32, and then compute the percentage change in the two averages. This

is the estimated change between ages 31 and 32. Then do the same for ages 32 and

33, where the sample is now somewhat different because the players have had to

play at both ages 32 and 33. Continue for each pair of ages. Section 5 argues that

this approach is likely to lead to biased estimates—to estimated rates of decline

at the older ages that are too large. The delta approach does not appear to be a

reliable way of estimating aging effects.

Schultz, Musa, Staszewski, and Siegler (1994) use a sample of 235 batters and

153 pitchers, players who were active in 1965. They compute averages by age.

Using these averages for a variety of performance measures, they ﬁnd the peak-

performance age to be about 27 for batters and 29 for pitchers. As will be seen, the

27 age for batters is close to the estimates in this paper, but the 29 age for pitchers

is noticeably larger. As they note (pp. 280–281), their averages cannot be used to

estimate rates of decline because of selection bias (better players on average retire

later). Schell (2005, Chapter 4) also computes averages by age and also notes (p.

46) the selection bias problem. He presents plots of these averages for various

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performance measures, but does not use them because of the bias problem. He

adjusts his performance measures using data on the ages at which players reached

various milestones, like 1000 at bats, 2000 at bats, etc. He does not attempt to

estimate rates of decline.

The two studies closest to the present one are Berry, Reese, and Larkey (1999)

and Albert (2002). Albert (2002), using LW (linear weights) as the measure of

performance, estimates a quadratic aging function for each player separately and

then combines the regression estimates using a Bayesian exchangeable model.

The estimates are made separately by decade. Albert assumes that the quadratic

is symmetric around the peak age. Barry, Rees, and Larkey (1999) postulate an

asymmetric, nonparametric aging function that is the same for all players. They are

also concerned with player differences across decades, and they use hierarchical

models to model the distribution of players for each decade. More will be said

about both of these studies in the next section.

2 The Model

Let y

denote the measure of performance for player i in year t (either OBP, OPS,

or ERA), and let x

denote the age of player i in year t. The model for player i is:



+ β

+ γ

+

, x

≤ δ

+ β

+ γ

+

, x

≥ δ

(1)

δ is the peak-performance age, and 

is the error term. As noted in the Introduction,

the two quadratic equations are constrained to have zero derivatives and touch at

= δ. This imposes the following three constraints on the coefﬁcients:

= −2γ

= α

+(γ

− γ

)δ

(2)

Figure 1 presents a plot of what is being assumed.

There is quadratic improvement

up to δ and quadratic decline after δ, where the two quadratics can differ. The

unconstrained coefﬁcients to estimate are γ

, γ

, δ, and α

For batters large values of OBP and OPS are good, and for pitchers small values of ERA

are good. Figure 1 and the discussion in this section assumes that large values are good. It is

straightforward to adjust the discussion for ERA.

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20 22 24 26 28 30 32 34 36 38 40

Figure 1

Postulated Relationship Between Performance and Age

Performance

Measure

Age

peak-performance

age

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Each player is assumed to have his own α

(and thus his own α

from equation

(2)). Let p

jit

be a dummy variable for player j that is equal to 1 if j = i and 0

otherwise, and let d

be a dummy variable that is equal to 1 if x

≤ δ and 0

otherwise. Then the equation to be estimated is:



j=1

jit

+ γ

[(δ

− 2δx

+ x

]

+γ

[−δ

+(x

− 2δx

)(1 − d

)] +

= 1 if x

≤ δ and 0 otherwise

(3)

where J is the total number of players. In this equation i runs from 1 to J. For

each player, t runs over the years that he played. 

is assumed to be iid and to be

uncorrelated with the age variables.

The coefﬁcients to estimate in equation (3) are the J values of the alphas, γ

, and δ. If δ is known, the two terms in brackets are known, and so the equation

is linear in coefﬁcients. The equation can then be estimated by the standard ﬁxed-

effects procedure of time-demeaning the data. Overall estimation can thus be done

by trying many values of δ to ﬁnd the value that minimizes the sum of squared

residuals. This does not, however, produce correct standard errors because the

uncertainty of the estimate of δ is not taken into account. Equation (3) must be

estimated by nonlinear least squares to get correct standard errors. This is a large

nonlinear maximization problem because of the large number of dummy variable

coefﬁcients estimated.

The key assumption of the model is that all players have the same β



s and



s, i.e., the same improving and declining rates. Given this, the speciﬁcation is

fairly ﬂexible in allowing the improving rate to differ from the declining rate and

in allowing the peak-performance age to be estimated. Each player has, of course,

his own constant term, which in Figure 1 determines the vertical position of the

curve.

In the table of results below, estimates of γ

, γ

, and δ are presented. In addition,

some implied values by age are presented. Consider the following:

= ˆy

|(x

= k) − ˆy

|(x

δ) (4)

The ﬁrst term on the right hand side is the predicted value for player i at age

k, and the second term is the predicted value for player i at the estimated peak-

performance age

δ. R

is the same for all players because a player’s constant

term appears additively in both predicted values and so cancels out. R

thus does

not need an i subscript. It is the amount by which a player at age k is below his

estimated peak. Va lu es of R

for different values of k are presented in the table

below.

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The derivative of y

with respect to x

∂y

/∂x

= 2γ

− δ)d

+2γ

− δ)(1 − d

)

(5)

This derivative is not a function of a player’s constant term, and so it is the same

for all players of the same age. Let

= 100

(∂y

/∂x

)|(x

= k)

¯y

(6)

where ¯y is the mean of y

over all the observations. D

is roughly the percentage

change in y for a player at age k. It is only roughly the percentage change because

¯y is used in the denominator rather than a speciﬁc player’s predicted value at the

relevant age. Val ue s of D

for different values of k are also presented in the table

below.

This model relative to the models of Berry, Reese, and Larkey (1999) and

Albert (2002), discussed at the end of the Introduction, is parsimonious. Only

three coefﬁcient estimates are estimated aside from the constant term for each

player. It will be seen that this leads to very precise coefﬁcient estimates—a

precisely estimated age proﬁle. Although Albert (2002) restricts the quadratic to

be symmetric around the peak age, which according to the results below is not the

case, his method has the advantage of not having to assume that the aging proﬁle

is the same for all players. The disadvantage is that even with the Bayesian model

that he uses, many parameters are in effect being estimated, and so the precision

may be low. Berry, Reese, and Larkey (1999) assume, as is done in this paper, that

the age proﬁle is the same for all players, but they also in effect estimate many

more parameters because, among other things, of their assumption that players

differ across decades.

A potential cost of the present approach is that the assumption of a constant

age proﬁle across players and over time may not be accurate, which means that

the model may be misspeciﬁed. One way in which the model may be misspeciﬁed

is the following. Say there is a variable like body mass that is different for each

player but that does not change for a given player across his career. If, say, body

mass has no effect on a player’s performance until age 37, at which point a larger

body mass has a negative effect on performance, then 

, which includes the effects

of omitted variables like body mass, will be correlated with age from age 37 on,

thus violating the assumption about the error term. Another possibility is that there

may be “ageless wonders,” who simply decline at slower rates as they age relative

to other players. These players will have positive values of 

at older ages, and so

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

will be correlated with age at older ages, again violating the assumption about

the error term. One check of the quantitative importance of these types of bias

is to examine the sensitivity of the results to the exclusion of older players. As

discussed in the next section, regressions were also run excluding players older

than 37. It will be seen that the results are not sensitive to this exclusion, and so

these potential biases do not appear large. Also, the results in Section 8 show that

the estimates are fairly stable using each half of the sample period.

One selection issue that is not a problem in the present model is the following.

Say that an older player is considering retiring, but in the current year he is doing

better than might be expected given his age—his error term is positive. He may

then choose to play another year, and so the next year will be in the sample. This

does not violate the assumption that age and the error term are uncorrelated as long

as the error term is not serially correlated. In this example, last year’s error affects

the decision to play this year, but this has no effect on this year’s error term, again

assuming no serial correlation.

One ﬁnal issue concerns experience. If the improvement of a player up to the

peak-performance age is interpreted as the player gaining experience (as opposed

to, say, just getting physically better), this experience according to the assumptions

of the model comes with age, not with the number of years played in the major

leagues. A player coming into the major leagues at, say, age 26 is assumed to be

on the same age proﬁle as an age-26 player who has been in the major leagues for

4 years. In other words, minor league experience must be assumed to be the same

as major league experience.

3 The Data

Yearly data on every player who played major league baseball from 1871 on are

available from http://baseball1.com. As noted in the Introduction, the period used

is 1921–2004 and only players who have played at least 10 full-time years in this

period are included in the sample, where a full-time year is a year in which a batter

played in at least 100 games and a pitcher pitched at least 450 outs. Almost all relief

pitchers are excluded from the sample because almost no relief pitcher pitches as

many as 450 outs in a year. The sample for batters included 5,596 observations

and 441 players, and the sample for pitchers included 1,809 observations and 144

players. These players are listed in Tables A.1 and A.2.

Players who are included in the sample may have played non full-time years,

but these years for the player are not in the sample. Players who played beyond

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2004 are included in the sample if they have 10 full-time years from 2004 back.

Players who began playing prior to 1921 are included if they have 10 full-time years

from 1921 forward, but their observations prior to 1921 are not included even if

the observations are for full-time years because no observations before 1921 are

used.

On-base percentage (OBP) is equal to (hits + bases on balls + hit by pitch)

divided by (at bats + bases on balls + hit by pitch + sacriﬁce ﬂies). Slugging

percentage is equal to (hits + doubles + 2 times triples + 3 times home runs) divided

by at bats. OPS is equal to OBP + slugging percentage. Earned run average (ERA)

is equal to the number of earned runs allowed divided by (the number of outs made

divided by 27). These are all standard deﬁnitions. The age of the player was

computed as the year in question minus the player’s birth year.

Some alternative regressions were run to examine the sensitivity of the esti-

mates, and these are reported below. For batters the exclusion restrictions were

changed to 80 games rather than 100 and 8 years rather than 10. This gave 10,605

observations for 932 players. For pitchers the exclusion was changed to 8 years

rather than 10. This gave 2,775 observations for 260 players. Another change was

to drop all observations in which a player was older than 37 years (but keeping a

player in even if this resulted in fewer than 10 full-time years for the player). This

resulted in 5,308 observations for the 441 batters and 1,615 observations for the

144 pitchers.

4 The Results

All the estimates are presented in Table 1. The ﬁrst set of three uses OPS, the

second set uses OBP, and the third set uses ERA. The ﬁrst estimate for each set is

the basic estimate; the second estimate is for the larger number of observations; and

the third estimate excludes observations in which the player is over 37. Estimated

standard errors for the coefﬁcient estimates are presented for the basic estimate for

each set. As noted above, the model is nonlinear in coefﬁcients, and for present

purposes the DFP algorithm was used to obtain the estimates.

The implied values

This is a large nonlinear maximization problem. There are 444 coefﬁcients to estimate: γ

, γ

δ, and the 441 dummy variable coefﬁcients. These calculations were done using the Fair-Parke

program (2003). The standard errors of the coefﬁcient estimates were computed as follows. Let

f(y

,α) = u

be the equation being estimated, where y

is the dependent variable, x

is the

vector of explanatory variables, α is the vector of coefﬁcients to estimate, and u

is the error term.

j indexes the number of observations; assume that it runs from 1 to J. Let K be the dimension of

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Tab l e 1

Coefﬁcient Estimates and Implied Aging Valu es

Estimate of #obs R

,(D

)byage

δ SE (#plys) 22 25 28 31 34 37 40

OPS

1 -0.001618 -0.000508 27.59 .0757 5596 -0.051 -0.011 0.000 -0.006 -0.021 -0.045 -0.078

(.000205) (.000021) (0.23) (441)

(2.28) (1.06) (-0.05) (-0.44) (-0.82) (-1.21) (-1.59)

2 -0.001617 -0.000550 27.60 .0758 10605

-0.051 -0.011 0.000 -0.006 -0.023 -0.049 -0.085

(932) (2.36) (1.10) (-0.06) (-0.49) (-0.92) (-1.35) (-1.78)

3 -0.001483 -0.000609 27.90 .0749 5308 -0.052 -0.012 0.000 -0.006 -0.023 -0.050 -0.089

(441) (2.20) (1.08) (-0.02) (-0.47) (-0.93) (-1.39) (-1.85)

OBP

1 -0.0005289 -0.0001495 28.30 .0276 5596 -0.021 -0.006 0.000 -0.001 -0.005 -0.011 -0.020

(.0000621) (.000074) (0.26) (441)

(1.88) (0.99) (0.09) (-0.23) (-0.48) (-0.73) (-0.99)

2 -0.0005252 -0.0001634 28.30 .0281 10605

-0.021 -0.006 0.000 -0.001 -0.005 -0.012 -0.022

(932)

(1.91) (1.00) (0.09) (-0.26) (-0.54) (-0.82) (-1.11)

3 -0.0005032 -0.0001742 28.50 .0271 5308

-0.021 -0.006 0.000 -0.001 -0.005 -0.013 -0.023

(441)

(1.84) (0.99) (0.14) (-0.25) (-0.54) (-0.83) (-1.13)

ERA

1 0.006520 0.002872 26.54 .6845 1809

0.134 0.015 0.006 0.057 0.160 0.314 0.520

(.005388) (.000658) (1.40) (144) (-1.69) (-0.57) (0.24) (0.73) (1.22) (1.72) (2.21)

2 0.021474 0.002265 24.00 .6910 2775 0.086 0.002 0.036 0.111 0.226 0.383 0.580

(260)

(-2.40) (0.13) (0.51) (0.89) (1.27) (1.64) (2.02)

3 0.011821 0.001926 25.20 .6848 1615 0.121 0.000 0.015 0.065 0.149 0.268 0.422

(144)

(-2.17) (-0.14) (0.31) (0.64) (0.97) (1.31) (1.64)

Notes:

• Standard errors are in parentheses for the coefﬁcient estimates.

• lines 1 and 3: 10 full-time years between 1921 and 2004; full-time year: 100 games for batters, 150 innings for pitchers.

• lines 3: player observation excluded if player aged 38 or over.

• lines 2: 8 full-time years between 1921 and 2004; full-time year: 80 games for batters, 150 innings for pitchers.

• R

deﬁned in equation (4); D

deﬁned in equation (6).

• Dummy variable included for each player. Dummy variable coefﬁcient estimates presented in Table A.1 for OPS line 1

and OBP line 1 and in Table A.2 for ERA line 1 under the heading CNST.

• The mean of all the observations (¯y in the text) is .793 OPS, line 1, .766 OPS, line 2, .795 OPS, line 3, .354 OBP, line 1,

.346 OPS, line 2, .355 OPS, line 3, 3.50 ERA, line 1, 3.58 ERA, line 2, 3.48 ERA, line 3.

ˆ ˆ

for R

and D

are presented for k equal to 22, 25, 28, 31, 34, 37, and 40. Remember

that R

is the amount by which a player at age k is below his estimated peak and

that D

is roughly the percentage change in the performance measure at age k.

A general result in Table 1 is that the estimates are not sensitive to the increase

in the number of players (by using 8 years as the cutoff instead of 10 years and by

using for batters 80 games played in a year instead of 100) and to the exclusion

of observations in which the player was older than 37. Compare, for example, the

values of R

and D

for k = 40 in lines 1, 2, and 3 for each of the three measures.

The following discussion will thus concentrate on the basic estimate—line 1—for

each set.

α (K coefﬁcients to estimate). Let G



be the K × J matrix whose jth column is ∂f(y

,α)/∂α.

The estimated covariance matrix of ˆα is ˆσ



−1

, where ˆσ

is the estimate of the variance of u

and

G is G evaluated at α = ˆα. For regression 1 for batters J is 5596 and K is 444. For regression

1 for pitchers J is 1809 and K is 147.

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Another general result in Table 1 is that the estimated rate of improvement

before the peak-performance age is larger than the estimated rate of decline after

the age. In other words, the learning curve at the beginning of a player’s career is

steeper than the declining curve after the peak-performance age.

Turning now to the basic estimates, for OPS δ is 27.6 years and by age 37 the

percentage rate of decline is 1.21 percent. For OBP the respective numbers are

28.3 years and 0.73 percent. The peak-performance ages are thus quite similar

for the two measures, but OPS declines somewhat more rapidly than OBP. To get

a sense of magnitudes, if a player’s peak OPS is 0.800 (the mean of OPS in the

sample is 0.793), then the -0.045 value for R

means that his predicted OPS at age

37 is 0.755, a decrease of 5.6 percent. Similarly, if a player’s peak OBP is 0.350

(the mean of OBP is the sample is 0.354), then the -0.011 value for R

means that

his predicted OBP at age 37 is 0.339, a decrease of 3.1 percent.

For ERA δ is 26.5 and by age 37 the percentage rate of decline is 1.72 percent.

If a pitcher’s peak ERA is 3.50 (the mean of ERA in the sample is 3.50), then the

0.314 value for R

means that his predicted ERA at age 37 is 3.814, an increase

of 9.0 percent. The estimated decline for pitchers is thus somewhat larger than for

batters, and the peak-performance age is slightly lower.

The precision of the estimates is fairly good, although better for batters than for

pitchers. The estimated standard error for the estimated peak-performance age is

0.23 years for OPS and 0.26 years for OBP. For ERA it is 1.40 years. The sample

period for pitchers is about a third the size of the period for batters, which at least

partly accounts for the less precision for pitchers.

5 Comparison to the Delta Approach

As discussed in the Introduction, the delta approach has been used to measure

aging effects. For example, Silver (2006, Table 7-3.4, p. 263) has used it for post

World War II batters and the EqR measure. To examine this approach further,

Table 2 presents estimated decline rates using the delta approach for the sample

of 441 batters used in this paper and the OBP measure. For example, there were

344 of the 441 batters who played full time when they were both 32 and 33. The

average OBP for this group was .3609 for age 32 and .3560 for age 33, which is

a decline of 1.36 percent. There were then 315 of the 441 batters who played full

time when they were both 33 and 34. The average OBP for this group was .3577

for age 33 and .3537 for age 34, which is a decline of 1.12 percent.

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Table 2

Estimated Decline Rates Using the Delta Approach

First Age Second Age

Ages # obs. OBP ave. OBP ave. % change

21–22 55 .3404 .3468 1.88

22–23 133 .3412 .3493 2.37

23–24 226 .3491 .3533 1.20

24–25 273 .3483 .3544 1.75

25–26 333 .3516 .3570 1.54

26–27 362 .3559 .3569 0.28

27–28 380 .3576 .3582 0.17

28–29 381 .3588 .3586 −0.00

29–30 368 .3590 .3577 −0.36

30–31 375 .3585 .3610 0.70

31–32 359 .3609 .3599 −0.28

32–33 344 .3609 .3560 −1.36

33–34 315 .3577 .3537 −1.12

34–35 263 .3578 .3511 −1.87

35–36 210 .3544 .3508 −1.02

36–37 146 .3545 .3525 −0.56

37–38 96 .3543 .3480 −1.78

38–39 64 .3599 .3530 −1.92

39–40 39 .3685 .3597 −2.39

40–41 22 .3585 .3439 −4.07

Comparing Tables 1 and 2, it is obvious that the decline rates are larger in

Table 2. In Table 1 for OBP, line 1, the decline rate is 0.48 percent for age 34,

0.73 percent for age 37, and 0.99 percent for age 40. In Table 2 the decline rate

is 1.87 percent for age 34, 1.78 percent for age 37, and 4.07 percent for age 40.

What can account for these large differences? A likely answer is that the delta

approach overestimates decline rates at the older ages—that the delta-approach

decline-rate estimates are biased. The reason is the following. First, note in Table

2 that the sample size drops fairly rapidly after age 32. Now consider a player

who is thinking about retiring and who has had a better than average year for him.

“Better-than-average” means that his error term in equation (1) is positive. This

is likely to increase the chances that he chooses to play the next year. If players’

error terms are uncorrelated across years, then a positive error in one year does not

increase the chances of a positive error the next year. Our player is expected to

have an average year (for him) the next year—an expected zero error term. If it

turns out that he in fact has an average (or below average) year, this may lead him to

retire at the end of the season. So error terms for players in their penultimate year

are likely to be on average higher than the error terms in their last year. Players

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don’t retire as often when error terms are large. The delta approach will thus be

biased at the older ages because the paired sample that is used will have on average

larger errors for the younger of the two ages.

This bias can in fact been seen in the sample used in this paper. Of the 441

batters in the sample, 401 had retired. The average of the error terms for the

last observation for each of these 401 players, using the error terms for the OBP

regression in Table 1, line 1, is −0.00954, which is smaller than the average of the

error terms from the second-to-last observation of −0.00448. (The last observation

in the sample for a player is usually the year in which he retired.) So there is

evidence that a player’s error term is lower in the year in which he retires than in

the year before he retires, thus leading the delta approach to be biased. Comparing

the estimates in Tables 1 and 2 suggests that the bias is quite large.

6 Unusual Age-Performance Proﬁles

Since there is a dummy variable for each player, the sum of a player’s residuals

across the years that he played is zero. Under the assumption that the errors, 

are iid, they should lie randomly around the age-performance curve in Figure 1 for

each player. It is interesting to see if there are players whose patterns are noticeably

different. For example, if a player got better with age, contrary to the assumptions

of the model, one would see in Figure 1 large negative residuals at the young ages

and large positive residuals at the old ages.

Using OPS regression 1 in Table 1, the following procedure was followed to

choose players who have a pattern of large positive residuals in the second half

of their careers. First, all residuals greater than one standard error (.0757) were

recorded. Then a player was chosen if he had four or more of these residuals from

age 28, the estimated peak-performance age, on. There were a total of 17 such

players. In addition, for reasons discussed below, Rafael Palmeiro was chosen,

giving a total of 18 players. The age-performance results for these players are

presented in Table 3. The residuals in bold are greater than one standard error. The

players are listed in alphabetic order except for Palmeiro, who is listed last.

The most remarkable performance by far in Table 3 is that of Barry Bonds.

Three of his last four residuals (ages 37–40) are the largest in the sample period, and

the last one is 5.5 times the estimated standard error of the equation. Not counting

Bonds, Sammy Sosa has the largest residual (age 33, 2001) and Luis Gonzalez

has the second largest (age 34, 2001). Mark McGwire has three residuals that are

larger than two standard errors (age 33, 1996; age 35, 1998; age 36, 1999). Larry

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Table 3

Age-Performance Results for Eighteen Players: OPS

Yea r Age Pred. Act. Resid. Year Age Pred. Act. Resid.

Albert Belle Bob Boone

1991 25 0.946 0.863 -0.083 1973 26 0.700 0.675 -0.025

1992 26 0.952 0.797 -0.155 1974 27 0.704 0.617 -0.087

1993 27 0.956 0.922 -0.034 1976 29 0.703 0.713 0.010

1994 28 0.956 1.152 0.196 1977 30 0.701 0.780 0.079

1995 29 0.955 1.091 0.136 1978

31 0.698 0.772 0.074

1996 30 0.954 1.033 0.079 1979 32 0.694 0.789 0.094

1997 31 0.951 0.823 -0.128

1980 33 0.689 0.637 -0.052

1998 32 0.947 1.055 0.108 1982 35 0.676 0.647 -0.029

1999 33 0.942 0.941 0.000 1983 36 0.668 0.641 -0.027

2000 34 0.936 0.817 -0.119

1984 37 0.659 0.504 -0.155

1985 38 0.649 0.623 -0.026

1986 39 0.638 0.593 -0.046

1987 40 0.626 0.615 -0.011

1988 41 0.613 0.739 0.126

1989 42 0.599 0.675 0.076

Barry Bonds Ken Caminiti

1986 22 1.035 0.746 -0.289

1989 26 0.803 0.685 -0.118

1987 23 1.051 0.821 -0.231

1990 27 0.807 0.611 -0.196

1988 24 1.065 0.859 -0.206

1991 28 0.807 0.695 -0.113

1989 25 1.075 0.777 -0.298

1992 29 0.807 0.790 -0.016

1990 26 1.081 0.970 -0.111

1993 30 0.805 0.711 -0.093

1991 27 1.085 0.924 -0.161

1994 31 0.802 0.847 0.046

1992 28 1.085 1.080 -0.006 1995 32 0.798 0.894 0.096

1993 29 1.084 1.136 0.051 1996

33 0.793 1.028 0.236

1994 30 1.083 1.073 -0.009

1997 34 0.787 0.897 0.110

1995 31 1.080 1.009 -0.071

1998 35 0.780 0.862 0.082

1996 32 1.076 1.076 0.000 2001

38 0.753 0.719 -0.033

1997 33 1.071 1.031 -0.040

1998 34 1.065 1.047 -0.018

1999 35 1.058 1.006 -0.051

2000 36 1.050 1.127 0.078

2001 37 1.041 1.379 0.338

2002 38 1.031 1.381 0.350

2003 39 1.019 1.278 0.258

2004 40 1.007 1.422 0.414

Chili Davis Dwight Evans

1982 22 0.786 0.719 -0.067 1973 22 0.806 0.703 -0.103

1983 23 0.802 0.657 -0.145 1974 23 0.823 0.756 -0.067

1984 24 0.816 0.875 0.059 1975 24 0.836 0.809 -0.027

1985 25 0.825 0.761 -0.065 1976 25 0.846 0.755 -0.091

1986 26 0.832 0.791 -0.041 1978 27 0.856 0.784 -0.072

1987 27 0.836 0.786 -0.049 1979 28 0.857 0.820 -0.036

1988 28 0.836 0.757 -0.079 1980 29 0.856 0.842 -0.014

1989 29 0.835 0.775 -0.060 1981 30 0.854 0.937 0.083

1990 30 0.833 0.755 -0.078 1982 31 0.851 0.936 0.085

1991 31 0.830 0.892 0.062 1983 32 0.847 0.774 -0.072

1992 32 0.827 0.825 -0.002 1984 33 0.842 0.920 0.078

1993 33 0.822 0.767 -0.055 1985 34 0.836 0.832 -0.004

1994 34 0.816 0.971 0.156 1986 35 0.829 0.853 0.024

1995 35 0.809 0.943 0.135 1987

36 0.821 0.986 0.166

1996 36 0.801 0.884 0.083 1988 37 0.812 0.861 0.050

1997 37 0.791 0.896 0.104 1989 38 0.802 0.861 0.059

1999 39 0.770 0.812 0.041 1990 39 0.791 0.740 -0.051

1991 40 0.779 0.771 -0.007

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Table 3 (continued)

Yea r Age Pred. Act. Resid. Year Age Pred. Act. Resid.

Steve Finley Julio Franco

1990 25 0.801 0.632 -0.169

1983 25 0.824 0.693 -0.131

1991 26 0.808 0.737 -0.071 1984 26 0.831 0.679 -0.152

1992 27 0.811 0.762 -0.049 1985 27 0.834 0.723 -0.111

1993 28 0.812 0.689 -0.123 1986 28 0.835 0.760 -0.074

1995 30 0.809 0.786 -0.023 1987 29 0.834 0.818 -0.016

1996 31 0.806 0.885 0.079 1988

30 0.832 0.771 -0.061

1997 32 0.802 0.788 -0.014 1989 31 0.829 0.848 0.019

1998 33 0.797 0.702 -0.096

1990 32 0.825 0.785 -0.040

1999 34 0.791 0.861 0.070 1991 33 0.820 0.882 0.062

2000 35 0.784 0.904 0.120 1993 35 0.807 0.798 -0.009

2001 36 0.776 0.767 -0.009

1994 36 0.799 0.916 0.117

2002 37 0.767 0.869 0.102 1996 38 0.780 0.877 0.097

2003 38 0.757 0.863 0.105 1997

39 0.769 0.730 -0.039

2004 39 0.746 0.823 0.077 2002

44 0.698 0.739 0.040

2003 45 0.681 0.824 0.143

2004 46 0.663 0.818 0.155

Gary Gaetti Andres Galarraga

1982 24 0.744 0.723 -0.021

1986 25 0.866 0.743 -0.123

1983 25 0.754 0.724 -0.030

1987 26 0.873 0.821 -0.052

1984 26 0.761 0.665 -0.095

1988 27 0.876 0.893 0.017

1985 27 0.764 0.710 -0.054

1989 28 0.877 0.761 -0.116

1986 28 0.765 0.865 0.101 1990

29 0.876 0.715 -0.161

1987 29 0.764 0.788 0.024 1991

30 0.874 0.604 -0.270

1988 30 0.762 0.905 0.143 1993 32 0.867 1.005 0.138

1989 31 0.759 0.690 -0.069

1994 33 0.862 0.949 0.087

1990 32 0.755 0.650 -0.105

1995 34 0.856 0.842 -0.014

1991 33 0.750 0.672 -0.078

1996 35 0.849 0.958 0.109

1992 34 0.744 0.610 -0.134

1997 36 0.841 0.974 0.133

1993 35 0.737 0.738 0.001 1998

37 0.832 0.991 0.159

1995 37 0.720 0.846 0.126 2000

39 0.811 0.895 0.084

1996 38 0.710 0.799 0.090 2001

40 0.799 0.784 -0.014

1997 39 0.699 0.710 0.011 2002 41 0.785 0.738 -0.047

1998 40 0.687 0.852 0.165 2003

42 0.771 0.841 0.069

1999 41 0.673 0.599 -0.074

Charlie Gehringer Luis Gonzalez

1926 23 0.862 0.721 -0.141 1991 24 0.842 0.753 -0.088

1927 24 0.875 0.824 -0.052 1992 25 0.852 0.674 -0.177

1928 25 0.885 0.846 -0.039 1993 26 0.858 0.818 -0.040

1929 26 0.892 0.936 0.045 1994 27 0.862 0.782 -0.080

1930 27 0.895 0.938 0.043 1995 28 0.862 0.812 -0.051

1931 28 0.896 0.790 -0.106 1996 29 0.861 0.797 -0.065

1932 29 0.895 0.867 -0.028 1997 30 0.859 0.722 -0.138

1933 30 0.893 0.862 -0.031 1998 31 0.857 0.816 -0.041

1934 31 0.890 0.967 0.077 1999 32 0.853 0.952 0.099

1935 32 0.886 0.911 0.025 2000 33 0.848 0.935 0.088

1936 33 0.881 0.987 0.106 2001 34 0.842 1.117 0.275

1937 34 0.875 0.978 0.102 2002 35 0.835 0.896 0.061

1938 35 0.868 0.911 0.043 2003

36 0.827 0.934 0.107

1939 36 0.860 0.967 0.107 2004 37 0.818 0.866 0.048

1940 37 0.851 0.875 0.024

1941 38 0.841 0.666 -0.175

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Table 3 (continued)

Yea r Age Pred. Act. Resid. Year Age Pred. Act. Resid.

Mark McGwire Paul Molitor

1987 24 0.981 0.987 0.007 1978 22 0.805 0.673 -0.132

1988 25 0.991 0.830 -0.161 1979 23 0.822 0.842 0.020

1989 26 0.997 0.806 -0.191 1980 24 0.835 0.809 -0.025

1990 27 1.001 0.859 -0.142 1982 26 0.851 0.816 -0.035

1991 28 1.002 0.714 -0.288

1983 27 0.855 0.743 -0.112

1992 29 1.001 0.970 -0.031 1985 29 0.855 0.764 -0.091

1995 32 0.992 1.125 0.134 1986 30 0.853 0.765 -0.087

1996 33 0.987 1.198 0.211 1987 31 0.850 1.003 0.153

1997 34 0.981 1.039 0.058 1988

32 0.846 0.836 -0.010

1998 35 0.974 1.222 0.249 1989

33 0.841 0.818 -0.023

1999 36 0.966 1.120 0.155 1990

34 0.835 0.807 -0.028

1991 35 0.828 0.888 0.060

1992 36 0.820 0.851 0.031

1993 37 0.811 0.911 0.101

1994 38 0.801 0.927 0.127

1995 39 0.790 0.772 -0.017

1996 40 0.778 0.858 0.081

1997 41 0.764 0.786 0.022

1998 42 0.750 0.718 -0.033

Sammy Sosa B.J. Surhoff

1990 22 0.854 0.687 -0.167

1987 23 0.732 0.773 0.041

1991 23 0.870 0.576 -0.294

1988 24 0.745 0.611 -0.134

1993 25 0.893 0.794 -0.099

1989 25 0.755 0.626 -0.129

1994 26 0.900 0.884 -0.016

1990 26 0.762 0.706 -0.056

1995 27 0.904 0.840 -0.063

1991 27 0.766 0.691 -0.075

1996 28 0.904 0.888 -0.016

1992 28 0.766 0.635 -0.131

1997 29 0.903 0.779 -0.124 1993 29 0.765 0.709 -0.056

1998 30 0.901 1.024 0.122 1995

31 0.760 0.870 0.109

1999 31 0.898 1.002 0.103 1996

32 0.756 0.834 0.078

2000 32 0.894 1.040 0.145 1997

33 0.751 0.803 0.052

2001 33 0.889 1.174 0.285 1998

34 0.745 0.789 0.044

2002 34 0.883 0.993 0.110 1999

35 0.738 0.839 0.101

2003 35 0.876 0.911 0.035 2000 36 0.730 0.787 0.057

2004 36 0.868 0.849 -0.020 2001 37 0.721 0.726 0.004

2004 40 0.688 0.785 0.097

Larry Walker Rafael Palmeiro

1990 24 0.967 0.761 -0.207

1988 24 0.893 0.785 -0.108

1991 25 0.977 0.807 -0.170 1989 25 0.903 0.728 -0.175

1992 26 0.984 0.859 -0.125 1990 26 0.910 0.829 -0.081

1993 27 0.988 0.841 -0.147 1991 27 0.914 0.922 0.008

1994 28 0.988 0.981 -0.007 1992 28 0.914 0.786 -0.128

1995 29 0.987 0.988 0.001 1993 29 0.913 0.926 0.013

1997 31 0.982 1.172 0.189 1994 30 0.911 0.942 0.031

1998 32 0.978 1.075 0.096 1995 31 0.908 0.963 0.055

1999 33 0.974 1.168 0.195 1996 32 0.904 0.927 0.023

2001 35 0.961 1.111 0.151 1997 33 0.899 0.815 -0.085

2002 36 0.952 1.023 0.071 1998 34 0.893 0.945 0.051

2003 37 0.943 0.898 -0.046 1999 35 0.886 1.050 0.164

2000 36 0.878 0.954 0.076

2001 37 0.869 0.944 0.075

2002 38 0.859 0.962 0.103

2003 39 0.848 0.867 0.019

2004 40 0.836 0.796 -0.040

• Act. = actual OPS, Pred. = predicted OPS, Resid. = Act. - Pred.

• Resid. sums to zero across time for each player.

• Valu e s of Resid. greater than one standard error are in bold.

•

Equation is OPS line 1 in Table 1. Standard error is .0757.

• Resid. in 2001 for Palmeiro is .0750.

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Walker has two residuals that are larger than two standard errors (age 31, 1997;

age 33, 1999) and one that is nearly two standard errors (age 35, 2001). Aside

from the players just mentioned, 8 other players have one residual greater than two

standard errors: Albert Belle (age 28, 1994), Ken Caminiti (age 33, 1996), Chili

Davis (age 34, 1994), Dwight Evans (age 36, 1987), Julio Franco (age 46, 2004),

Gary Gaetti (age 40, 1998), Andres Galarraga (age 37, 1998), and Paul Molitor

(age 31, 1987).

There are only 3 players in Table 3 who did not play more than half their

careers in the 1990s and beyond: Bob Boone (1973–1989), Dwight Evans (1973–

1991), and Charlie Gehringer (1926–1941). Remember that the period searched

was 1921–2004, so this concentration is unusual. An obvious question is whether

performance-enhancing drugs had anything to do with this concentration. In 2005

Palmeiro tested positively for steroids, and so it is of interest to see what his age-

performance results look like. He is listed last in Table 3. Palmeiro’s pattern looks

similiar to that of many of the others in the table. He has three residuals greater

than one standard error in the second half of his career, one of these greater than

two standard errors (age 35, 1999; age 36, 2000; age 38, 2002). In addition, his

residual in 2001 was .0750, which is very close to the standard error of .0757. He

was thus very close to being chosen the way the other players were. No other

players were this close to being chosen.

Since there is no direct information about drug use in the data used in this

paper, Table 3 can only be interpreted as showing patterns for some players that

are consistent with such use, not conﬁrming such use. The patterns do not appear

strong for the three pre-1990 players: Boone, Evans, and Gehringer. For the other

players, some have their large residuals spread out more than others. The most

spread out are those for Gaetti, Molitor, and Surhoff. Regarding Galarraga, four of

his six large residuals occurred when he was playing for Colorado (1993–1997).

Walker played for Colorado between 1995 and 2003, and his four large residuals

all occurred in this period. Colorado has a very hitter-friendly ball park. Regarding

the results in Table 3, there are likely to be different views on which of the patterns

seem most suspicious, especially depending on how one weights other information

and views about the players. This is not pursued further here.

From the perspective of this paper, the unusual patterns in Table 3 do not ﬁt the

model well and thus are not encouraging for the model. On the other hand, there

are only at most about 15 players out of the 441 in the sample for which this is

true. Even star players like Babe Ruth, Ted Williams, Rogers Hornsby, and Lou

Gehrig do not show systematic patterns. In this sense the model works well, with

only a few key exceptions.

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7 Comparison to Other Events

In Fair (2007) rates of decline were estimated for various athletic events and for

chess. Deterioration rates were estimated from age 35 on using world records

by age. Given the results in Table 1, one way to compare the present results to

the earlier ones is to compute what percent is lost by age 40 in each event. For

example, for OPS in line 1, the percent lost is .078 divided by the mean (.793),

which is 9.8 percent. For OPB in line 1, the percent lost is .020 divided by .354,

which is 5.6 percent. Finally, for ERA in line 1, the percent lost is .520 divided by

3.50, which is 14.9 percent. As discussed in Section 4, pitchers are estimated to

decline more rapidly than batters.

The above three percents can be compared to the percents for the other events.

This is done in Table 4. The results for the other events are taken from Table 3

in Fair (2007). Two ways of comparing the results are presented in Table 4. The

ﬁrst is simply to list the percent lost by age 40 for each event. The second is to

take, say, the 9.8 percent at age 40 for OPS and list the age at which this percent is

reached for each of the other events. This second way is done for OPS, OBP, and

ERA.

It should be kept in mind that the percent declines for the other events are

declines from age 35. If decline in fact starts before age 35, as it is estimated to do

for baseball, then the percents for the other events are too low.

The events are listed in the notes to Table 4. The rates of decline for baseball

are larger than they are for the other events. For OBP, non-sprint running (“Run”),

and the high jump, the results are not too far apart: 5.6 percent versus 4.1 percent

and 4.5 percent, with Run and the high jump being only 2 years ahead of OBP

(42 years versus 40). The rate of decline for Sprint is smaller, even smaller for

the swimming events, and very small for chess. The most extreme case is ERA

versus Chess1, where the 14.9 percent decline for ERA at age 40 is not reached

until age 85 for chess! Remember, however, that the ERA results are based on a

smaller sample than the OPS and OBP results, and so the 14.9 percent ﬁgure is

less reliable than the others. Nevertheless, other things being equal, chess players

do seem to have a considerable advantage over pitchers.

The estimates for the other events have the advantage of being based on age

records up to very old ages, in some cases up to age 100. Because of the way

professional baseball works, it is not possible to get trustworthy estimates at ages

The aging estimates in Fair (2007) are not affected if decline starts before age 35. The estimates

just require that decline has begun by age 35. Although the ﬁrst age of decline is not estimated, for

the events considered in the paper there does not appear to be much decline before age 35.

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Table 4

Comparison of Aging Effects Across Events

Age at Age at Age at

% loss at 9.8% 5.6% 14.9%

age 40 loss loss loss

OPS 9.8 40

OBP 5.6 40

ERA 14.9 40

Sprint 3.0 51 45 59

Run 4.1 47 42 53

High Jump 4.5 46 42 51

M50 2.1 57 48 68

M100 2.5 54 46 63

M200+ 1.8 59 50 64

Chess1 0.9 79 64 85

Chess2 0.8 71 63 78

Notes:

• Sprint = 100, 200, and 400 meter track.

• Run = all running except 100, 200, and

400 meter track.

• M50 = 50 meter and yard swimming events.

• M100 = 100 meter and yard swimming events.

• M200+ = all other swimming events.

• Chess1 = Chess, best rating.

• Chess2 = Chess, second best rating.

Non baseball results taken from Table 3 in

Fair (2007).

much beyond 40. In events like running and swimming people of all ages can

participate. An elite runner, for example, can continue to run even when he (or

she) is past the age at which he has any chance of placing in the top group. There

are thus many observations on performances of old elite runners. This is not true of

professional baseball, where once a player is out of the top group, he is not allowed

to play. (Even Roger Clemens is not likely to be playing when he is 60.) There

is thus no way of estimating the rate of decline of professional baseball players

beyond the age of about 40. It may be if players were allowed to play into old age,

their rates of decline would not be much different from those in, say, running or

the high jump, but this cannot be tested.

It is interesting to speculate why rates of decline might be larger in baseball.

One possibility is that baseball skills, like fast hand/eye coordination and bat speed,

decline faster than skills in the other events. Another possibility is that this reﬂects

players’ responses to the fact that once they are out of the top group they can’t play.

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Assume that a player has some choice of his age-performance proﬁle. Assume in

particular that he can choose curve A or B in Figure 2, where, contrary to the

assumptions of the model, neither curve is quadratic after the peak-performance

age. The two curves reﬂect a trade-off between yearly performances and decline

rates. It may be, as in curve A, that a player can stay near his peak-performance

value for a number of years after his peak-performance age, but at a cost of faster

bodily deterioration later. An alternative strategy may be, as in curve B, not to push

as hard after the peak-performance age and have a slower decline rate. If bmin

in Figure 2 is the minimum performance level for a player to stay in the major

leagues, then the player is forced to retire at age k

if he chooses curve A and at

age k

if he chooses curve B. Which curve a player chooses if he is maximizing

career income depends on the wage rate paid at each performance level.

Now say that the wage rate is simply proportional to the performance measure

and that curves A and B are such that the player is indifferent between them. If

bmin is then lowered to bmin



, it is clear that the player will now prefer B to

A since the added area under B between k



and k

is greater than that under A

between k



and k

. There is thus an incentive to choose ﬂatter age-performance

proﬁles as the minimum performance level is lowered. If this level is lower for

the other events than it is for baseball, this could explain at least part of the larger

estimated decline rates for baseball.

If players do have some choice over their age-performance proﬁle,the estimates

in this paper reﬂect this choice, although, contrary to the curves in Figure 2, the

functional form is restricted to be quadratic. The assumption of the model that

, β

, γ

, and δ are the same for all players is stronger in this case because it

reﬂects the assumption that players all make the same choice.

8 Possible Changes Over Time

The regressions in Table 1 span a period of 84 years, a period in which a number

of important changes occurred in baseball. Mention has already been made of the

designated hitter rule in the American League. Another change is that beginning in

the early 1970s, the reserve clause was eliminated and players got more bargaining

power. Under the reserve clause, most contracts were one-year contracts, and

players were required to negotiate with their current team. The main bargaining

weapon of players was to hold out. After the reserve clause was eliminated, many

contracts became multi year and players had more freedom to move around. This

all resulted in a larger fraction of baseball revenues going to the players. There

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Figure 2

Alternative Age-Performance Profiles

Performance

Measure

Age

bmin

bmin'

1 1

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may also have been technical progress over this period, with advances in medical

procedures, increased training knowledge, and the like.

It is thus of interest to see if the coefﬁcient estimates in Table 1 are stable

over time. The sample was divided into two periods, the ﬁrst consisting of players

who began playing in the major leagues in 1965 or earlier and the second of those

who began playing in 1966 or later. For batters, the ﬁrst period consisted of 212

players and 2674 observations and the second consisted of 229 players and 2922

observations. For pitchers, there were 65 players and 807 observations in the ﬁrst

period and 79 players and 1002 observations in the second. The ﬁrst equation for

each of the three performance measures in Table 1 was tested. A χ

test was made

of the hypothesis that the coefﬁcients are the same in the two periods. There are 3

degrees of freedom, since 6 age coefﬁcients are estimated instead of 3. The critical

value is 7.83 at the 95 percent conﬁdence level and 11.34 at the 99 percent level.

For OBP the χ

value is 1.72, and so the stability hypothesis is not rejected.

For OPS the results are somewhat sensitive to whether Barry Bonds and Mark

McGwire are included. With the two included the χ

value is 12.72, and so the

stability hypothesis is rejected at the 99 percent level. When the two are not

included, the χ

value is 11.13, and so the stability hypothesis is rejected at the

95 percent level but not the 99 percent level. For ERA the χ

value is 17.15, a

rejection at the 99 percent level.

The results are thus mixed, especially considering that the ERA results are

less reliable because of the smaller sample sizes. It is the case, however, that the

estimates using the second period only imply lower decline rates than those in

Table 4 for all three measures of performance. For OPS the percent loss at age

40 is 6.5 percent instead of 9.8 percent. For OBP the loss is 4.0 percent instead

of 5.6 percent. For ERA the loss is 12.9 percent instead of 14.9 percent. The

4.0 percent ﬁgure for OBP is now close to the ﬁgures for Run and High Jump in

Table 4: 4.1 percent and 4.5 percent.

If the decline rates in baseball are now smaller than they used to be, this could

simply be due to technical progress mentioned above. If, for example, curve A

in Figure 2 is shifted to the right from the peak-performance age, the cumulative

decline at any given age will be smaller. This may be all that is going on. However,

if, as discussed in Section 6, players have the option of choosing different age-

performance proﬁles, an interesting question is whether the elimination of the

reserve clause has led them, other things being equal, to choose a proﬁle with a

smaller decline rate? Quirk and Fort (1992, pp. 235–239) show that the salary

distribution in baseball has gotten more unequal with the elimination of the reserve

clause. This, however, works in the wrong direction regarding decline rates. If the

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relative reward to doing well has increased, this should, other things being equal,

lead to players choosing curve A over curve B in Figure 2, since curve A has more

years of very high performance than does curve B. So it is unclear whether the

elimination of the reserve clause has anything to do with a fall in the decline rate.

A related question is why teams moved in the more recent period to a ﬁve-man

pitching rotation from a four-man rotation, thus possibly decreasing the decline

rate for pitchers. Has this something to do with the change in structure in the

1970s? These are left as open questions. The main result here is that there is some

evidence of slightly smaller decline rates in the second half of the 84-year period,

but the rates are still generally larger than those for the other events.

9 Ranking of Players

As noted in the Introduction, the regressions can be used to rank players on the

basis of the size of the estimated dummy variable coefﬁcients. Each player has his

own estimated constant term. The 441 batters are ranked in Table A.1, and the 144

pitchers are ranked in Table A.2. Remember that a player is in the sample if he has

played 10 or more full-time years between 1921 and 2004, where “full time” is

deﬁned as 100 or more games per year for batters and 450 or more outs for pitchers.

In Table A.1 batters are ranked by the size of the player constant terms in the basic

OPS regression—OPS line 1 in Table 1. The constant terms are denoted “CNST.”

Each player’s lifetime OPS is also presented for comparison purposes along with

his ranking using this measure. Table A.1 also presents the player constant terms

in the basic OBP regression—OBP line 1 in Table 1—and each player’s lifetime

OBP. In Table A.2 pitchers are ranked by the size of the player constant terms in

the basic ERA regression—ERA line 1 in Table 1. Each player’s lifetime ERA is

also presented for comparison purposes along with his ranking using this measure.

A number of caveats are in order before discussing these tables. Baseball

aﬁcionados have strong feelings about who is better than whom, and it is important

to be clear on what criterion is being used in the present ranking. First, what counts

in the present ranking is the performance of a player in his full-time years, not all

years. (The lifetime values also presented in the tables are for all years, not just

full-time years.) Second, the present ranking adjusts for age effects. A player’s

dummy variable coefﬁcient determines the position of his graph in Figure 1, and

the present ranking is simply a ranking by the height of the player’s graph in this

ﬁgure. Lifetime values do not account for possible differences in ages played.

The present ranking thus answers the following question: How good was player i

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age corrected when he played full time? The population consists of players who

played full time for 10 or more years between 1921 and 2004.

A useful way to think about the present ranking is to consider when a player will

be ranked higher in the present ranking than in the lifetime ranking. One possibility

is that his performance when he played part time was on average worse than when

he played full time, possibly because he was injured. The present ranking does not

use part time performances, but lifetime values do. Another possibility, focusing

only on full-time years,is that he played full time much longer than average and thus

played more years beyond the peak-performance age. The present ranking adjusts

for this, but lifetime values do not. Therefore, whether one likes the present ranking

depends on the question he or she is interested in. If one feels that performances

during part-time years should count, the present ranking is not relevant. Also, of

course, if one does not want to adjust for age differences, the present ranking is

not relevant.

As a ﬁnal point before turning to the rankings, issues like ball park differences

and the designated hitter rule in the American League are more important potential

problems in the ranking of players than they are in the estimation of aging effects

in Table 1. Consider a pitcher who pitched his entire life in the American League

under the designated hitter rule. If because of this he had on average larger ERAs

than he would have had in the National League, this does not matter in the estima-

tion of aging effects. It just means that his constant term is larger than otherwise.

The assumption upon which the estimation is based is that aging effects are the

same between the two leagues, not that the players’ constant terms are. However,

in ranking players by the size of their constant terms, it does matter if the desig-

nated hitter rule leads to larger ERAs in the American League, since the estimated

constant terms are affected by this. Likewise, if a batter played in a hitter-friendly

ball park his entire career, this will affect his constant term but not the estimated

aging coefﬁcients. It should thus be kept in mind that the present ranking does not

take into account issues like ball park differences and the designated hitter rule and

this may be important in some cases.

Turning now to Table A.1, for OPS the ranking is Babe Ruth 1 and Ted Williams

2 using both CNST and Lifetime. The order is reversed using OBP. A real winner

in the table is Henry Heilmann, who ranks 8 using CNST for both OPS and OBP.

The Lifetime rankings, however, are 25 and 16, respectively. Heilmann played

14 full-time years, 4 of them before 1921. It turns out that he did noticeably

better beginning in 1921 (the live ball?). He is thus ranked higher using CNST

than Lifetime since CNST counts only performances from 1921 on. Apparently

he was a very nice person, possessing “many virtues, including loyalty, kindness,

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tolerance and generosity.”

Most of the large differences between the CNST and Lifetime rankings can

be traced to the length of the player’s career. For example, for OPS Ralph Kiner

is ranked 19 using Lifetime but only 27 using CNST. Kiner played exactly 10

years (all full time), ages 24-33, which is below average regarding the number of

years played beyond the peak-performance age (27.59 for OPS). Thus his lifetime

performance is more impressive than his performance age corrected. On the other

side, for OPS Carl Yastrzems ki is ranked 75 using CNST but only 99 using Lifetime.

Yastrzemski played 23 years, ages 22-44, all but age 42 full time, which is way

above average regarding the number of years played both before and after the

peak-performance age. Remember, however, that not all the differences between

the CNST and Lifetime rankings are due to length-of-career differences. Some

are due to the different treatments of part-time and full-time performances, where

Lifetime counts part-time years and CNST does not.

There are large differences between the OPS rankings and the OBP rankings

for both CNST and Lifetime. Using CNST, Manny Ramirez is 7 OPS and 15 OBP,

Mark McGwire is 11 OPS and 41 OBP, Willy Mays 19 OPS and 56 OBP, Ken

Griffey Jr. 20 OPS and 72 OBP, Hank Aaron 22 OPS and 87 OBP, Albert Belle

25 OPS and 121 OBP, and so on. On the other side, Edgar Martinez is 9 OBP and

17 OPS, Mickey Cochrane is 13 OBP and 45 OPS, Jackie Robinson is 23 OBP

and 60 OPS, Arky Va ug han is 18 OBP and 67 OPS, Wade Boggs is 16 OBP and

82 OPS, and so on. Within OBP, the differences between CNST and Lifetime are

similar to those within OPS.

Pitchers are ranked in Table A.2. Similar considerations apply here as applied

for batters. Whitey Ford ranks ﬁrst in both rankings. Mike Cuellar ranks 5 using

CNST but 14 using Lifetime. Cuellar played 10 full-time years, ages 29-38, which

is above average regarding the number of years played after the peak-performance

age (26.54 for ERA). Thus, age corrected (i.e., using CNST), he looks better.

Even more extreme is Phil Niekro, who ranks 10 CNST and 48 Lifetime. Niekro

pitched 24 years, ages 25-48, with all but ages 25, 26, 27, 42, and 48 being full time.

This is way above average regarding the number of years played after the peak-

performance age, and so age correcting his performance makes a big difference.

On the other side, Juan Marichal ranks 4 Lifetime but only 11 CNST. Marichal

played 13 full-time years, ages 24-36, which is somewhat below average regarding

the number of years played after the peak-performance age. Hal Newhouser ranks

9 Lifetime but only 18 CNST. He played 11 full-time years, ages 20-31 except for

Ira Smith, Baseball’s Famous Outﬁelders, as quoted in James (2001), p. 798.

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age 30. Another noticeable case is Steve Rogers, who ranks 17 Lifetime but only

46 CNST. He played 11 full-time years, ages 25-35. (Sandy Koufax is not in the

rankings because he played only 9 full-time years.)

Hopefully the rankings in Tables A.1 and A.2 will serve as food for thought

for baseball fans.

10 Conclusion

The estimated aging effects in Table 1 are based on the sample of players who played

10 or more full-time years in the major leagues between 1921 and 2004. The peak-

performance age is around 28 for batters and 26 for pitchers. The (percentage)

rates of decline after the peak-performance age are greater for pitchers than for

batters and greater for OPS than for OBP. Overall, the estimated rates of decline are

modest, although even a small decline in a highly competitive sport like baseball

can be important. Table 4 shows that the losses in baseball are larger than the

losses in track and ﬁeld, running, and swimming events and considerably larger

than the losses in chess. The results reported in Section 8 suggest that decline rates

in baseball may have decreased slightly in the more recent period. The results in

Section 7 show that there are 18 batters whose performances in the second half of

their careers noticeably exceed what the model predicts they should have been. All

but 3 of these players played from 1990 on. It is not possible from the data used

in this study to determine whether any of these performances are due to illegal

drug use. From the perspective of evaluating the model used in this paper it is

encouraging that there are only 18 batters out of 441 who deviate noticeably from

the model’s predictions.

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Table A.1

Ranking of Batters

OPS OBP

Full time & Full time &

age corrected Lifetime age corrected Lifetime

Rank CNST Rank OPS Rank CNST Rank OPS

Babe Ruth 1 0.822 1 1.164 2 0.368 2 0.474

Ted Williams 2 0.756 2 1.115 1 0.371 1 0.482

Rogers Hornsby 3 0.718 6 1.010 3

0.341 5 0.434

Lou Gehrig 4 0.706 3 1.080 4 0.332 3 0.447

Barry Bonds 5 0.699 4 1.053 5

0.329 4 0.443

Jimmie Foxx 6 0.668 5 1.038 7 0.315 7 0.428

Manny Ramirez 7 0.649 7 1.010 15 0.301 14 0.411

Harry Heilmann 8 0.638 25 0.930 6

0.321 16 0.409

Frank Thomas 9 0.628 8 0.996 8 0.314 6 0.429

Jim Thome 10 0.626 10 0.979 14

0.301 15 0.410

Mark McGwire 11 0.615 9 0.982 41

0.279 36 0.394

Mickey Mantle 12 0.613 11 0.977 10 0.309 8 0.420

Stan Musial 13 0.612 13 0.976 12

0.302 11 0.417

Joe DiMaggio 14 0.606 12 0.977 30

0.283 29 0.398

Larry Walker 15 0.602 14 0.969 26

0.286 24 0.401

Mel Ott 16 0.598 17 0.947 11 0.308 13 0.414

Edgar Martinez 17 0.595 24 0.933 9

0.311 10 0.418

Johnny Mize 18 0.590 15 0.959 27

0.286 32 0.397

Willie Mays 19 0.587 20 0.941 56

0.274 62 0.384

Ken Griffey Jr. 20 0.584 22 0.937 72

0.269 85 0.377

Jeff Bagwell 21 0.582 16 0.951 21

0.293 18 0.408

Hank Aaron 22 0.578 26 0.928 87

0.265 100 0.374

Gary Shefﬁeld 23 0.577 28 0.928 20

0.293 26 0.400

Mike Piazza 24 0.576 18 0.947 69 0.269 59 0.386

Albert Belle 25 0.570 23 0.933 121

0.256 121 0.369

Frank Robinson 26 0.561 29 0.926 45

0.277 48 0.389

Ralph Kiner 27 0.559 19 0.946 42

0.279 31 0.398

Earl Averill 28 0.558 27 0.928 40

0.279 35 0.395

Chipper Jones 29 0.557 21 0.937 32 0.283 25 0.401

Duke Snider 30 0.553 31 0.919 81 0.266 75 0.380

Al Simmons 31 0.551 33 0.915 76 0.267 74 0.380

Dick Allen 32 0.545 34 0.912 84 0.266 79 0.378

Mike Schmidt 33 0.543 35 0.907 79 0.267 72 0.380

Juan Gonzalez 34 0.542 37 0.904 240 0.234 268 0.343

Bob Johnson 35 0.539 38 0.899 38 0.280 40 0.393

Bill Terry 36 0.538 39 0.899 34 0.281 41 0.393

Mo Vau ghn 37 0.532 36 0.906 86 0.265 68 0.383

Chuck Klein 38 0.530 30 0.922 113 0.259 77 0.379

Fred McGriff 39 0.529 48 0.886 85 0.266 86 0.377

Willie McCovey 40 0.528 42 0.889 92 0.263 97 0.374

Babe Herman 41 0.528 32 0.915 91 0.263 67 0.383

Rafael Palmeiro 42 0.528 43 0.889 104

0.260 106 0.372

Tim Salmon 43 0.524 47 0.886 55 0.274 55 0.386

Goose Goslin 44 0.523 46 0.887 51 0.274 53 0.387

Mickey Cochrane 45 0.521 40 0.897 13 0.302 9 0.419

Sammy Sosa 46 0.518 41 0.892 272 0.229 242 0.348

Willie Stargell 47 0.518 44 0.889 190 0.243 169 0.360

Ellis Burks 48 0.518 60 0.874 143

0.252 146 0.363

Moises Alou 49 0.517 54 0.880 135 0.253 132 0.367

Eddie Mathews 50 0.515 49 0.885 98 0.262 90 0.376

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Table A.1 (continued)

Ranking of Batters

OPS OBP

Full time & Full time &

age corrected Lifetime age corrected Lifetime

Rank CNST Rank OPS Rank CNST Rank OPS

Harmon Killebrew 51 0.514 50 0.884 102 0.261 92 0.376

Darryl Strawberry 52 0.513 68 0.862 164 0.247 197 0.356

Bernie Williams 53 0.513 59 0.875 53

0.274 52 0.388

Charlie Gehringer 54 0.510 51 0.884 25 0.287 21 0.404

Ryan Klesko 55 0.509 45 0.888 117

0.258 104 0.373

Paul Waner 56 0.508 57 0.878 24 0.289 20 0.404

Will Clark 57 0.508 53 0.880 66 0.270 63 0.384

Larry Doby 58 0.508 58 0.876 58

0.273 57 0.386

Gabby Hartnett 59 0.507 72 0.858 108 0.259 120 0.370

Jackie Robinson 60 0.505 52 0.883 23

0.291 17 0.409

Jack Clark 61 0.505 80 0.854 61

0.271 78 0.379

David Justice 62 0.503 56 0.878 97 0.262 84 0.378

Al Kaline 63 0.502 78 0.855 75

0.267 93 0.376

George Brett 64 0.501 76 0.857 109

0.259 122 0.369

Joe Cronin 65 0.501 75 0.857 39

0.279 46 0.390

Jose Canseco 66 0.500 63 0.867 223 0.238 216 0.353

Arky Vaug ha n 67 0.499 70 0.859 18

0.295 19 0.406

Jeff Heath 68 0.499 55 0.879 131

0.254 117 0.370

Norm Cash 69 0.498 67 0.862 112

0.259 99 0.374

Bill Dickey 70 0.497 62 0.868 89

0.265 70 0.382

Joe Medwick 71 0.496 64 0.867 162

0.248 153 0.362

Jim Bottomley 72 0.495 61 0.870 129

0.254 123 0.369

George Grantham 73 0.495 82 0.854 29

0.284 42 0.392

Heinie Manush 74 0.494 77 0.856 83 0.266 88 0.377

Carl Ya str ze m sk i 75 0.493 99 0.842 63

0.270 76 0.380

Kiki Cuyler 76 0.491 69 0.860 67

0.269 54 0.386

Minnie Minoso 77 0.491 88 0.848 48

0.276 49 0.389

Andres Galarraga 78 0.490 93 0.846 226

0.237 247 0.347

Tony Gwynn 79 0.490 91 0.847 46 0.277 50 0.388

Orlando Cepeda 80 0.490 87 0.849 208 0.240 233 0.350

John Olerud 81 0.488 65 0.864 33 0.282 28 0.399

Wade Boggs 82 0.488 73 0.858 16 0.298 12 0.415

Reggie Jackson 83 0.488 95 0.846 181 0.244 202 0.356

Reggie Smith 84 0.487 79 0.855 139 0.252 134 0.366

Shawn Green 85 0.486 66 0.864 213 0.240 196 0.357

Rudy Yo rk 86 0.483 96 0.846 161 0.248 154 0.362

Jim Rice 87 0.482 81 0.854 221 0.238 224 0.352

Billy Williams 88 0.480 83 0.853 165 0.247 160 0.361

Enos Slaughter 89 0.479 107 0.835 62 0.271 71 0.382

Kent Hrbek 90 0.479 90 0.848 133 0.253 131 0.367

Fred Lynn 91 0.479 97 0.845 178 0.245 166 0.360

Eddie Murray 92 0.479 105 0.836 155

0.249 175 0.359

Rico Carty 93 0.478 110 0.833 120 0.257 124 0.369

Sid Gordon 94 0.476 98 0.843 95 0.263 89 0.377

Luis Gonzalez 95 0.476 71 0.859 147 0.250 119 0.370

Rickey Henderson 96 0.476 135 0.820 19 0.295 23 0.401

Dave Winﬁeld 97 0.476 120 0.827 192 0.243 218 0.353

Jeff Kent 98 0.474 74 0.858 251

0.233 222 0.352

Rocky Colavito 99 0.473 89 0.848 186 0.243 177 0.359

Sam Rice 100 0.473 183 0.801 71 0.269 101 0.374

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Table A.1 (continued)

Ranking of Batters

OPS OBP

Full time & Full time &

age corrected Lifetime age corrected Lifetime

Rank CNST Rank OPS Rank CNST Rank OPS

Ted Kluszewski 101 0.472 86 0.850 234 0.234 217 0.353

Ray Lankford 102 0.472 100 0.841 152 0.249 142 0.364

Gene Woodling 103 0.472 142 0.817 44

0.278 56 0.386

Dwight Evans 104 0.470 102 0.840 122 0.256 118 0.370

Roy Sievers 105 0.470 119 0.829 198

0.242 211 0.354

Harold Baines 106 0.470 133 0.820 154 0.249 201 0.356

Tony Lazzeri 107 0.470 92 0.846 93 0.263 73 0.380

Frank Howard 108 0.470 85 0.851 235

0.234 219 0.352

Paul Molitor 109 0.469 143 0.817 110 0.259 126 0.369

Bobby Bonds 110 0.469 123 0.824 195

0.242 214 0.353

Paul O’Neill 111 0.467 111 0.833 158

0.249 148 0.363

Roberto Clemente 112 0.467 108 0.834 176 0.245 176 0.359

Greg Luzinski 113 0.466 101 0.840 166

0.247 149 0.363

Bobby Doerr 114 0.466 126 0.823 132

0.253 159 0.362

Bob Meusel 115 0.464 84 0.852 228

0.236 199 0.356

Vic Wertz 116 0.463 109 0.833 142 0.252 141 0.364

Dante Bichette 117 0.463 106 0.835 320

0.219 309 0.336

Keith Hernandez 118 0.463 132 0.821 47

0.276 61 0.384

Andre Thornton 119 0.462 154 0.811 140

0.252 172 0.360

Joe Morgan 120 0.462 137 0.819 35

0.281 43 0.392

Ben Chapman 121 0.462 125 0.823 57

0.273 66 0.383

Kirby Puckett 122 0.461 104 0.837 183

0.244 171 0.360

Gil Hodges 123 0.461 94 0.846 220

0.239 181 0.359

Ernie Banks 124 0.461 116 0.830 341 0.215 337 0.330

Reggie Sanders 125 0.461 112 0.832 282

0.228 260 0.344

Ivan Rodriguez 126 0.461 103 0.837 245

0.233 246 0.347

Boog Powell 127 0.460 128 0.822 153

0.249 165 0.360

Yog i Berra 128 0.460 114 0.830 244

0.234 241 0.348

Rod Carew 129 0.458 129 0.822 36 0.280 39 0.393

Bing Miller 130 0.458 134 0.820 172 0.245 180 0.359

Mark Grace 131 0.457 122 0.825 70 0.269 65 0.383

Joe Judge 132 0.457 189 0.798 65 0.270 80 0.378

Ron Santo 133 0.457 121 0.826 151 0.250 151 0.362

Carlton Fisk 134 0.456 191 0.797 266 0.230 287 0.341

Bobby Bonilla 135 0.456 118 0.829 184 0.244 189 0.358

Tony Oliva 136 0.455 117 0.830 225 0.238 215 0.353

George Foster 137 0.454 138 0.818 283 0.228 295 0.339

Dixie Walker 138 0.454 136 0.820 64 0.270 69 0.383

Roberto Alomar 139 0.453 150 0.814 100 0.261 111 0.371

Barry Larkin 140 0.452 147 0.815 124 0.255 113 0.370

Luke Appling 141 0.451 188 0.798 22 0.292 27 0.399

Tony Perez 142 0.451 170 0.804 277

0.228 286 0.341

Harlond Clift 143 0.451 113 0.831 50 0.275 45 0.390

Ver n Stephens 144 0.450 146 0.815 203 0.241 204 0.355

Chili Davis 145 0.450 155 0.811 160 0.248 173 0.360

Don Mattingly 146 0.450 115 0.830 216 0.239 186 0.358

Dave Parker 147 0.449 159 0.810 287 0.227 294 0.339

Ron Gant 148 0.449 173 0.803 288

0.226 312 0.336

Tino Martinez 149 0.449 141 0.817 257 0.232 256 0.345

Julio Franco 150 0.449 228 0.785 101 0.261 133 0.366

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Table A.1 (continued)

Ranking of Batters

OPS OBP

Full time & Full time &

age corrected Lifetime age corrected Lifetime

Rank CNST Rank OPS Rank CNST Rank OPS

Ernie Lombardi 151 0.449 139 0.818 182 0.244 187 0.358

Elmer Val o 152 0.448 213 0.790 17 0.296 30 0.398

Johnny Bench 153 0.448 140 0.818 279

0.228 281 0.342

Joe Adcock 154 0.448 130 0.822 313 0.221 302 0.337

Vinny Castilla 155 0.447 151 0.813 371

0.208 367 0.324

Hal McRae 156 0.447 168 0.805 206 0.240 228 0.351

Bob Watson 157 0.446 156 0.811 136 0.253 144 0.363

Andre Dawson 158 0.446 167 0.805 352

0.212 368 0.323

Ken Singleton 159 0.445 124 0.824 68 0.269 51 0.388

Matt Williams 160 0.444 169 0.805 381

0.205 390 0.317

Frankie Frisch 161 0.444 182 0.801 107

0.259 127 0.369

Dale Murphy 162 0.443 148 0.815 259 0.231 253 0.346

Andy Pafko 163 0.443 185 0.799 215

0.240 232 0.350

Tim Raines 164 0.443 157 0.810 60

0.271 60 0.385

Joe Gordon 165 0.443 127 0.823 212

0.240 192 0.357

Cesar Cedeno 166 0.443 212 0.790 196 0.242 251 0.346

Craig Biggio 167 0.442 160 0.807 106

0.260 102 0.373

Bob Elliott 168 0.441 149 0.815 105

0.260 95 0.375

Joe Torre 169 0.441 144 0.817 159

0.249 139 0.365

Joe Vo s m i k 170 0.440 163 0.807 115

0.258 125 0.369

Del Ennis 171 0.439 153 0.812 289

0.226 292 0.340

Cecil Cooper 172 0.438 175 0.802 303

0.223 305 0.337

Jeff Conine 173 0.438 184 0.799 233

0.235 239 0.348

Carl Furillo 174 0.438 152 0.813 219 0.239 205 0.355

Brian Downing 175 0.438 195 0.796 111

0.259 116 0.370

Jimmy Wynn 176 0.437 181 0.801 130

0.254 136 0.365

Lonnie Smith 177 0.437 207 0.791 90

0.264 110 0.371

Wally Joyner 178 0.436 180 0.802 149

0.250 152 0.362

Pete Rose 179 0.436 230 0.784 80 0.267 94 0.375

Ken Boyer 180 0.435 158 0.810 256 0.232 238 0.349

Mickey Ver non 181 0.435 220 0.787 148 0.250 184 0.359

Rusty Staub 182 0.434 203 0.793 137 0.253 150 0.362

Gary Matthews 183 0.434 177 0.802 145 0.250 143 0.364

Greg Vaug hn 184 0.433 162 0.807 322 0.219 303 0.337

Rick Monday 185 0.432 172 0.804 170 0.246 162 0.361

Bobby Murcer 186 0.432 179 0.802 179 0.245 190 0.357

Bobby Grich 187 0.432 200 0.794 118 0.258 114 0.370

Phil Cavarretta 188 0.431 217 0.788 94 0.263 105 0.372

Brady Anderson 189 0.431 219 0.787 144 0.251 155 0.362

Darrell Evans 190 0.431 204 0.792 163 0.247 161 0.361

Dom DiMaggio 191 0.431 178 0.802 74 0.268 64 0.383

Al Oliver 192 0.430 199 0.795 254

0.232 262 0.344

Bobby Higginson 193 0.430 145 0.816 207 0.240 179 0.359

Kenny Lofton 194 0.430 194 0.797 116 0.258 108 0.372

Richie Hebner 195 0.430 210 0.790 205 0.241 221 0.352

Garret Anderson 196 0.430 165 0.806 345 0.214 344 0.329

Robin Ve ntu ra 197 0.429 166 0.806 174 0.245 156 0.362

Pie Traynor 198 0.429 192 0.797 157

0.249 157 0.362

Roger Maris 199 0.429 131 0.822 286 0.227 255 0.345

Sam West 200 0.428 196 0.796 127 0.255 112 0.371

Journal of Quantitative Analysis in Sports, Vol. 4 [2008], Iss. 1, Art. 1

DOI: 10.2202/1559-0410.1074

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Table A.1 (continued)

Ranking of Batters

OPS OBP

Full time & Full time &

age corrected Lifetime age corrected Lifetime

Rank CNST Rank OPS Rank CNST Rank OPS

Buddy Myer 201 0.427 198 0.795 52 0.274 47 0.389

Ryne Sandberg 202 0.426 197 0.795 269 0.230 264 0.344

Steve Finley 203 0.426 224 0.787 302

0.224 304 0.337

Joe Sewell 204 0.425 171 0.804 54 0.274 44 0.391

Pinky Higgins 205 0.425 187 0.798 126

0.255 115 0.370

Bill Madlock 206 0.425 161 0.807 171 0.246 137 0.365

George Hendrick 207 0.425 246 0.775 312 0.221 343 0.329

Bill Skowron 208 0.424 206 0.792 319

0.219 327 0.332

Bill White 209 0.424 164 0.806 247 0.233 229 0.351

J.T. Snow 210 0.424 208 0.791 168

0.246 185 0.358

Chet Lemon 211 0.423 193 0.797 210

0.240 206 0.355

Ken Griffey Sr. 212 0.423 211 0.790 191 0.243 183 0.359

Stan Hack 213 0.423 209 0.791 43

0.278 37 0.394

Earl Torgeson 214 0.422 176 0.802 73

0.268 58 0.386

Ken Caminiti 215 0.421 202 0.793 274

0.229 250 0.347

Ted Simmons 216 0.421 227 0.785 248 0.233 240 0.348

Ron Cey 217 0.421 186 0.798 232

0.235 212 0.354

Hank Bauer 218 0.421 229 0.785 250

0.233 254 0.346

Willie Horton 219 0.421 216 0.789 327

0.218 325 0.332

Lu Blue 220 0.421 174 0.803 31

0.283 22 0.402

Ben Oglivie 221 0.420 225 0.786 306

0.222 310 0.336

Cal Ripken Jr. 222 0.420 218 0.788 284

0.228 291 0.340

Jimmie Dykes 223 0.419 271 0.764 114

0.258 138 0.365

Ron Fairly 224 0.419 262 0.768 134 0.253 168 0.360

Charlie Jamieson 225 0.418 276 0.763 59

0.272 83 0.378

Marty McManus 226 0.418 221 0.787 173

0.245 191 0.357

Don Baylor 227 0.417 242 0.777 273

0.229 280 0.342

Gary Carter 228 0.417 251 0.773 300

0.224 313 0.335

Chuck Knoblauch 229 0.416 231 0.783 88 0.265 82 0.378

Travis Jackson 230 0.415 257 0.770 276 0.228 306 0.337

Lou Whitaker 231 0.415 215 0.789 146 0.250 147 0.363

Al Smith 232 0.415 222 0.787 201 0.241 188 0.358

George Kell 233 0.414 232 0.781 128 0.254 129 0.367

Bobby Thomson 234 0.413 201 0.794 348 0.214 329 0.332

Robin Yo unt 235 0.413 255 0.772 253 0.232 279 0.342

Andy Van Slyke 236 0.413 205 0.792 242 0.234 235 0.349

George McQuinn 237 0.413 233 0.781 187 0.243 193 0.357

Edgardo Alfonzo 238 0.412 190 0.797 197 0.242 158 0.362

Wally Moses 239 0.412 236 0.779 156 0.249 145 0.363

Travis Fryman 240 0.411 237 0.779 305 0.222 311 0.336

Steve Garvey 241 0.411 243 0.775 334 0.216 340 0.329

Dusty Baker 242 0.411 238 0.779 231

0.235 248 0.347

Amos Otis 243 0.411 261 0.768 243 0.234 270 0.343

Alan Trammell 244 0.410 264 0.767 188 0.243 226 0.351

Ray Durham 245 0.410 214 0.789 229 0.236 210 0.354

Richie Ashburn 246 0.409 240 0.778 28 0.284 33 0.396

Sam Chapman 247 0.408 234 0.780 275 0.229 277 0.343

Eric Karros 248 0.406 239 0.779 364

0.210 364 0.325

Mike Hargrove 249 0.405 223 0.787 49 0.275 34 0.396

George Bell 250 0.405 226 0.785 405 0.198 395 0.316

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Table A.1 (continued)

Ranking of Batters

OPS OBP

Full time & Full time &

age corrected Lifetime age corrected Lifetime

Rank CNST Rank OPS Rank CNST Rank OPS

Ruben Sierra 251 0.405 260 0.769 379 0.206 391 0.317

Johnny Callison 252 0.404 253 0.772 317 0.220 333 0.331

Jose Cruz 253 0.404 247 0.774 227

0.236 207 0.354

Ken Keltner 254 0.404 241 0.778 308 0.222 298 0.338

Frank Thomas 255 0.403 249 0.774 8

0.314 6 0.429

Joe Carter 256 0.403 256 0.771 420 0.192 421 0.306

Dave Kingman 257 0.403 235 0.779 431 0.185 428 0.302

Tony Phillips 258 0.401 274 0.763 103

0.260 98 0.374

Jay Bell 259 0.401 282 0.759 255 0.232 266 0.343

Billy Herman 260 0.401 250 0.774 138

0.252 130 0.367

Joe Kuhel 261 0.400 270 0.765 175

0.245 182 0.359

Kevin McReynolds 262 0.400 245 0.775 355 0.212 346 0.328

Brett Butler 263 0.399 294 0.753 82

0.266 87 0.377

Doug DeCinces 264 0.398 248 0.774 351

0.213 342 0.329

Eddie Yos t 265 0.398 268 0.765 37

0.280 38 0.394

Graig Nettles 266 0.398 301 0.750 324 0.219 341 0.329

Gregg Jefferies 267 0.398 267 0.765 237

0.234 261 0.344

Todd Zeile 268 0.398 258 0.769 271

0.229 252 0.346

Bret Boone 269 0.397 252 0.773 354

0.212 355 0.327

Lee May 270 0.397 254 0.772 411

0.196 405 0.313

Roy White 271 0.396 272 0.764 169

0.246 170 0.360

Dan Driessen 272 0.395 265 0.767 211

0.240 200 0.356

Carlos Baerga 273 0.395 286 0.757 310

0.221 328 0.332

Tom Brunansky 274 0.395 278 0.761 332 0.216 351 0.327

Sal Bando 275 0.395 280 0.760 214

0.240 220 0.352

Claudell Washington 276 0.395 316 0.745 321

0.219 362 0.325

Harvey Kuenn 277 0.393 269 0.765 200

0.241 195 0.357

Sherm Lollar 278 0.393 283 0.759 193

0.243 194 0.357

Dave Henderson 279 0.392 289 0.756 378 0.206 379 0.320

Lou Brock 280 0.392 296 0.753 264 0.230 275 0.343

Vad a Pinson 281 0.391 259 0.769 363 0.210 353 0.327

Darrell Porter 282 0.391 275 0.763 202 0.241 209 0.354

Jim Eisenreich 283 0.391 312 0.746 239 0.234 283 0.342

Gil McDougald 284 0.390 266 0.766 209 0.240 198 0.356

Larry Parrish 285 0.390 285 0.757 382 0.205 385 0.318

Rick Ferrell 286 0.389 322 0.741 77 0.267 81 0.378

Toby Harrah 287 0.388 281 0.760 150 0.250 140 0.365

Carney Lansford 288 0.387 292 0.753 236 0.234 273 0.343

Gus Bell 289 0.387 244 0.775 357 0.211 336 0.330

Lance Parrish 290 0.386 293 0.753 396 0.200 404 0.313

Buddy Bell 291 0.385 306 0.747 267 0.230 289 0.341

Lou Piniella 292 0.385 320 0.741 311

0.221 324 0.333

Billy Goodman 293 0.385 290 0.754 96 0.263 91 0.376

Eric You ng 294 0.385 297 0.753 167 0.247 163 0.361

Charlie Grimm 295 0.385 325 0.738 246 0.233 285 0.341

Bob Bailey 296 0.384 300 0.750 230 0.236 244 0.347

George Scott 297 0.384 263 0.767 344 0.214 321 0.333

Tony Gonzalez 298 0.383 273 0.763 260

0.231 231 0.350

Jorge Orta 299 0.383 308 0.746 307 0.222 315 0.334

Bruce Bochte 300 0.382 287 0.756 180 0.245 167 0.360

Journal of Quantitative Analysis in Sports, Vol. 4 [2008], Iss. 1, Art. 1

DOI: 10.2202/1559-0410.1074

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Table A.1 (continued)

Ranking of Batters

OPS OBP

Full time & Full time &

age corrected Lifetime age corrected Lifetime

Rank CNST Rank OPS Rank CNST Rank OPS

B.J. Surhoff 301 0.380 302 0.749 318 0.219 317 0.334

Felipe Alou 302 0.380 279 0.760 368 0.208 348 0.328

Tony Fernandez 303 0.379 309 0.746 241

0.234 245 0.347

Willie Kamm 304 0.379 288 0.756 123 0.256 109 0.372

Gary Gaetti 305 0.379 321 0.741 415

0.194 417 0.308

Dave Martinez 306 0.378 339 0.730 252 0.232 284 0.341

Gee Walker 307 0.378 277 0.761 360 0.211 334 0.331

Hector Lopez 308 0.378 317 0.745 331

0.216 339 0.330

Rico Petrocelli 309 0.378 298 0.752 333 0.216 326 0.332

Pinky Whitney 310 0.377 284 0.758 290

0.226 274 0.343

Pee Wee Reese 311 0.377 319 0.743 141

0.252 135 0.366

Dick Bartell 312 0.377 307 0.747 217 0.239 203 0.355

Greg Gross 313 0.376 351 0.723 78

0.267 107 0.372

Bill Freehan 314 0.376 299 0.752 299

0.224 293 0.340

Pete Runnels 315 0.376 295 0.753 119

0.257 96 0.374

Willie Jones 316 0.375 291 0.753 296 0.225 269 0.343

Jerry Mumphrey 317 0.374 315 0.745 238

0.234 237 0.349

Lloyd Waner 318 0.374 305 0.747 224

0.238 213 0.353

Tony Cuccinello 319 0.373 327 0.737 265

0.230 271 0.343

Devon White 320 0.372 324 0.739 384

0.204 380 0.320

Bill Buckner 321 0.372 343 0.729 356

0.212 375 0.321

Delino DeShields 322 0.372 341 0.729 189

0.243 223 0.352

Elston Howard 323 0.372 304 0.749 391

0.202 374 0.322

Terry Steinbach 324 0.371 311 0.746 362 0.210 359 0.326

Chris Chambliss 325 0.371 303 0.749 326

0.218 316 0.334

Curt Flood 326 0.371 337 0.732 258

0.231 278 0.342

Alvin Dark 327 0.370 318 0.744 329

0.217 322 0.333

Dick McAuliffe 328 0.370 310 0.746 285

0.227 272 0.343

Lloyd Moseby 329 0.369 313 0.746 335 0.216 332 0.332

Tommy Davis 330 0.369 333 0.734 342 0.215 345 0.329

Roy Smalley 331 0.367 323 0.740 280 0.228 258 0.345

Marquis Grissom 332 0.366 329 0.736 392 0.201 384 0.319

Jim Fregosi 333 0.365 328 0.736 295 0.225 297 0.338

Davey Lopes 334 0.364 326 0.737 263 0.231 236 0.349

Jose Offerman 335 0.364 330 0.734 177 0.245 164 0.361

Willie McGee 336 0.363 342 0.729 314 0.220 319 0.333

Willie Randolph 337 0.363 350 0.724 99 0.262 103 0.373

Tim Wallach 338 0.362 336 0.732 395 0.200 397 0.316

Garry Maddox 339 0.361 334 0.733 386 0.203 378 0.320

Brooks Robinson 340 0.360 353 0.723 358 0.211 373 0.322

Don Money 341 0.360 331 0.734 347 0.214 349 0.328

Al Bumbry 342 0.359 356 0.721 261

0.231 267 0.343

Lonny Frey 343 0.359 332 0.734 194 0.243 178 0.359

Pete O’Brien 344 0.359 314 0.745 330 0.216 307 0.336

Bill Bruton 345 0.358 357 0.720 336 0.216 347 0.328

Willie Montanez 346 0.358 340 0.729 349 0.213 352 0.327

Doc Cramer 347 0.357 360 0.716 278 0.228 290 0.340

Deron Johnson 348 0.356 338 0.731 416

0.194 411 0.311

Willie Davis 349 0.356 352 0.723 404 0.198 408 0.311

Benito Santiago 350 0.354 354 0.722 414 0.194 419 0.307

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Table A.1 (continued)

Ranking of Batters

OPS OBP

Full time & Full time &

age corrected Lifetime age corrected Lifetime

Rank CNST Rank OPS Rank CNST Rank OPS

Whitey Lockman 351 0.354 335 0.732 292 0.226 282 0.342

Red Schoendienst 352 0.353 349 0.724 304 0.222 300 0.337

Jose Cardenal 353 0.352 344 0.728 340

0.216 323 0.333

Bill Doran 354 0.352 345 0.728 222 0.238 208 0.354

Ken Oberkfell 355 0.351 364 0.713 204

0.241 230 0.351

Charlie Hayes 356 0.348 363 0.714 383 0.204 392 0.316

Ed Kranepool 357 0.346 382 0.693 353 0.212 393 0.316

Jim Piersall 358 0.345 358 0.718 346

0.214 331 0.332

Tim McCarver 359 0.345 347 0.725 337 0.216 301 0.337

Phil Garner 360 0.344 366 0.711 375

0.207 370 0.323

Jim Gilliam 361 0.344 361 0.715 185

0.244 174 0.360

Ron Hunt 362 0.344 362 0.715 125 0.255 128 0.368

Al Cowens 363 0.342 355 0.722 388

0.202 383 0.319

Vic Power 364 0.342 348 0.725 413

0.195 400 0.315

Ossie Bluege 365 0.341 370 0.707 218

0.239 225 0.352

Matty Alou 366 0.340 346 0.726 297 0.225 259 0.344

Peanuts Lowrey 367 0.339 376 0.698 298

0.225 308 0.336

Mark McLemore 368 0.338 384 0.690 199

0.241 234 0.349

Nellie Fox 369 0.338 368 0.710 262

0.231 243 0.347

Denis Menke 370 0.336 365 0.713 270

0.230 276 0.343

Tommy Harper 371 0.334 359 0.717 323

0.219 296 0.338

Dave Philley 372 0.333 367 0.710 325

0.218 318 0.334

Phil Rizzuto 373 0.330 372 0.706 249

0.233 227 0.351

Omar Vizquel 374 0.329 375 0.699 291 0.226 288 0.341

Rabbit Maranville 375 0.329 420 0.658 339

0.216 386 0.318

Dick Groat 376 0.328 380 0.696 343

0.215 338 0.330

Terry Pendleton 377 0.327 371 0.707 408

0.197 398 0.316

Stan Javier 378 0.327 369 0.708 293

0.226 257 0.345

Johnny Roseboro 379 0.326 378 0.697 361 0.210 360 0.326

Willie Wilson 380 0.321 373 0.702 369 0.208 357 0.326

Joe Orsulak 381 0.321 377 0.698 370 0.208 366 0.324

Mike Scioscia 382 0.320 374 0.700 294 0.225 263 0.344

Mike Bordick 383 0.320 387 0.685 366 0.209 369 0.323

Frank White 384 0.319 399 0.675 434 0.182 436 0.293

Steve Sax 385 0.319 383 0.692 309 0.221 314 0.335

Bob Boone 386 0.318 416 0.661 376 0.206 399 0.315

Granny Hamner 387 0.318 386 0.686 417 0.192 425 0.303

Bill Virdon 388 0.317 381 0.696 397 0.200 394 0.316

Chris Speier 389 0.317 398 0.676 328 0.217 354 0.327

Paul Blair 390 0.315 390 0.684 422 0.191 426 0.302

Enos Cabell 391 0.313 396 0.677 410 0.196 418 0.308

Royce Clayton 392 0.313 391 0.684 402

0.198 406 0.312

Bob Kennedy 393 0.311 413 0.664 394 0.201 413 0.309

Dave Concepcion 394 0.310 393 0.679 365 0.209 372 0.322

Greg Gagne 395 0.309 389 0.684 429 0.186 427 0.302

Tom Herr 396 0.308 379 0.697 281 0.228 249 0.347

Clete Boyer 397 0.307 407 0.670 428 0.186 432 0.299

Scott Fletcher 398 0.307 402 0.674 316

0.220 330 0.332

Frank Bolling 399 0.307 394 0.679 403 0.198 403 0.313

Maury Wills 400 0.306 415 0.661 315 0.220 335 0.330

Journal of Quantitative Analysis in Sports, Vol. 4 [2008], Iss. 1, Art. 1

DOI: 10.2202/1559-0410.1074

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Table A.1 (continued)

Ranking of Batters

OPS OBP

Full time & Full time &

age corrected Lifetime age corrected Lifetime

Rank CNST Rank OPS Rank CNST Rank OPS

Bill Mazeroski 401 0.303 410 0.667 427 0.187 431 0.299

Leo Cardenas 402 0.302 395 0.678 412 0.195 407 0.311

Jim Gantner 403 0.302 406 0.671 389

0.202 381 0.319

Jim Davenport 404 0.301 388 0.684 399 0.199 387 0.318

Brad Ausmus 405 0.301 392 0.680 367

0.209 361 0.326

Otis Nixon 406 0.301 419 0.658 268 0.230 265 0.344

Jim Sundberg 407 0.300 401 0.674 359 0.211 356 0.327

Ozzie Smith 408 0.299 411 0.666 301

0.224 299 0.338

Russ Snyder 409 0.298 385 0.688 390 0.202 363 0.325

Al Lopez 410 0.298 414 0.663 350

0.213 358 0.326

Lenny Harris 411 0.298 412 0.665 398

0.199 388 0.317

Tommy McCraw 412 0.297 408 0.670 424 0.190 416 0.309

Del Unser 413 0.295 397 0.677 393

0.201 382 0.319

Garry Templeton 414 0.294 403 0.673 426

0.187 423 0.304

Chico Carrasquel 415 0.294 400 0.674 338

0.216 320 0.333

Manny Trillo 416 0.294 417 0.660 387 0.202 396 0.316

Marty Marion 417 0.293 409 0.668 373

0.208 371 0.323

Tony Taylor 418 0.293 405 0.673 385

0.204 376 0.321

Billy Jurges 419 0.291 418 0.660 374

0.207 365 0.325

Tony Pena 420 0.291 404 0.673 423

0.190 415 0.309

Luis Aparicio 421 0.288 423 0.653 407

0.197 410 0.311

Hughie Critz 422 0.287 421 0.656 421

0.191 424 0.303

Derrel Thomas 423 0.287 426 0.649 377

0.206 389 0.317

Bert Campaneris 424 0.286 424 0.653 401 0.199 409 0.311

Bill Russell 425 0.282 427 0.648 406

0.198 412 0.310

Jim Hegan 426 0.281 429 0.639 432

0.184 435 0.295

Julian Javier 427 0.278 425 0.651 435

0.182 434 0.296

Cookie Rojas 428 0.276 428 0.643 418

0.192 422 0.306

Tito Fuentes 429 0.275 422 0.653 425 0.189 420 0.307

Roy McMillan 430 0.262 430 0.635 400 0.199 401 0.314

Ozzie Guillen 431 0.259 434 0.626 437 0.174 437 0.287

Aurelio Rodriguez 432 0.256 433 0.626 441 0.161 441 0.275

Larry Bowa 433 0.252 435 0.620 430 0.185 430 0.300

Freddie Patek 434 0.250 431 0.633 419 0.192 414 0.309

Don Kessinger 435 0.249 432 0.626 409 0.197 402 0.314

Leo Durocher 436 0.244 436 0.619 433 0.184 433 0.298

Alfredo Grifﬁn 437 0.233 438 0.604 439 0.170 438 0.285

Bud Harrelson 438 0.231 437 0.616 372 0.208 350 0.327

Tim Foli 439 0.225 439 0.593 438 0.171 439 0.283

Ed Brinkman 440 0.210 440 0.580 440 0.165 440 0.280

Mark Belanger 441 0.200 441 0.580 436 0.181 429 0.300

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Table A.2

Ranking of Pitchers

ERA

Full time &

age corrected Lifetime

Rank CNST Rank ERA

Whitey Ford 1 4.692 1 2.745

Tom Seaver 2 4.694 3 2.862

Bob Gibson 3 4.696 5 2.915

Jim Palmer 4 4.741 2 2.856

Mike Cuellar 5 4.815 14 3.138

Lefty Grove 6 4.835 10 3.058

Warren Spahn 7 4.873 12 3.086

Gaylord Perry 8 4.874 13 3.105

Greg Maddux 9 4.886 7 2.949

Phil Niekro 10 4.887 48 3.351

Juan Marichal 11 4.897 4 2.889

Carl Hubbell 12 4.906 8 2.978

Randy Johnson 13 4.927 11 3.068

Don Drysdale 14 4.932 6 2.948

Nolan Ryan 15 4.962 20 3.193

Dazzy Va nce 16 4.977 27 3.240

Roger Clemens 17 4.986 18 3.181

Hal Newhouser 18 4.998 9 3.055

Dutch Leonard 19 5.029 29 3.250

Dave McNally 20 5.048 26 3.237

Luis Tiant 21 5.051 40 3.304

Tommy John 22 5.055 45 3.342

Catﬁsh Hunter 23 5.065 31 3.256

Don Sutton 24 5.092 32 3.261

Steve Carlton 25 5.098 23 3.215

Jim Bunning 26 5.100 35 3.269

Curt Schilling 27 5.107 43 3.325

Dolf Luque 28 5.109 28 3.245

Curt Davis 29 5.123 57 3.422

Vida Blue 30 5.135 33 3.265

Kevin Brown 31 5.138 21 3.201

Bob Lemon 32 5.139 25 3.234

Bert Blyleven 33 5.141 41 3.314

Bucky Walters 34 5.152 38 3.302

Jerry Koosman 35 5.153 50 3.359

Ed Lopat 36 5.162 22 3.206

Rick Reuschel 37 5.168 51 3.373

Claude Passeau 38 5.171 42 3.319

Red Faber 39 5.176 16 3.149

Lon Warneke 40 5.199 19 3.183

Billy Pierce 41 5.199 34 3.269

John Smoltz 42 5.205 36 3.274

Joe Niekro 43 5.207 84 3.593

Dizzy Trout 44 5.219 24 3.233

Robin Roberts 45 5.221 56 3.405

Steve Rogers 46 5.221 17 3.175

Fergie Jenkins 47 5.225 44 3.338

Dwight Gooden 48 5.230 70 3.506

Eppa Rixey 49 5.251 15 3.148

Allie Reynolds 50 5.252 39 3.304

Journal of Quantitative Analysis in Sports, Vol. 4 [2008], Iss. 1, Art. 1

DOI: 10.2202/1559-0410.1074

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Table A.2 (continued)

Ranking of Pitchers

ERA

Full time &

age corrected Lifetime

Rank CNST Rank ERA

Bob Feller 51 5.252 30 3.255

Claude Osteen 52 5.259 37 3.298

Charley Root 53 5.263 81 3.586

Lefty Gomez 54 5.265 47 3.344

Orel Hershiser 55 5.274 67 3.482

Jerry Reuss 56 5.287 88 3.637

Al Leiter 57 5.298 90 3.654

Bret Saberhagen 58 5.300 46 3.343

Jim Perry 59 5.302 62 3.446

Dave Stieb 60 5.313 58 3.438

Hal Schumacher 61 5.315 49 3.357

Milt Pappas 62 5.318 54 3.398

Virgil Trucks 63 5.332 53 3.385

Curt Simmons 64 5.334 76 3.543

Larry Jackson 65 5.344 55 3.401

Bob Buhl 66 5.347 78 3.545

Camilo Pascual 67 5.353 87 3.633

Burt Hooton 68 5.356 52 3.380

Tom Glavine 69 5.362 60 3.438

Ken Holtzman 70 5.369 68 3.487

Jim Kaat 71 5.373 63 3.453

Paul Derringer 72 5.373 64 3.459

Lew Burdette 73 5.374 91 3.656

Danny Darwin 74 5.376 114 3.837

Bob Welch 75 5.383 66 3.467

David Cone 76 5.396 65 3.462

Mickey Lolich 77 5.410 59 3.438

Murry Dickson 78 5.412 92 3.656

Dennis Martinez 79 5.419 97 3.697

Fernando Va le nzu el a 80 5.421 77 3.545

Charlie Hough 81 5.426 106 3.746

Jimmy Key 82 5.439 71 3.507

Bill Lee 83 5.467 74 3.542

Freddie Fitzsimmons 84 5.469 72 3.509

Tom Candiotti 85 5.475 103 3.732

Ted Lyons 86 5.482 94 3.668

Larry French 87 5.485 61 3.444

Early Wynn 88 5.487 75 3.542

Tommy Bridges 89 5.508 79 3.573

Rick Rhoden 90 5.523 85 3.595

Herb Pennock 91 5.543 86 3.598

Bob Friend 92 5.553 80 3.584

Kevin Appier 93 5.553 105 3.738

Doyle Alexander 94 5.555 107 3.757

Waite Hoyt 95 5.560 82 3.588

Frank Tanana 96 5.580 93 3.662

Ver n Law 97 5.584 109 3.766

Mike Mussina 98 5.586 83 3.593

Frank Viola 99 5.591 101 3.728

Tom Zachary 100 5.592 100 3.728

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Table A.2 (continued)

Ranking of Pitchers

ERA

Full time &

age corrected Lifetime

Rank CNST Rank ERA

Doug Drabek 101 5.601 104 3.735

Rick Wise 102 5.602 96 3.687

Burleigh Grimes 103 5.613 73 3.527

Charlie Leibrandt 104 5.626 98 3.712

Bruce Hurst 105 5.628 121 3.917

Bob Forsch 106 5.629 108 3.765

Bob Knepper 107 5.650 95 3.676

Chuck Finley 108 5.663 116 3.845

Mark Langston 109 5.670 125 3.967

Red Lucas 110 5.673 99 3.721

Ned Garver 111 5.685 102 3.731

Dennis Eckersley 112 5.691 69 3.501

Jamie Moyer 113 5.718 135 4.148

Paul Splittorff 114 5.722 112 3.812

Jim Lonborg 115 5.727 118 3.857

Sam Jones 116 5.729 115 3.838

Red Rufﬁng 117 5.732 110 3.798

Mel Harder 118 5.734 111 3.801

Jesse Haines 119 5.756 89 3.641

David Wells 120 5.766 131 4.035

Jack Billingham 121 5.768 113 3.829

Bobo Newsom 122 5.771 127 3.984

Ron Darling 123 5.789 119 3.874

Guy Bush 124 5.790 117 3.855

Jack Morris 125 5.806 120 3.900

Danny MacFayden 126 5.860 123 3.961

Bill Gullickson 127 5.862 122 3.930

Andy Benes 128 5.888 126 3.973

Steve Renko 129 5.904 130 3.995

George Uhle 130 5.909 129 3.993

Tim Belcher 131 5.938 136 4.163

Mike Torrez 132 5.944 124 3.962

Rick Sutcliffe 133 5.968 133 4.080

Mike Hampton 134 5.974 128 3.991

Kenny Rogers 135 5.978 139 4.269

Kevin Gross 136 6.060 134 4.113

Wes Ferrell 137 6.073 132 4.039

Bump Hadley 138 6.085 138 4.244

John Burkett 139 6.104 140 4.309

Mike Moore 140 6.159 143 4.389

Earl Whitehill 141 6.227 142 4.358

Steve Trachsel 142 6.285 137 4.231

Kevin Tapani 143 6.327 141 4.347

Bobby Witt 144 6.616 144 4.834

Journal of Quantitative Analysis in Sports, Vol. 4 [2008], Iss. 1, Art. 1

DOI: 10.2202/1559-0410.1074

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