A new approach for analyzing running and swimming 
data provides an interesting view of sex differences. 

Assessing Time Trends in 
Sex Differences in 
Swimming & Running 

Howard Wainer, Catherine Njue, and Samuel Palmer 

Introduction 

The Boston Marathon has been staged 
annually since 1897. Until the early 
1960s only 200 to 300 runners competed, 
but the number of runners has 
increased steadily since then. Now 
race organizers impose stiff qualifying 
standards to limit the field. Even with 
such restrictions in effect, the Boston 
Marathon’s starting field numbered 
over 38,000 in 1996. Before the 1970s 
few women ran for recreation, and 
amateur regulations on competitive 
racing barred them from distances 
longer than 4 km. All that has changed 
as shown by the increase in entrants in 
several for-women-only races that have 
drawn more than 8,000 starters. In Fig. 
1 all of the winning times for the 
Boston Marathon are shown augmented 
by the best fitting linear trend lines. 

Three natural questions that are suggested 
by such data are the following: 

1. How far behind men’s marathon 
performances are women? 
2. How long before women catch up? 
3. Why are women lagging? 
Most would agree that the extrapolation 
implicit in Fig. 1 is unwarranted 
and does not contribute much toward 
understanding the issues that underlie 
these three questions. In this article 
we shall explore these issues more 

Figure 1. Winning times (in minutes) in the Boston Marathon since 1897 with linear 
trends indicated for both men and women. 


carefully and suggest a measure that 
seems to provide help. 

Over the past century women’s participation 
in all competitive sports has 
increased markedly. At the same time 
the disparities in performance between 
men and women have shrunk substan


tially. One cannot help but suspect 
that the former is a principal cause of 
the latter. To illustrate this phenomenon 
further, consider the changes in 
the world record for 100-m dash 
shown in Fig. 2 as mean speeds for the 
distance (in meters/second). We see 

10 VOL. 13, NO. 1, 2000 


Figure 2. World record speeds for 100 meters for both men Figure 3. The ratio of women's world record speed in 100


and women since 1910. 

that men have increased their mean 
speed from less than 9.5 m/sec in 1910 
to slightly over 10 m/sec by 1970. The 
women’s record speed improved from 
about 8.5 m/sec in 1934 to almost 9.5 
m/sec by 1990. 

If we consider the ratio of these 
speeds, women to men, we find (in 
Fig. 3) that in 1934 the women’s 
record was about 14% slower than the 
men’s, but that this difference had 
shrunk to about 7% by 1990. Figure 3 
is entirely derivative. It was obtained 
by fitting a linear function to each of 
the two time series in Fig. 2 and then 
dividing those two functions to yield 
the function shown as Fig. 3. Because 
a linear fit is rather good, this rough 
and ready approach is accurate so far 
as it goes, although it would be a capital 
mistake to extrapolate from it. In no 
way do we mean to imply that this ratio 

meter dash to men's (derived from the linear fits in Fig. 2). 

will reach unity sometime in the middle 
of the next century. We recognize 
that the underlying trends are nonlinear 
and one possible explanation for 
the dramatic gains of women, relative 
to men, is that their records were taking 
place at a different portion of the 
curve. We shall return to this issue 
below, but for now merely note that 
the women’s 100-m world record, set 
in 1984, is about the same as the men’s 
record set 75 years earlier. 

To gain momentum on what will be 
one of our major points, that women 
are improving at a faster rate in longer 
distances, consider the world records 
for the 800-m and 10,000-m runs 
shown in Figs. 4 and 5 respectively. 
These two plots are parallel in structure 
to Fig. 1 albeit with somewhat 
slower speeds and hence steeper 
slopes characterizing women’s 

improvement. Once again we can plot 
the ratio of men’s to women’s speeds, 
as we did in Fig. 3. This is shown for all 
three distances in Fig. 6. 

Figure 6 shows several aspects of 
the phenomena clearly. Women world 
record holders run slower than their 
male counterparts, but over the past 30 
years the difference between them has 
shrunk considerably. The longer the 
distance the more rapidly has the difference 
shrunk. Thus, we ought to 
expect that for really long races the 
slope of improvement should be steeper 
still. 

What About Swimming? 

Swimming seems somewhat different, 
at least superficially. The data shown 
in Fig. 7 are a parallel to those in Fig. 

Figure 4. World record speeds for 800 meters for both men 
and women since 1910. 


Figure 5. World record speeds for 10,000 meters for both men 
and women since 1910. 


CHANCE 11 


Figure 6. The ratio of women's world record speed in 100-Figure 7. The ratio of women's world record speed in 100meter, 
800-meter and 10,000-meter runs to men's (derived meter and 400-meter freestyle swimming to men's record 

from the linear fits in Figs. 2, 4, and 5). 

6; there are similarities and differences. 
First we note that women’s 
world record speeds for the 100-m and 
400-m freestyle events has been 
approaching those of men, although 
somewhat faster for the 100-m event 
than for the 400-m event. Women’s 
performances for the longer event, 
however, appear closer to men’s performances. 


Two Problems With This 
Approach 

What can we infer from these results? 
Surely we can see that women are 
making substantial improvements 
in their athletic 
achievements, but can we 
project from these data what 
will be happening in the 
future. If we were to extrapolate 
from Figs. 6 and 7 we 
might say that women will 
be running and swimming 
faster than men within the 
next few decades. Most 
experts would be very surprised 
if this were to occur. 
Where have we been 
misled? 

There are two problems. 
One is with the data that we 
have examined so far, and 
the second is with the 
method we have chosen to 
analyze them. Let us look at 

speeds. 

each of these problems in turn and try 
to find a solution. First the method of 
analysis. 

The trends over time of athletic 
record setting are nonlinear. When a 
sport (or event) is new, records 
improve very quickly, but as the sport 
matures the records change much 
more slowly. There are many causes for 
this, but the most obvious one is participation 
rate. When a sport is new 
the record holder might be the best 
among hundreds, whereas for a mature 
sport the world record holder might be 
the best among millions; as participation 
increases, the record improves 
apace. 

Figure 8. Winning speeds (in meters/second) in the Boston 
Marathon since 1897 with the women's curve translated left 
75 years and upward 2%. 


We can see this effect clearly in 
data from the Boston Marathon, which 
has been run over essentially the same 
course for more than a century. In Fig. 
8 we can see that the curve that 
describes the improvements in men’s 
performance at the very beginning of 
the race, when participation among 
men was limited, is quite similar to the 
curve that describes the women’s performances 
since 1972, when they were 
first allowed to compete. As women’s 
participation increased so too did the 
speed necessary to win. How can we 
characterize this simply? We have 
duplicated the data points associated 
with the women’s record speed and 
moved them back in time to where 

they roughly coincide with 
the men’s records. This suggests 
a different approach to 
representing the difference 
between men’s and women’s 
performances. 

Let us ask the following 
question: “How many years 
ago was the men’s record 2% 
faster than the women’s 
record is today?” This question 
has two pieces to it: 
“How many years ago” is a 
translation of the curve to the 
left; “2% faster” is a translation 
upward. The key 
assumption here is that the 
improvement in performance 
for both men and women follows 
the same curve, so to 
make accurate comparisons 

12 VOL. 13, NO. 1, 2000 


Figure 9. The number of years since the women's record in the Figure 10. Bronze-medal speeds for three Olympic track 
Boston Marathon (+2%) was the men's record has remained events for both men and women, with linear fits. 

roughly the same for the last 20 years. 

we must translate one curve to overlay 
the other. The choice of 2% is somewhat 
arbitrary but provides a criterion 
for prediction. Cashmore (1999) estimated 
that this difference, which he 
ascribed to “the strength gap,” is about 
5%. We agree that this is a reasonable 
ballpark estimate but that there is 
some variation as a function of kind of 
exercise. We return to the issue of how 
to choose the size of this performance 
gap later. 

When we plot this difference (in 
Fig. 9) we find that it is remarkably flat 
suggesting that between 60 and 80 
years ago men were running the 
Boston Marathon about 2% faster than 
women and that the improvements 
that women have shown over 
the past 20 years were mirrored 
by improvements that 
men made over a half century 
earlier. This result more 
closely matches expert opinion 
in estimating when 
women will overtake men in 
the Boston Marathon. 

The second problem is the 
statistical instability of inferences 
about the main body of 
a distribution that are made 
from an extreme value. Such 
values are always unstable 
and hence such inferences 
are bound to be equivocal. 
We could improve stability if 
we could get the entire distribution 
of world class performance. 
Sadly, such informa


tion is not available. We note, however, 
that third-place times are also the 
median of the top five finishers. Thus, 
if we focus our attention on the performance 
of Olympic bronze medalists, 
we would have a measure that represents 
the median performance of the 
five best athletes in that Olympic year. 

Some Results 

In Fig. 10 the bronze medal times are 
shown for men and women for three 
track events (100 m, 800 m and 
10,000 m) in all Olympics for which 
these events were run. Superimposed 
over each data series is the best fitting 

Figure 11. The number of years since the women's bronze-
medal speed (+2%) was the men's bronze-medal speed in 
three track events. 


straight line, which adequately represents 
the time trends seen in these 
events. Using the fitted values we can 
calculate the number of years ago that 
the women’s bronze speed (+2%) was 
the men’s bronze speed. This result is 
shown in Fig. 11. The results suggest 
quite a different inference than that 
we would have drawn from the results 
in Fig. 6 but are consistent with what 
we observed with the Boston 
Marathon data. Specifically, we see 
that women’s performances in sprints 
(100 m) is closer to men’s performances 
(less than 50 years behind) 
than for longer distances. We will not 
pay much attention to the 10,000 m 
results because the race has only been 

run three times, and hence 
we have too little information 
to draw reliable inferences 
about trends. 

It is appropriate at this 
time in the narrative to 
describe briefly the analytic 
methodology used to assess 
time lags and why we chose 
2% as the vertical difference 
in men’s and women’s performances. 
To illustrate both 
what we did and how it 
turned out, let us consider 
the Olympic bronze medal 
records in swimming 100 
meters freestyle shown in 
Fig. 12. 

The analytic problem is 
to translate the women’s 
curve horizontally and verti-

CHANCE 13 


Figure 13. The number of years since the women's Olympic 
Figure 12. Olympic bronze-medal speeds for the 100-meter bronze-medal speed (+9%) was the men's Olympic bronze-
freestyle swimming event for both men and women. medal speed in 100-meter freestyle swimming event. 

cally until it coincides as closely as possible 
to the men’s curve. We did a conditional 
minimization by sequentially 
first moving the women’s curve to the 
left four years and then upward until 
the mean squared difference between 
to the two curves was minimized; then 
we moved it back four more years and 
repeated the minimization. When this 
was done we found (summarized in 
Table 1) that the error was minimal 
when there was a lag of between 8 and 
12 years and with women swimming 
about 8% slower than men. 

This result is somewhat different 
from what we found for track using the 
same methodology, where it yielded a 
lag of about 75 years with a 2% difference. 
Data limitations as well as the 
rather flat response surface requires us 
to interpret these results broadly. 
Obviously the estimates of the two 
parameters of the curve translation will 

be negatively correlated, so interpretations 
are muddied a bit. But it seems 
clear that the data suggest that, in 
swimming sprints, women seem to be 
limited to achieving speeds about 8% 
slower than men’s and that women’s 
improvement curve lags men’s by 
about a decade. In Fig. 13 we show the 
trend in the number of years that 
women’s speeds (+9%) have lagged 
men’s in the 100-m freestyle. There is 
clearly a time trend suggesting substantial 
gains over the last few decades 
(as well as an anomalous lump surrounding 
World War II). We shall suggest 
one plausible explanation for the 
gains in the next section. 

When we repeat the same analysis 
(see summary in Table 2) for the 400m 
freestyle event, we find very similar 
results except that minimal error of fit 
seems to occur 
with only a four 

Table 1 — Results of Optimization Curve 
Registration for 100-m Freestyle 


Number of Percent Mean squared 
years back upward error (× 1,000) 
4 13% 5.47 
6 11% 3.95 
8 9% 2.86 
12 7% 2.97 
16 5% 4.59 

year delay. Once again (see Figs. 14 
and 15) we see a remarkably similar 
structure in the improvements in 
bronze medal times (Fig. 14) as well as 
a swooping reduction in the time lag 
between the sexes. One interpretation 
of this result is that women’s training in 
distance swimming seems to have 
reached approximately the same level 
as men’s. 

Discussion 

In this article we have examined sex 
differences in performance in track 
and swimming over time. We noticed 
that the men’s and women’s curves of 
improvement in performance were 
very similar, except that the women’s 
curves were offset horizontally by a 

Table 2 — Results of Optimization Curve 
Registration for 400-m Freestyle 


Number of Percent Mean squared 
years back upward error (× 1,000) 
0 11% 2.16 
4 9% 1.86 
8 7% 2.11 
12 5% 2.53 
16 3% 3.22 
18 0% 3.78 
20 0% 4.49 

14 VOL. 13, NO. 1, 2000 


Figure 14. Olympic bronze-medal speeds for the 400-meter 
freestyle swimming event for both men and women. 
Figure 14. Olympic bronze-medal speeds for the 400-meter 
freestyle swimming event for both men and women. 
varying number of years and vertically 
by some percentage. The vertical offset 
seems to vary between 2% (for the 
marathon) and 9% (the 100-m 
freestyle). The range of this offset 
melds well with the average 5% estimate 
that others have found (e.g., 
Cashmore 1999). This offset is usually 
interpreted as representing some sort 
of physiological strength difference. 
Such an interpretation is supported by 
our findings. It is generally agreed that 
the biggest sex differences in strength 
are in the upper body, so we should 
expect to see larger differences in the 
vertical offset in swimming than in 
running because the former uses 
upper body strength to a greater 
degree. Our analyses only partially support 
the emerging consensus that sex 
differences are small to nonexistent in 
physical endurance. We found smaller 
sex differences in the longer swimming 
event, but in running women seem to 
do relatively better in the sprints. 

After making the vertical physiological 
adjustment, we used, as our principal 
measure of women’s improvements, 
the number of years that have 
elapsed between when the adjusted 
women’s record was the men’s record. 
This measure provided some interesting 
insights. Most notable was the shift 
in the message seen in Fig. 1 to that 
seen in Fig. 8; women’s performances 
in the marathon seem to lag men’s by 
about 75 years, not the 5 or 6 that a 
naive linear extrapolation might suggest. 
The lag is not much greater in the 
sprints; Florence Griffith Joiner’s world 
record for 100 m (10.54 seconds), 
though much better than any other 
competitor, is about the same Harold 

Figure 15. The number of years since the women's Olympic 
bronze-medal speed (+7%) was the men's Olympic bronzeAbraham’s 
1924 medal speed in 400-meter freestyle swimming event. 

Olympic time. 

In swimming the lag is much shorter, 
being measured in years not 
decades. Moreover it has been shrinking 
quickly. Why? Palmer (in press) 
suggested that changes in attitudes 
toward what women are physically 
capable of has given rise to vastly 
increased rigor of training. Women’s 
training in swimming now is indistinguishable 
from men’s in terms of both 
content and frequency. But, more 
important is participation. Palmer (in 
press) reports that “girls start swimming 
younger than ever before. There has been 
a dramatic increase in the enrollment in 
swim clubs and other swim teams to the 
point that now fully 60% of swimmers 
under age 17 are female. Secondly, more 
girls than ever before are sticking with 
the sport. Since Title IX was enacted in 
1972, more women have been able to 
continue swimming through college, 
and not have to stop at the age of eighteen 
due to lack of opportunity.” 

We are well aware of the fragility of 
the analytic procedure of curve registration 
we used to translate one performance 
curve to another. It relies on the 
untested assumption that the size of 
the talent-pool within each sex is time 
shifted, but otherwise essentially 
equal. Moreover, the response surface 
is quite flat, and other solutions, with 
quite different interpretations may be 
almost as likely. We are currently doing 
a more rigorous sensitivity analysis that 
should help illuminate this issue. But 
nonetheless, we believe that the 
results obtained provide us with a 
much clearer view of sex differences in 
athletic performance. We anticipate 

that others will augment both our data 
and our methodology to yield better 
answers yet. 

[Howard Wainer’s work on this project was 
partially supported by the Senior Scientist Award 
given to him by the trustees of the Educational 
Testing Service. Catherine Njue’s work on this 
project was principally done while she was a 
summer pre-doctoral fellow at the Educational 
Testing Service. Samuel Palmer’s work was done 
as part of an independent study project in statistics 
at Princeton High School. We would like to 
thank Phil Whitten of Swimming World 
Magazine for the bronze-medal swimming data, 
Cathy Durance of USA Swimming for the data 
on women’s participation, and John Tukey.] 

References and Further Reading 

Bronze medal Olympic track times 
for men and women in 100 m, 800 
m, and 10,000 m were obtained 
from University of Exeter Centre 
for Innovation in Mathematics 
Teaching on the World Wide Web: 
www.ex.ac.uk/cimt/data/. 

Cashmore, E. (1999), “Women’s 
Greatest Handicaps: Sex, 
Medicine, and Men,” British 
Journal of Sports Medicine, 33, 76. 

Palmer, S. (in press), “Sex 
Differences in Swimming Then 
and Now,” Swimming World, 35. 

Track World Record progression 
data for men and women in 100 
m, 800 m and 10,000 m were 
obtained from Mika Perkiomaki’s 
Great Track and Field Statistics 
Page on the World Wide Web: 
www.utu.fi/~mikkoski/yu/ 
links.html. 

CHANCE 15 


Comment: Studying Trends in Sport 
Participation by Modeling Results 
of Elite-Level Athletic Performance 

David E. Martin 


Study of the results of athletic competition 
is useful for identifying the contemporary 
limitations of human performance 
(Hill 1924). Sport sociologists 
in particular will thus delight in reading 
Wainer, Njue, and Palmer’s 
“Assessing Time Trends in Sex 
Differences in Swimming and 
Running.” We all know that women’s 
participation in sports has lagged 
behind that of men, and sociologists 
have offered many explanations: 
Cultural and religious influences are 
but two. Sport psychologists suggest 
that women as athletes are equally 
trainable in comparison to men, with 
similar competitiveness and enthusiasm. 
Thus, as the many barriers to 
women’s participation in sports gradually 
disappear, the increasing size of 
the gene pool that is participating has 
substantially increased the rate of performance 
improvement. The challenge 
has been to identify and portray these 
changing dynamics over time. 

Wainer, Njue, and Palmer provide a 
novel approach for quantifying such 
changing patterns, and illustrate their 
concepts using selected swimming and 
running events that have been contested 
since the modern Olympic Games 
began. Those interested in other sports 
will enjoy using this strategy to study 
other Olympic events contested annually 
over this past century that have 
had both male and female participation. 
Results for each Olympic Games 
are easily obtained (Wallechinsky 
1996), and each sport publishes its 
own annual top performance sum


maries. There are a few points of possible 
misinterpretation or confusion in 
Wainer et al.’s presentation, however — 
not in their study strategy but in their 
data collection and interpretation — 
that deserve mention. 

First, although the Boston 
Marathon does in fact impose qualifying 
time standards that must be met by 
those wishing to run, the example used 
to indicate the number of such competitors 
— namely, the 35,868 finishers 
in 1996 — gives an inflated picture. 
That was the race’s centenary, 
and a large number of race participants 
were permitted to run as a result of lottery 
selection without meeting the 
qualifying standard. A better indication 
of recent race participation at Boston is 
given by examining the number of finishers 
during those recent years in 
which participation was invitationally 
restricted — 1995 (8,258), 1997 
(8,893), 1998 (10,289), and 1999 
(11,274). 

Second, although it is correct that 
the Boston Marathon has been run 
continuously since 1897 over basically 
the same route, extensive road construction 
has produced four different 
course distances. Figure 1 of Wainer et 
al. plots winning times from 1897 
through 1999, and Fig. 8 plots pace in 
meters per second over the same period. 
For these plots to portray meaningful 
comparisons of performance, the 
finish times and race paces need to be 
normalized to a single appropriate distance. 
Logically, this ought to be the 
standard marathon distance used 

presently (26 miles, 285 yards; 42,195 
meters). Awareness of these course 
variations is not mentioned in the article. 
The various distances, and the 
years these distances were used 
(Martin and Gynn 1979) are as follows: 
1897 through 1923: 39,742 meters; 24 

miles 1,232 yards 
1924 through 1926: 42,025 meters; 26 

miles, 209 yards 
1927 through 1952: 42,195 meters; 26 

miles, 385 yards 
1953 through 1956: 41,083 meters; 25 

miles, 938 yards 
1957 through 1999: 42,195 meters; 26 

miles, 385 yards 

Third, it is mentioned that “third 
place (Olympic) times are also the 
median of the top five finishers,” and 
thus, we should “focus our attention 
on the performance of the Olympic 
bronze medallists” for developing “a 
measure that represents the median 
performance of the five best athletes in 
that Olympic year.” Though it seems 
reasonable to assume that bronze-
medal times are more representative of 
the entire distribution of world class 
performances than gold-medal times, 
the bronze-medal times are suspect as 
well. For one thing, some Olympic 
tracks (or swimming pools) are better 
suited to fast times than are others. 
Thus, in some Olympic years the best 
world class performances don’t occur 
at the Olympics at all. The data problems 
identified here point out the 
intrinsic difficulties in studies of this 
type. 

16 VOL. 13, NO. 1, 2000 


Finally, although it is quite interesting 
from a sport sociology viewpoint to 
compare the changing dynamics of 
participation by women and men, it is 
inappropriate to extend the comparison 
to performance dynamics. This was 
alluded to briefly in suggesting correctly 
that extrapolation of the data in Fig. 
1 to predict that women will run faster 
than men by mid-2000 is nonsensical. 
Others have suggested that this will 
indeed occur (Whipp and Ward 1992), 
and have been duly criticized (Julin 
1992). One must be careful when 
using such phrases as “when women 
will catch up to men” or “why women 
are lagging behind” lest they be misinterpreted 
to mean that women ought 
to be able to “catch up” to men in sport 
performance. There’s ample reason 
why they should not: They are two 
physiologically different populations, 
not two subsets of a single population, 
one of which is somehow “lagging 
behind.” A brief review of how the 
physiological differences contribute to 
the performance difference in both the 
strength-and endurance-oriented 
events makes this clear. 

In the shorter sprint distances contested 
by both runners and swimmers, 
the performance differences between 
the sexes are affected by two independent 
variables. One is the larger skeletal 
muscle mass among men, caused 
by both human growth hormone 
and testosterone, which exist 
in higher concentration in 
men than in women. In 
addition, however, 
women are under 
a greater estrogenic 
influence 
than 
men, and 
this hormone 
promotes 
greater 
fat storage in tis


sues. A champion male 100-metersdash 
man may have 6% body fat, but 
his female counterpart will have 11%. 
This fat slows both athletes because it 
must be carried along as excess baggage, 
but the slowing effect in women 
is greater. Among swimmers, however, 
the increased body fat among women 
makes them more buoyant, giving 
them a performance advantage. This 
explains the smaller male/female performance 
difference for swimming 
records than for running records. 
Unfortunately, the variable extent of 
clandestine steroid hormone usage 
among both sexes in both running and 
swimming over the years has added a 
confounding element to meaningful 
analysis of record-quality data. 

As the competition distance lengthens, 
in both swimmers and runners, 
although the muscle and fat differences 
remain, the endurance component 
of performance contributes to performance 
more critically. The name of 
the game in successful longer distance 
competition is sustained high-volume 
oxygen utilization to generate energy, 
with minimal performance inhibition 
due to accumulating tissue acidity. In 
the bloodstream, 98.5% of the oxygen is 
transported via attachment to hemoglobin 
molecules located in red blood 
cells. Hemoglobin is a large protein 
molecule. In turn, red 
blood cell production 
is regulated by 
another protein 
hormone 

— erythropoietin. 
The higher 
level of 
testosterone 
in men, 
which is an 
anabolic 

(protein


building) hormone, provides them with 
both a measureably greater total hemoglobin 
and red-blood-cell mass. A typical 
mean value for blood hemoglobin 
concentration in men is 15 gm/dl; for 
women this is 13.5 gm/dl — a 10% difference. 
Calculation of blood oxygen 
(O2) transport using these values gives 

17.3 ml O2/dl of blood for women and 
19.2 ml O2/dl for men — again a 10% 
difference. In addition to the blood oxygen 
concentration difference, there is a 
blood volume difference. Women have 
roughly 66 ml of blood volume/kg body 
weight (BW), compared to 77 ml/kg 
BW for men — a 14.3% difference. 
Physiologists enjoy using such facts 
to estimate the influence of these sex 
differences on the existing performance 
differences seen in sport. Such differences 
explain clearly why women will 
never “catch up” to men, and indeed, 
why there is no need to. They are different 
people — vive la difference! 

References and Further Reading 

Hill, A.V. (1924), “The Physiological 
Basis of Athletic Records,” 
Lancet, 2, 481-486. 

Julin, A.L. (1992), “No 
Improvement in Running?” 
Nature, 356, 21. 

Martin, D.E., and Gynn, R.W.H. 
(1979), The Marathon Footrace, 
Springfield, IL: Charles C. 
Thomas. 

Wallechinsky, D. (1996), The 
Complete Book of the Summer 
Olympics, 1996 Edition, New 
York: Little Brown. 

Whipp, B.J., and Ward, S.A. (1992), 
“Will Women Soon Outrun 
Men?” Nature, 355, 25. 

CHANCE 17 


Comment: Can We Trust a Method 
Just Because We Like Its Predictions? 


Phillip N. Price 


If the analysis and conclusions of 
Wainer, Njue, and Palmer (2000) are 
right, then we live in a special time: 
After 70 years in which men’s and 
women’s performances (both absolute 
and relative to one another) could be 
accurately extrapolated using straight 
lines, this method will stop working 
very soon. I happen to agree. But I 
wonder if there has ever been a time 
when people didn’t feel that the runners 
and swimmers of their time were 
finally approaching the limits of 
human performance. Who knows, perhaps 
changes in training or diet, or in 
the rules regarding the use of steroids 
and other drugs, will allow improvements 
to continue this way for several 
more decades. I doubt it, but history 
should make us cautious. 

An interesting feature of the analysis 
by Wainer et al. is that it attempts 
to predict differences between men’s 
and women’s speeds (in some future 
year in which records for both sexes 
have stabilized) without predicting the 
limiting speed for either sex. That’s a 
very good thing, as it’s clearly impossible 
to extract a reliable plateau speed 
based on historical data, at least for 
running: Just take a look at Fig. 10! In 
fact, as can be seen in Fig. 10, and as 
noted by Whipp and Ward (1992), not 
only are running speeds (still) improving 
linearly for most running events, 

but they are doing so at about the same 
rate at all distances — of the order of 7 
m/min per decade for men, and 16 
m/min for women, within about 10% 
in each case. (These estimates do not 
quite agree with Fig. 10, which shows 
bronze-medal Olympic speeds rather 
than world records, but they are close 
enough). Of course, these improvements 
cannot keep up forever and perhaps 
not even for long, though the 70to 
100-year stretch covered in the figure 
is quite impressive as it is. In the 
swimming data shown in the article, 
there is evidence of a plateau already 
being approached, and the marathon 
records also appear to be stabilizing 
(the winning times for both men and 
women in the 1999 Boston Marathon 
were little changed from the 1992 
records shown in Fig. 1, in contrast to 
the extrapolation that this should have 
been the year the women beat the 
men). 

By coincidence, I began reading 
Full House (Gould 1996) shortly after 
being asked to comment on this article. 
Full House is about some consequences 
of physiological plateaus in 
athletics and other contexts, and 
Gould’s Chapter 8 discusses men’s and 
women’s athletic performances. Gould 
agrees with most of the points of 
Wainer et al., particularly with regard 
to the error of predicting women’s ver


sus men’s performances based on linear 
extrapolation from the past: he calls 
extrapolation “a dangerous, generally 
invalid, and often foolish game,” which 
perhaps overstates the case a bit. He 
also says “In the most popular and 
established men’s [running] events, we 
note the pattern of rapid initial 
improvement followed by flattening of 
the curve” and suggests that the same 
will happen soon in the women’s 
events. Judging from Fig. 10, I think 
Gould and I must have different ideas 
about what “flattening of the curve” 
means. 

In any case, Wainer et al. have set 
themselves a difficult task — to extract 
sensible estimates of relative future 
performances for the sexes without any 
way of predicting absolute performances. 
This is analogous to Cox’s proportional 
hazards model (Cox 1972), in 
which it is possible to draw inferences 
about treatment and covariate effects 
without having to specify or estimate a 
baseline hazard function that quantifies 
the probability of dying before a 
given age. 

Wainer et al. suggest a clever 
method for extrapolating relative performances 
without delving into the 
more difficult issue of absolute performances: 
Shift the women’s record-versus-
year curve in both dimensions 
(speed and year) to extract two para


18 VOL. 13, NO. 1, 2000 


meters rather than one. These shifts 
have simple interpretations: the shift in 
speed represents the amount by which 
women are intrinsically slower (or 
faster) than men, and the shift in year 
represents the amount by which 
women’s approach to their top speed 
lags that of men due to dramatically 
unequal participation and training of 
men and women in the past and, possibly, 
the present. I think this separation 
of the variation into two components, 
a lag in years and a gap in speed, 
is a substantial conceptual advance. 

All of which is fine as far as it goes, 
but the actual application is more in 
the realm of curve fitting than model 
fitting. I certainly agree that the rapid 
increases in speed in each sex’s first 15 
years of the Boston Marathon are 
largely attributable to increases in the 
pool of participants so that qualitatively 
these 15-year periods are comparable. 
But the method suggested by 
Wainer et al. requires more than qualitative 
comparability: It attempts to 
extrapolate based on the exact shapes 
of the speed-versus-year curves. Does 
the growth in the number and skill of 
male Boston Marathon participants 
around 1900 really match that of 
female participants 80 years later so 
well that we can use the race results 
from these widely separated periods to 
estimate differences in male–female 
future performances that are of the 
order of 2%? I’m rather skeptical, even 
though the results are in line with 
expert opinion (as they are for the 
other races considered in the article). 

Furthermore, notice the standard 
that I (along with Wainer et al.) am 
applying here: we are judging the 
applicability of the method not by 
comparison to data, but through comparison 
to expert opinion regarding the 
future speed gap. Of course, there is 
nothing wrong with including expert 
opinion in making predictions, and 
indeed this is often done in Bayesian 
modeling by using informative prior 
distributions. But there is a distinction: 
in Bayesian models, the prior distribution 
can affect the inferences directly, 
but that is not the case here. For example, 
suppose we obtain data for some 
race not considered by Wainer et al. 
(the 1500-m run, say, or the 100-m 
backstroke), apply their method, and 
discover that the best fit is obtained if 

the women’s curve is shifted back 40 
years and down 5% rather than up. 
Experts would judge it implausible that 
women will eventually be 5% faster 
than men in these races, but the current 
method has no way of allowing 
this judgment to influence the prediction. 
Would we then conclude that the 
experts are wrong and that the model 
results are right and women will be 5% 
faster than men in these races? I doubt 
it; at least, I wouldn’t bet that way. 

It is intriguing, though, that the 
results of applying Wainer’s method to 
a variety of races, in both running and 

Wainer et al. have set 
themselves a difficult 
task — to extract 
sensible estimates of 
relative future 
performances for the 
sexes without any 
way of predicting 
absolute 
performances. 

swimming, all yield reasonable results 
(in the sense of being in line with 
expert opinion) for persistent 
male–female performance differences. 
This is particularly interesting because 
the method doesn’t just tolerate curvature 
in the speed-versus-year curve; it 
demands it: if improvements for both 
sexes were truly linear, then the trade-
off between left-right and up-down 
shifts would be undefined. There is 
some evident downward concavity in 
the swimming data presented, but 
none (to my eye) in the running data, 
so I find it surprising that the method 
generates sensible results for those 
cases too. Thus, in spite of being skeptical 
for the reasons mentioned in the 
previous paragraph, I admit the possibility 
that Wainer et al. might be onto 
something. 

And of course, it’s easier to criticize 
than to suggest improvements. 
Suppose I wanted to estimate future 
male–female performance differences, 

what would I do? Realistically, for now 
I would poll the experts, just as, if I 
were interested in predicting the outcome 
of a future football game, I 
would rely on the point spread rather 
than a statistical model. But that’s a 
pretty unsatisfying answer to publicly 
state in a statistics magazine. And of 
course, just because someone is an 
expert, he or she is not necessarily 
right. How many track and field 
experts who witnessed Bannister 
breaking the four-minute mile would 
have thought that the mile record 
would someday be under 3:44, as it is 
now? 

OK, so I’m not convinced of 
Wainer’s method, nor do I want to just 
rely on expert opinion. What’s the 
alternative? I’ll start by suggesting 
some future work that builds on the 
work of Wainer et al. No prediction is 
complete without an estimate of 
uncertainty. Even if the method of 
Wainer et al. happens to be perfectly 
suited to the actual application, stochastic 
variation in the performances 
will lead to variation in the predictions 
depending on which years of records 
are considered. The sensitivity to such 
stochastic variation could be assessed 
by examining the stability of the predictions 
for various sets of data: What 
are the predictions if silver-medal 
speeds are used instead of bronze, if 
data after 1980 (or prior to 1960) are 
excluded, and so on. The resulting 
variation in predictions would set a 
lower bound on the uncertainty; of 
course, it wouldn’t address the major 
issue of model (or rather method) misspecification. 


Beyond that, I can only suggest the 
more difficult task of modeling physiology 
(or at least the relevant 
male–female physiological differences) 
directly. Whipp and Ward 
(1972) discussed relationship 
between speed improvements and 
changes in metabolic rate, a possibly 
helpful way of looking at the issue. 
Wainer et al. point out in their discussion 
that there appear to be inherent 
differences in upper-body strength 
between men and women, and that 
this ought to translate into speed differences, 
particularly in upper-bodyintensive 
sports such as swimming. A 
more precise estimate of the persistent 
male-female strength difference 

CHANCE 19 


or power might be obtained (how?) 

and converted into an expected differ


ence in swimming speed by regressing 

speed on strength for male and female 

swimmers (it would be interesting if 

the regression slopes turn out to be 

different). Alternatively, scaling laws 

based on weight, muscle mass, fat 

mass, and so forth, could perhaps be 

used, somewhat along the lines of 

Schmidt-Nielsen (1972), which is, by 

the way, the best short data-based 

book ever written. 

None of these suggestions have the 

elegance and simplicity of the method 

suggested by Wainer et al., and in fact, 

they aren’t likely to provide much bet


ter estimates of eventual sex differ


ences in performance either. It’s just a 

tough problem. I think that Wainer et 

al. have made a significant contribu


tion to this area of study by suggesting 

that performance differences should 

be decomposed into two components 

(year and speed) rather than one, but I 

don’t really trust the results of their 

suggested method. I suspect we won’t 

have better data-based estimates (as 

opposed to expert opinions) until some 

events start showing evidence of 

approach to a speed asymptote, a phe


nomenon that must surely occur some


time, and possibly soon, but which (at 

least for running) is not apparent in the 

data so far. 

References and Further Reading 

Cox, D.R. (1972), “Regression 
Models and Life-Tables,” Journal 
of the Royal Statistical Society, 
Ser. B, 34, 187–220. 

Gould, S.J., (1996), Full House: The 
Spread of Excellence From Plato 
to Darwin, New York: Three 
Rivers Press. 

Schmidt-Nielsen, K. (1972), How 

Animals Work, New York: 

Cambridge University Press. 

Wainer, H., Njue, C., and Palmer, 

S.J. (2000), “Assessing Time 
Trends in Sex Differences in 
Swimming and Running,” 
Chance, 13(1), 10–15. 
Whipp, B.J., and Ward, S.A. (1992), 

“Will Women Soon Outrun 

Men?” Nature, 355, 25. 

20 VOL. 13, NO. 1, 2000 

Harvard University Press 
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Sex and Sports: A Rejoinder 

Howard Wainer, Samuel Palmer, and Catherine Njue 

We would like to thank Professors 
Martin and Price for their careful reading 
and encouraging suggestions. They 
have provided for us an inkling of what 
Picasso must’ve felt like when a recreational 
scribbling of his was sold for 
thousands of dollars. We are delighted 
that something that was so much fun 
to do has elicited the praise and 
thoughtful consideration of such 
accomplished scholars. 

Professor Martin has provided 
important physiological facts that can 
be used to explain the effects of our 
model. It might be useful in the future 
to try to include them as covariates and 
see to what extent the accuracy of the 
model’s predictions are improved. In 
addition Professor Martin’s facts help to 
answer Professor Price’s concern about 
the efficacy of our approach. Price feels 
that the principal value of our approach 
is that it seems to match expert opinion 
(not a bad first step, we reckon). 
Martin’s description of the inexorable 
chain of physiological events that begins 
with sex differences in hemoglobin and 
testosterone concentrations, affecting 
differences in muscle mass and oxygen-
carrying capacity, yielding differences in 
speed and endurance, provides a compelling 
supporting argument for our 
separation of the effects of physiology 
and participation. 

Implicit in Professor Price’s praise 
of our two-parameter conceptualization 
is the notion that if sufficiently 
good data can be obtained (see Palmer 
(in press) for a fuller discussion of this) 
we can get better adjustments for participation 
and so isolate the truly physiological 
effect. At this point trying to 
connect the estimates of measured 
physiological variables with adjusted 
performance could be enormously 
informative. 

Of course, participation rates are 
inadequate to accurately measure what 
is fundamentally a self-selected 
process (Wainer, Palmer, and Bradlow 
1998; Wainer 1986), but they do provide 
a beginning. When coupled with 
the sort of mixture modeling proposed 
by Rubin (1977), they can help us to 
obtain good estimates of the precision 
of our parameters, which reflect more 
than what is available from traditional 
sampling error. 

Price also correctly points to the flat 
response surface explicit in our fitting 
procedure as a source of plausibly substantial 
sampling variability. As this is 
being written, we are trying to debug a 
computer program designed to assess 
this variability. It uses a specialized 
variation of the jackknife and when 
complete should provide us with a 
fuller notion of the stability of our estimates. 
Unfortunately, it appears that 
the publishing deadline will reach us 
before the results. 

What if the women’s curve needs to 
be shifted down 5%? Professor Price 
suggests this as an outcome that would 
invalidate our model. Because such an 
outcome (if it were not the result of a 
sampling anomaly) would violate the 
physiological implications that form 
the basis of our model, he is correct. 
But any saltworthy model must be 
potentially disprovable. The model 
then gains credence as additional data 
are gathered that are still supportive. 
Note, however, that our model does 
allow women to exhibit superior performances 
to men when they have 
greater participation and the gains due 
to participation outweigh the losses 
associated with physiological differences. 
An example of such a situation 
might be in synchronized swimming, 
which is almost totally dominated by 

women. If men suddenly decided to 
compete, we would expect that, for a 
while at least, women’s performances 
would be superior. 

Of course, we have focused on 
sporting events that are scored in a 
completely objective way (a stop 
watch), but it is not inconceivable 
that sports based on human judges 
could also fit into this mold. Our preference 
for extending these results 
would not be in this direction but 
rather toward those sports (like 
archery) in which strength and 
endurance play a vastly reduced role. 
Here we would expect that participation 
rates would be the sole determiner 
of relative performance. 

We are not unaware of the plausible 
applicability of this methodology to the 
study of sex differences on performance 
in various academic skills. 

References and Further Reading 

Palmer, S. (in press), “Sex 
Differences in Swimming Then 
and Now,” Swimming World, 35. 

Rubin, D.B. (1977), “Formalizing 
Subjective Notions About the 
Effects of Nonrespondents in 
Sample Surveys,” Journal of the 
American Statistical Association, 
72, 538–543. 

Wainer, H. (1986), Drawing 
Inferences From Self-selected 
Samples, New York: Springer-
Verlag. 

Wainer, H., Palmer, S.J., and 
Bradlow, E.T. (1998), “A 
Selection of Selection 
Anomalies,” Chance, 11(2), 3–7. 

CHANCE 21