Why are (the best) women so good at chess? Participation rates and gender differences in intellectual domains

Abstract

A popular explanation for the small number of women at the top level of intellectually demanding activities from chess to science appeals to biological differences in the intellectual abilities of men and women. An alternative explanation is that the extreme values in a large sample are likely to be greater than those in a small one. Although the performance of the 100 best German male chess players is better than that of the 100 best German women, we show that 96 per cent of the observed difference would be expected given the much greater number of men who play chess. There is little left for biological or cultural explanations to account for. In science, where there are many more male than female participants, this statistical sampling explanation, rather than differences in intellectual ability, may also be the main reason why women are under-represented at the top end.

Full text

Why are (the best) women so good at chess?
Participation rates and gender differences
in intellectual domains
Merim Bilalic
´
1,
*, Kieran Smallbone
2
, Peter McLeod
1
and Fernand Gobet
3
1
Department of Experimental Psychology, University of Oxford, South Parks Road, Oxford OX1 3UD, UK
2
Manchester Centre for Integrative Systems Biology, University of Manchester,
131 Princess Street, Manchester M1 7DN, UK
3
Centre for Cognition and Neuroimaging, Brunel University, Uxbridge, Middlesex UB8 3PH, UK
A popular explanation for the small number of women at the top level of intellectually demanding activities
from chess to science appeals to biological differences in the intellectual abilities of men and women. An
alternative explanation is that the extreme values in a large sample are likely to be greater than those in a
small one. Although the performance of the 100 best German male chess players is better than that of the
100 best German women, we show that 96 per cent of the observed difference would be expected given
the much greater number of men who play chess. There is little left for biological or cultural explanations to
account for. In science, where there are many more male than female participants, this statistical sampling
explanation, rather than differences in intellectual ability, may also be the main reason why women are
under-represented at the top end.
Keywords: gender differences; participation (base) rates; chess; intellectual activities;
intelligence; science
1. INTRODUCTION
The former president of Harvard University, Lawrence
Summers, expressed a view widely held by both academic
researchers (e.g. Geary 1998;Kimura 1999;Pinker 2002)
and laypeople when he suggested that innate biological
differences in intellectual abilities may explain why the
most successful scientists and engineers are predomi-
nantly male (Summers 2005). Recent research confirms
that, although there is little difference between the average
scores of men and women on intelligence and aptitude
tests, highly intelligent people are predominantly male
(Deary et al. 2003;Irwing & Lynn 2005). As Irwing &
Lynn (2005) said: ‘different proportions of men and
women with high IQs.may go some way to explain the
greater numbers of men achieving distinctions of various
kinds for which a high IQ is required, such as chess
Grandmasters, Fields medallists for mathematics, Nobel
prize winners and the like’ (p. 519; emphasis added).
Despite the increasing proportion of women in
intellectually demanding professions, they are still under-
represented at the top level in science (Long 2001;Xie &
Shauman 2003;Ceci & Williams 2007) and engineering
(Long 2001). There are various possible explanations for
this apart from innate biological differences (for critiques
and negative findings see Kerkman et al. 2000;Spelke
2005;Lachance & Mazzocco 2006). Socialization and
different interests (Benbow et al. 2000;Ayalon 2003),
gender roles ( Massa et al. 2005), gatekeeper effects
(Davidson & Cooper 1992;Steele 1997;Huffman &
Torres 2002), cultural differences (Andreescu et al. 2008)
and higher participation rates of men (Charness &
Gerchak 1996;Chabris & Glickman 2006) have all been
proposed. Here, we show that in chess, an intellectually
demanding activity where men dominate at the top level,
the difference in the performance of the best men and
women is largely accounted for by the difference that
would be expected, given the much greater number of
men who participate. Despite the clear superiority of the
top male players, there is, in reality, very little performance
gap in favour of men for non-statistical theories to explain.
Before considering cultural or biological explanations
for better performance by the top performers in one of two
groups of different size, a simple statistical explanation
must be considered. Even if two groups have the same
average (mean) and variability (s.d.), the highest perform-
ing individuals are more likely to come from the larger
group. The greater the difference in size between the two
groups, the greater is the difference to be expected
between the top performers in the two groups. Nothing
about underlying differences between the groups can be
concluded from the preponderance of members of the
larger group at the far ends of the distribution until one
can show that this preponderance is greater than would be
expected on statistical sampling grounds.
It is difficult to quantify how participation rates
influence the number of outstanding men and women in
fields such as science and engineering because both
achievement and participation rates are difficult to
measure. But it is straightforward in chess because there
is an objective measure of achievement and the number of
male and female participants is known. Chess has an
Proc. R. Soc. B (2009) 276, 1161–1165
doi:10.1098/rspb.2008.1576
Published online 23 December 2008
*Author and address for correspondence: Section for
Experimental MR, Department of Neuroradiology, Tu
¨bingen
University, Hoppe-Seyler-Street 3, 72076 Tu
¨bingen, Germany
(merim.bilalic@med.uni-tuebingen.de).
Received 31 October 2008
Accepted 1 December 2008 1161 This journal is q2008 The Royal Society

interval scale, the Elo rating, for measuring skill level.
Every serious player has an Elo rating that is obtained on
the basis of their results against other players of known
rating (see Elo 1986). National federations record the
ratings of all players who take part in competitions. Hence,
it is possible to test whether the difference between the
best male and female performers is any greater than would
be expected given the different numbers of male and
female chess players.
2. MATERIAL AND METHODS
We developed an analytic method to estimate the expected
difference between the top male and female performers based
on the overall male and female participation rates using the
parameters of central tendency (mean) and variability (s.d.)
of the underlying population. A novelty of this method is
that it can be used to estimate the values of the extreme
members of very large samples (such as ours, where
nZ120 399). This method is described in detail in the
appendix A. Our approach is based on the work of Charness &
Gerchak (1996) who showed that the difference between the
top male and female player in the world in 1994 was similar to
that predicted by the relative numbers of male and female
players in the ratings of the international chess federation
(FIDE). A limitation with this study was the statistical
problem that the estimate of the extreme value from a sample
tends to be highly variable (Glickman & Chabris 1996;
Glickman 1999). Also, the world’s top female player at the
time, the Hungarian Judit Polga
¯r, was a phenomenon, by far
the strongest female player the world has ever known. She is
currently ranked 27th of all players in the world but she is the
only female player in the top 100. The fact that the best
female player is an outlier in her population, combined with
the problem of high variability of the extreme value, means
that conclusions drawn on the basis of the performance of the
top player alone may not be applicable to top players in
general. The FIDE rating list used by Charness and Gerchak
only reports players of average strength and above. We used
the German rating list that lists all players. Most importantly,
instead of estimating just the top male and female player, we
estimated the expected performance of the best 100 male and
female performers.
We applied this method to the population of all German
players recorded by the German chess federation (Deutscher
Schachbund). With over 3000 rated tournaments in a year,
the German chess federation is one of the largest and the best
organized national chess federations in the world. Given that
almost all German tournaments are rated, including events
such as club championships, all competitive and most hobby
players in Germany can be found on the rating list. The rating
itself is based on the same assumptions as the Elo rating used
by the international chess federation. The two correlate
highly (rZ0.93).
We considered the players in the list published in April
2007. A small number of the best male and female chess players
from all over the world participate in tournaments rated by the
German chess federation and dominate the top of the rating
list. We have excluded all foreign players from the analyses so
that our conclusions about the expected performance of the
best male and female players, based on the total number and
performance of male and female German players, are only
applied to German players. Figure 1 shows the distribution of
ratings for the German chess population. The distribution is
approximately normal with mean of 1461 and s.d. of 342.
Rated men (113 386) greatly outnumber rated women (7013);
that is, there are 16 male chess players for every woman.
3. RESULTS
Figure 2 shows the real difference in rating for each of the
top 100 pairs of male and female players, and the
difference to be expected for each pair given the much
larger number of male players. The expected superiority of
male players varies from approximately 270 Elo points for
the best male player to approximately 440 Elo points for
the 100th. Figure 2 shows that, in fact, the top three
women are better than would be expected. The next 70
pairs show a small but consistent advantage for men—
their superiority over the corresponding female player is a
little greater than would be expected purely from
the relative numbers of male and female players.
From approximately the 80th pair the advantage shifts.
The female players are slightly better than would be
expected. Averaged over the 100 top players, the expected
male superiority is 341 Elo points and the real one is 353
points. Therefore 96 per cent of the observed difference
between male and female players can be attributed to a
simple statistical fact—the extreme values from a large
sample are likely to be bigger than those from a small one.
4. DISCUSSION
Chess has long been renowned as the intellectual activity
par excellence (Newell et al. 1958) and male dominance at
chess is frequently cited as an example of innate male
intellectual superiority (e.g. Howard 2005;Irwing & Lynn
2005). The reason seems obvious—the best male players
are indisputably better than the best female players. For
example: not a single woman has been world champion;
only 1 per cent of Grandmasters, the best players in the
world, are female; and there is only one woman among
the best 100 players in the world. When considering such
a seemingly convincing example of real world male
superiority, one can easily forget to consider the great
disparity in the number of participants and the statistical
consequences of this for the probable gender of the
best players.
0 500 1000 1500 2000 2500 3000
ratin
g
1000
2000
3000
4000
5000
no. of players
Figure 1. The distribution of the German chess rating
with the best-fit normal curve superimposed. nZ120 399,
mZ1461, sZ342, 16 : 1 men to women ratio.
1162 M. Bilalic
´et al. Gender differences in intellectual domains
Proc. R. Soc. B (2009)

This was the case when the chess portal ChessBase
asked some of the best female players to explain male
dominance in chess (Ahmadov 2007). None of the
interviewed women even mentioned the greatly differing
participation rates and its consequences on the probable
gender of the top performers. Similarly, at a recent
gathering of more than 20 experts on gender difference
to discuss the reasons for the paucity of women at the top
of science, a broad range of reasons was discussed, but
there was no mention of participation rates (Ceci &
Williams 2007).
One way to avoid the conclusion based on participation
rates would be to argue that the base participation rate for
womenusedinthisstudyunderestimatesthereal
participation rate. It is possible that there is a self-selection
process based on the innate biological differences in
intellectual abilities, and that the effects of this self-
selection are already observable in the rating list we used.
Women may be inferior in the intellectual abilities that are
important for successful chess playing. This innate
disadvantage may lead women to give up on chess in
greater numbers than more successful men. The small
number of women is then a consequence of their greater
drop-out, which in turn is produced by their innate lack of
the intellectual abilities required to succeed at chess.
Differential participation rates may explain the discre-
pancy at the top, but the difference is itself a direct product
of innate differences in intellectual abilities.
This argument sounds reasonable but it rests on a
controversial assumption. It requires that there should be
innate differences between men and women in the
intellectual abilities required for success at chess.
The topic of gender differences in cognitive abilities is
a hotly debated one, which lacks conclusive evidence
(for example, Geary 1998;Kimura 1999;Kerkman et al.
2000;Pinker 2002;Spelke 2005;Summers 2005;
Lachance & Mazzocco 2006;Ceci & Williams 2007).
Even if such differences exist, it is unclear which, if
any, intellectual abilities are associated with chess skill
(for a recent review, see Bilalic
´et al. 2007). Whatever the
final resolution of these debates, there is little empirical
evidence to support the hypothesis of differential drop-out
rates between male and females. A recent study of 647
young chess players, matched for initial skill, age and
initial activity found that drop-out rates for boys and girls
were similar (Chabris & Glickman 2006). Our study does
not deal directly with the reasons why there are so few
women in competitive chess. These may have to do with
selective drop-out before tournament play starts in the
early stages of learning to play chess. We can speculate
about the reasons for low participation rates of women
in competitive intellectual endeavours (as is often done,
e.g. Steele 1997;Benbow et al. 2000;Kerkman et al. 2000;
Massa et al. 2005;Spelke 2005;Summers 2005;
Lachance & Mazzocco 2006;Andreescu et al. 2008) but
empirical evidence is scarce.
This study demonstrates that the great discrepancy in
the top performance of male and female chess players can
be largely attributed to a simple statistical fact—more
extreme values are found in larger populations. Once
participation rates of men and women are controlled for,
there is little left for biological, environmental, cultural or
other factors to explain. This simple statistical fact is often
overlooked by both laypeople and experts. In other
domains such as science and engineering, where the
predominance of men at the top is offered as evidence of
the biological superiority of men, large differences
between the number of women and men engaged in
these activities are evident (Long 2001;Xie & Shauman
2003). In these areas of life, it is not possible to estimate
the performance of the top women and men and their
participation rates as precisely as it is in chess. But until
0
50
100
150
200
250
300
350
400
450
500
1 10192837465564738291100
no. of
p
air
rating difference
Figure 2. The differences between the real ratings of the best 100 female and male chess players and the differences expected on
the basis of the common distribution of male and female ratings and the number of male and female players. The expected
difference was obtained by subtracting the estimated rating for the nth female from the estimated rating for the nth male. The
ratings were estimated using the participation rates of men and women and the parameters of their shared population (mean and
s.d.). Black triangles, expected differences; white squares, real differences.
Gender differences in intellectual domains M. Bilalic
´et al. 1163
Proc. R. Soc. B (2009)

the effect of participation rates has been allowed for, the
greater number of men among the most successful people
should not be cited as evidence of innate differences
between male and female intellectual abilities.
We are grateful to Frank Hoppe for providing us with the
German database and Eric-Jan Wagenmakers for his
comments on an earlier draft of the paper. Supported by an
ESRC Post-doctoral Fellowship to M.B.
APPENDIX A
A.1 The expected difference between the kth
best male and female players
Standard methods (e.g. David & Nagaraja 2003) allow
exact calculation of the expected score of the kth best
player, but this proves numerically difficult for large
sample sizes (such as ours, where nZ120 399). Here we
use the size of the population to our advantage, deriving a
simple approximation valid for large n.
Suppose we have a population of independent,
identically distributed variables; in our case, the sample
is all ratings from the German chess federations for both
genders. We sort and relabel the sample such that
X
n,n
%/%X
n,1
; that is, the notation X
n,k
denotes the
kth best score from the sample, which has size n.For
example, X
n,1
and X
n,n
denote the best and worst ratings,
respectively, in our sample. Now defining E
n,k
ZE(X
n,k
),
the expected value (or mean) of X
n,k
, we have the
recurrence relationship (Harter 1961)
En;kC1Z1
kðnEnK1;kKðnKkÞEn;kÞ;
which is equivalent to
En;kC1Z
n!
k!ðnKkK1Þ!X
n
mZnKk
ðK1ÞkCmKnk
nKm

Em;1
m:
Thus we see that the expected value of the (kC1)th
best performer of a sample size nmay be written as a sum
of terms of the form E
m,1
—the best performer from a
sample of size m—where the indexing variable takes values
between nKkand n.
Suppose that X
i
wN(0,1) are drawn from a standard
normal distribution. If the sample size is large, then the
expected value of the best performer increases linearly
with the logarithm of sample size, i.e. E
m,1
zc
1
Cc
2
ln m,
where c
1
Z1.25 and c
2
Z0.287 (figure 3).
If nKkis large, then each value taken by min the
previous sum will be large. Hence, we may apply the log-
linear approximation to each occurrence of E
m,1
, obtaining
En;kC1zc1Cc2
n!
k!ðnKkK1Þ!X
n
mZnKk
ðK1ÞkCmKnk
nKm
!lnm
mZc1Cc2
n!
k!ðnKkK1Þ!ðK1ÞkVkfðnÞ:
Here, f(x)Zln x/xand VgðxÞZgðxÞKgðxK1Þdenotes
the backward difference operator (Abramowitz & Stegun
1972). Furthermore, the backward difference operator is a
close approximation of the differential operator, and hence
VkfðxÞz
dk
dxkfðxÞZðK1Þkk!ln xKHðkÞ
xkC1;
where HðkÞZPk
jZ1jK1denotes kth harmonic number
and we define H(0)Z0. Substituting, we find that the
expected value of the kth best performer is given by
En;kzc1Cc2
n!
ðnKkÞ!nkðln nKHðkK1ÞÞ:
This holds if X
i
wN(0,1) are drawn from a standard
normal distribution. Moving to a general normal
distribution, if X
i
wN(m,s
2
)ZmCsN(0,1), we find
En;kzðmCc1sÞCc2sn!
ðnKkÞ!nkðln nKHðkK1ÞÞ:
Given a distribution with known mean mand s.d. d,this
final formula defines the expectation of the kth highest value
within a sample of size n, valid provided nis large and k is
relatively small. As such, it affords us a method for estimating
the expected rating of a range of top players from the
German chess data for each gender; indeed, we use the
formula to calculate the expected ratings of the top 100 male
and female players using the mean and s.d. of the population
(the German chess data), in turn allowing us to determine
the expected difference in rating between those players.
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