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FORUM ARTICLE
Kouroi and Statistics
JANE
B. CARTER AND LAURA
J.
STEINBERG
Abstract
In a well-known series of articles,
Eleanor Guralnick
undertook statistical studies to compare
the proportions
of Greek (^) archaic kouroi with one (^) another and with the
proportions
of the Egyptian
second (^) canon; she concluded
that Greek sculptors
used the Egyptian
canon sporadically
for proportioning
kouroi during
most of the sixth century
B.C.E. (^) Here, we^ examine the results of Guralnick's analy
ses against
the backdrop
of current statistical method.
While we do not believe that her analyses convincingly
demonstrate any
Greek use of the Egyptian system,
we
agree
that the analyses
do distribute the kouroi included
in the studies into two main groups.
We argue
that this
division results from the influence of regional styles,
rather
than from the use of standardized proportional systems.
We also examine Guralnick's methodology
in (^) cluster,
principal components,^
and z-score^ analyses
and dem
onstrate that her studies do not provide statistically sig
nificant evidence for similarities among
Greek (^) kouroi or
between kouroi and the Egyptian canon,^
in part
because
of the limitations of the statistical techniques employed
and in part
because of problems
in her procedures
and
data. (^) Thus, we^ disassociate archaic Greek kouroi from a
dependence
on the Egyptian
standardized proportional
schemes and argue
instead that (^) the development
of re
gional styles
best explains
the proportional
similarities
documented by
Guralnick.*
INTRODUCTION
The
quest
to establish that
early
Greek statues used
the
Egyptian system
for
proportioning
human
figures
stems from a combination of two factors:
the
ap
proximate synchronism
of the earliest Greek statues in
stone and the resumption
of direct contacts between
Greece and
Egypt
around themiddle of the seventh
century B.C.E.;^
and (2)
the (^) visual resemblance be
tween Greek kouroi and Egyptian
statues. Standing
male
figures
in both Greece and
Egypt
face forward,
hold their arms
alongside
their
thighs,
and advance
the left
leg.
These factors have
suggested
thatGreeks
learned
sculptural techniques
from the
Egyptians;
ifso,
one could
expect
tofind
Egyptian proportions
in
early
Greek statues.^ Eleanor Guralnick's statistical studies of
kouroi and korai (^) are, in part,
an attempt
to demon
strate this hypothesis.
Guralnick concluded that "at
least
through
the third
quarter
of the sixth
century
Greek sculptors
made (^) conscious use (^) of the contempo
raryEgyptian
canon without
major
modification."2 For
the last 30
years,
Guralnick's studies have contributed
significantly
to a more or less orthodox view about the
Egyptian origins
ofGreek
sculpture.3However,^
a
strong
argument
can (^) be made that in the first instance,
the
Greeks
adopted techniques
and
sculptural types^
from
regions
in (^) the eastern Mediterranean, in^ particular
from
Syria-Palestine.
That
argument
iswell
beyond
the
scope
of this
paper.
Our
purpose
here is to show that
while Guralnick's articles contain much that is valuable
for the
study
of kouroi, theydo not in factdemonstrate
the likelihood that
early
Greek
sculptors
of kouroi used
an
Egyptian system
of
proportions. Thus,
we
hope
to
open
the way
for new discussion about (^) archaic Greek
statues (^) and their origins.
The
underlying premise
ofGuralnick's studies is that
archaic Greek sculptors
used one or more (^) standard
ized proportioning schemes,^
and her numerous^ articles
have served to reinforce this idea. We,
on (^) the contrary,
doubt that
early
Greek statues
embody any
formal
sys
tem of proportions.
Such proportional
similarities as
do exist among
kouroi are^ best explained,
we believe,
by
the evolution of
regional styles
and a consistent
type
of idealization embodied
by virtually
all kouroi.
We are^ grateful
for the helpful
comments of the two
anonymous
reviewers for the A]A.
We would also like to thank
Kapon Editions,^
Nikolaos Kaltsas (^) (National Archaeological
Museum of Athens),
Daria Lanzuolo (DAI Rome), (^) Joachim
Heiden (DAI Athens), (^) MatthewWesterby (Metropolitan
Mu
seum of Art),
Irene B?sel (Staatliche (^) Antikensammlungen
und Glyptothek
in Munich),
and Gay
Robins for their assis
tance and permission
to use^ the images reproduced
here.
'Guralnick 1976, 1978, 1981, 1982, 1985, 1996a, 1996b,
2Guralnick (^) 1985,409.
E.g.,
Osborne (1996, 209-11, (^) 371) cites^ only Guralnick
to support
his statement^ that the "size and proportions [of
kouroi] make^ it clear^ beyond
doubt that they
were directly
inspired by Egyptian
stone sculpture."
Hurwit (2007,274,
n.
- likewise^ cites^ only Guralnick^
as (^) evidence that the kou
ros type
was (^) invented "after Greeks had been exposed
... (^) to
Egyptian techniques";
cf. the newest edition of Pedley's (2007,
- (^) widely
used textbook: "computer
studies have now^ con
firmed the closeness of proportions
between the earliest kou
roi and Egyptian figurines."
4Cf. Boardman 2006,12,19-24.
American
Journal ofArchaeology
J.B.
CARTER AND
LJ.
STEINBERG
[AJA
We have (^) not, therefore, (^) attempted
new measure
ments or pursued
fresh statistical approaches
with the
aim of^ identifying
one or more proportional
schemes
in Greek kouroi. In our^ opinion,
the evidence pub
lished
by
Guralnick shows
convincingly
that archaic
Greek statues of men^ and women^ exhibit a varied
range
of proportions
and offers an^ explanation
for
the
proportions
of kouroi that does not
depend
on
abstract proportional systems.
The
Egyptian system
has been documented almost
entirely
in two-dimensional works rather than in
three-dimensional statues.^ Egyptian
artists frequently
employed guidelines
for their
preliminary
sketches
of human
figures,
and traces of their
guidelines
are
sometimes
preserved
in unfinished reliefs or under
painted
surfaces. From this evidence,
Egyptologists
have shown that
Egyptian
artists in a
given period
con
sistentlyplaced
certain
parts
of the human
anatomy
at
fixed
points, therebyproducing figures
with
fairly
uni
form
proportions, regardless
of their scale.
Surviving
guidelines
show that a new^ system
came into use in the
25th
Dynasty (i.e.,
the late
eighth century
or
early
sev
enth century B.C.E.).^
The new^ system, (^) usually
called
the second
canon,
was inuse
during
the 26th
Dynasty
(664-525 B.C.E.),^ when, judging
from
archaeological
and textual sources, Greeks
began
to visit
Egypt again
for the firsttime since the end of the Bronze
Age.
At
Theban Tomb
223, from the 26th^ Dynasty, preserves
a
good example
of the second canon^
grid
used to draw
a
standing
male
figure (fig. 1).
From the baseline, the
top
of the
eye
is at 21 units,
themouth is at 20 units,
the
junction
of neck and shoulder is at 19 units,
the
nipples
are at 16 units, the navel
(or
the small of the
back)
is at 13 units, the bottom of the^ buttocks^ isat 11
units, the top of the knee
is at 7 units, and the bottom
of the knee is at 6 units.
Guralnick's studies
apply
three different statisti
cal methods. We emphasize
that these studies con
tain valuable^ information for our
understanding
of
Greek kouroi and korai. At the same^ time, statistical
theory
has progressed considerably
in (^) the last quarter
century,
and important
new (^) work has been done on
Egyptian proportioning techniques.^
First we look at
what Guralnick's statistical analyses
tell us^ about kou
roi, thenwe
examine some
significant problems
with
her statistical
methodology
and her data.
Fig.
- An Egyptian standing
male figure
with origi
nal grid
lines from Tomb 223 at Thebes, 26th^ Dy
nasty.
This illustrates the so-called second canon or
later
grid (drawingby
A. Fowler; Robins 1994, 161,
fig. 7.2).
GURALNICK S RESULTS FOR KOUROI
In her initial studies of kouroi, Guralnick
employed
cluster
analysis
to find similarities between the
Egyp
tian canon and a
group
of 24 Greek kouroi.9 For the
cluster analyses,
Guralnick relied primarily
on two
similar data sets
(A
and
B)
ofmeasured dimensions.
Each dimension of each figure
was expressed
in terms
of its
proportion
to the distance between the
eyes
and
knee-tops
of the figure;
the ratios permitted compari
sons of
proportions among^ figures
of different sizes.
Table 1 shows Guralnick's results when the statistical
software sorted the
Egyptian
second canon, 17 Greek
kouroi, and^ the^ average dimensions^ of Greek, Turk
5Robins (^) 1994,160.
6 Robins (^) (1994, 228, 258-59) shows^ clearly
that the guide
lines used by Egyptian
artists did not^ always
determine the
proportions
of their figures. E.g.,
the same^ grid system
was
used for the (^) taller, more^ slender figures
of the 19th and 20th
Dynasties
as had been used for the shorter and stockier fig
ures (^) of the 5th and 6th Dynasties.
Robins thus correctly rejects
the use^ of the term "canon" in reference to the Egyptian grids.
Because "canon" is generally
used for the Egyptian system,
however, for convenience^
we (^) do so here.
7Hdt. (^) 2.152-54; Boardman 1999,111-53.
8Robins (^) 1994,160-61, fig.
Guralnick 1976,1978.
10 Guralnick (^) 1978, 464, fig. 2;^ 1996a,^ 41,^ fig.
4.1. Data set A
comprised
11 dimensions: top
of head to eyes, eyes
to chin,
chin to sternal (^) notch, sternal notch to nipples, nipples^
to na
J.B.
CARTER AND
LJ.
STEINBERG
[AJA
As the number of clusters is further (^) decreased, some
of the clusters must contain more and more^ objects.
In table
1, when
the
objects
were sorted into
only
three (^) clusters, cluster^ F^ contained^10 objects,
cluster
E had 2
objects (the
"twins"Kleobis and Biton), and
cluster A had 9
objects.
At a^ late
stage
in the
analysis,
the
degree
of
similarity among
the
objects
in each
cluster
may
be
quite
low. For
example,
with
only
four
clusters, the^
Dermys
and
Kitylos pair
was
assigned
to
cluster F, although
the
proportions
of
Dermys
and
Kitylos probably
have little in common with
any
of the
kouroi
assigned
to cluster F at that
stage.
In this
analysis,
no kouros clustered with the
Egyp
tian canon until the number of clusters was reduced
to 11,
about halfway
between the maximum number
of 21 clusters and theminimum of 1 cluster
(see
table
In the firsthalf of the
analysis?from
21 clusters to
12 clusters?the^
Egyptian
canon was the
only object
in its cluster.With 11 clusters, the New York kouros
and Ptoon 12 kouros were
assigned
to cluster F with
the
Egyptian
canon. At the next
stage?with
10 clus
ters?the Tenea kouros joined
cluster F. When the
clusters were reduced to 8,
the Thebes 3 kouros en
tered cluster (^) F, became again
the sole member of its
group
with 7 clusters, then
rejoined
cluster F with the
decrease to 6 clusters.With
only
5 clusters, theMelos
kouros and the Volomandra kouros also grouped
with
the
Egyptian
canon in cluster F.
The similarities indicated
by
the clusters are relative,
not absolute. In
general, objects
that
group together
when there are many
clusters are more^ alike than
objects grouped together
when the number of clus
ters is small.
Objects very^
different from one^ another
could be forced to
group together
as the number of
clusters is decreased. At no^ point,
however, does^ the
analysis
indicate the
degree
of
similarityamong^
clus
tered objects.
It is also necessary
to bear in mind that
the choice^ of variables included^ in the data^ set can
have considerable consequences
in the results. For
example,
with 21
objects
and data setA,
the Ptoon 12
kouros first
grouped
with the
Egyptian
canon when
the number of clusters was^ reduced to 11
(see
table
1). However,^ using
the same 21
objects
and data setB,
the Ptoon 12 kouros first
grouped
with the
Egyptian
canon much earlier,
with 16 clusters.13 In this (^) case, the
degree
to which the Ptoon 12 kouros
appears
to be
proportionally
related to the
Egyptian
canon
depends
on which variables are included. Cluster
analysis
does
not have an
underlying
theoretical model that
permits
the claim of
statistically significant similarity among
grouped objects.
In the
analysis
with 17 kouroi
using
data setA,
kouroi
(New York,^
Ptoon 12, and Tenea)
clustered
with the
Egyptian
canon when there were 10 clusters
(see
table
With the same^ 17 kouroi and data set
B,
the cluster
analysis grouped together
the
Egyptian
canon, these^3 kouroi, and^ the Melos^ kouros^ when
there were^11 clusters.15 Guralnick concluded that
these 4 kouroi
(New York, Melos,^ Tenea,^
and Ptoon
12) "closely
resemble the
Egyptian
Second Canon in
their
proportions
from the
eyes
to the
top
of the knees"
(fig.2).
In fact,we do not
actually
know how close the
resemblance isbetween these kouroi and the
Egyptian
canon.17 The analyses
have only
shown that,
in rela
tive (^) terms, these 4 kouroi seem to be more similar to
the
Egyptian
canon than the other 13 kouroi in terms
of the data sets used. There is no basis for
claiming
statistically significant similarityamong^ objects
in
any
cluster. Indeed,
the cluster analyses
indicate that the
Ptoon 12, Tenea,
and Melos kouroi are more similar
to one another than
any
kouros is to the
Egyptian
canon
(in
terms of data setA,
see table
1), because
these statues are
already grouped
with one^ another in
cluster B when there are 17 clusters,
six stages
before
any
kouros
groups
with the
Egyptian
canon.
Moreover,
it is important
to (^) remember that even if
some method could establish
significant proportional
similarity
between the
Egyptian
canon and certain
Greek kouroi,
different explanations
could be pro
posed
for the
similarity.
One
hypothesis, put
forward
byGuralnick,^
could be thatGreek
sculptors
sometimes
used the
Egyptian
canon. We
suggest
that there is a
Guralnick (^) 1978,465-66, figs. 3,4. 14
According
to the Panel on Discriminant Analysis,
Classi
fication,
and Clustering (1989,^ 35),^
the nature and composi
tion of the clusters "appear
to cause fundamental difficulties
for formal statistical inference and distribution theory";
see
also Punj
and Stewart (^) 1983,136; Norusis (^) 1985,183; Shennan
Guralnick (^) 1978, 466, fig.
- Guralnick (1978, 467,^ fig. 5;
468-69) again
used data set A^ in a^ cluster^ analysis
with a to
tal of 28 objects:
24 kouroi plus
the Egyptian
canon and the
Greek, Turkish,
and Italian men. In this (^) case, the Ptoon 12
and the New York^ kouroi^ clustered^ with^ the Egyptian
canon
when the objects
were (^) sorted into 14 and 13 clusters, (^) respec
tively.
Guralnick (^) (1978, fig. 5)^
does not show the results when
the number of clusters was^ greater
than 14. However, assum
ing
that the chart^ documents^ the highest
number of clusters
at which a kouros and the Egyptian
canon (^) clustered together,
then here, also,
the New York and Ptoon 12 kouroi joined
the
Egyptian
canon halfway
between the maximum number of 28
clusters and the minimum of 1 cluster.
Guralnick (^) 1978, 466.
17 As Guralnick (1978, 472) acknowledges,
the cluster and
principal components analyses
"cannot determine...^ how
closely
the actual measurements of the statues conform to the
proportional
schemes."
2010] KOUROI^ AND^ STATISTICS^
'4*
"
a b c^ d
Fig.
- Kouroi considered by
Guralnick to be proportionally
close to the Egyptian
second canon: (^) a, New York (^) kouros, probably
from
Attica, Naxian^ marble,
ca. 600-590 B.C.E. (?
The Metropolitan
Museum of Art; Fletcher
Fund 1932,32.11.1); b,
Melos (^) kouros, found on
Melos in 1891, Naxian (^) marble, ca. 550 B.C.E. (^) Athens, National^ Archaeological Museum,^
inv. no. 1558 (C.
Iosifidis and G. Moutevellis;
Kaltsas (^) 2002, cat. no.^ 48); c,Tenea (^) kouros, found in cemetery
of ancientTenea in 1846, Parian^ marble, ca. 550 B.C.E.^ Munich, Staadiche
Antikensammlungen
und Glyptothek,
inv. no.^ GL (^168) (H. Koppermann;
? Staatliche Antikensammlungen
und Glyptothek,
M?nchen); d, Ptoon
12 kouros, from the Sanctuary
of Apollo
at Ptoon in Boeotia, island^ marble,
ca. (^) 530-520 B.C.E. Athens,
National Archaeological
Museum, inv. no.^12 (G. Fafalis; Kaltsas^ 2002,
cat. no. 80).
more
probable explanation
for
proportional
similari
ties that
may
exist between the
Egyptian
canon and
certain Greek kouroi.
Guralnick tested her conclusion that four kouroi
?New York, Melos, Tenea,
and Ptoon 12 (see (^) fig.
2)?"closely
resemble" the
Egyptian
canon
by using
a
technique
known as
principal components analysis
with a
larger group
of 24 kouroi.18 Guralnick illustrat
ed these resultswith two
graphs
inwhich each
object
(kouroi,
second (^) canon, and Greek, Turkish,
and Ital
ianmen)
is
represented by
a
point;
circles are drawn
around the points
that seem^ to^ form^ groups.
Gur
alnick found agreement
between the^ cluster^ analyses
and the
principal components^ analysies
"in all essential
conclusions";
in particular,
she claims^ that^ the^ prin
cipal components^ analyses
validate "the existence of
a
group
of statues whose
proportions
are like those
of the
Egyptian
canon."20 This
group
consists of the
four kouroi thatwere most
closely
associated with the
Egyptian
canon
by
the cluster
analyses (New^
York, Me
los,Tenea, and Ptoon^ 12) (see fig. 2) plus
the Thera
kouros
(fig. 3). Guralnick
adds that threemore^ kou
roi?Florence, Ptoon^ 10, and^ Volomandra?are^ also
similar in
proportions
based on^ the
principal compo
nents analyses.
Guralnick's
graphs illustrating
the
principal compo
nents analyses
do not appear
to support
these claims.
In fact,
there are a number of striking
anomalies
Guralnick (^) 1978,469.
19 Guralnick (^) 1978, 470, fig.
(using
all the variables in
her data set A); 471, fig.
(using
the seven^ variables in data
set A that were most alike in the 24 kouroi and the Egyptian
canon).
20Guralnickl978,469.
2010]
KOUROI AND STATISTICS 109
canon
(NewYork, Melos, Tenea,
Ptoon
12, and Thera)
(see figs. 2, 3)
do not
represent
either a
chronological
or a
geographical group^
as one
might
have
expected
them to do.26 (^) Instead, the five statues span
most (^) of
the sixth
century,
with theNew York kouros
B.C.E.)
and the Ptoon 12 kouros
(530-520 B.C.E.)
at
opposite
ends of this time
span.
The
geographical
distribution of the five statues^ is
equally
wide: one
from Athens
(the New York
kouros),
two from the
Cyclades (Thera
and
Melos),
one from near Corinth
(Tenea),
and one found at the Ptoon
sanctuary
in
Boeotia
(Ptoon 12). By contrast, the
New York kou
ros and the Sounion
kouros, both made
inAttica and
probably
within a decade of each other, did not cluster
together using
data setA even at the lowest threshold
of
similarity
when the statues were divided into three
clusters (see
table l).
Guralnick first
explains
this
seemingly
random dis
tribution of the five kouroi in
space
and time as^ the
result of
sporadic tripsby
Greek
sculptors
to
study
in
Egypt; only sculptors
"who had learned the
system
under a master would apply
it."29 This explanation
does not fitwell with what we^ know of Greek kouroi
in
general. During
much of the sixth
century,
for ex
ample,
kouroi that are from the same region
and ap
proximately contemporary^
are (^) also visually
similar.
Apparently, sculptors working
near one another did
share technical and aesthetic preferences.
Guralnick
argues
that similar treatments of the surfaces of statues
may
obscure the
underlying proportional
differences
revealed by
her statistical studies.30 (^) However, there is
no obvious reason that
sculptors working
in the same
region
would
intentionally produce
statues thatwere
superficially
similar but
quite
different
proportionally.
Regional
similarities should
apply equally
to both un
derlying proportions
and surface treatment. If visually
similar statues
produced
in the same
region
within a
time frame of 10 or 15
years
do not have similar
pro
portions,
the
simplest explanation
is that their
sculp
tors (^) did not use a (^) standardized proportional system.
Guralnick also
suggests
that
pattern
books
might
have circulated
among
archaic
sculptors
and
might
have contained "copies
of or adaptations
from" the
Egyptian system.
There isno evidence that
sculptors
used
pattern
books in theArchaic
period
or later.We
know of no (^) treatise or (^) technical work about statues
until the Canon of
Polykleitos
in the second half of the
fifth century B.C.E.,^
and a book by
one sculptor
about
his own work is not the same
thing
as a
pattern
book
with a
variety
of
proportional systems.
If such
pattern
books had been
widely
used
by early
Greek
sculptors,
we would
expect
a mixture of different
styles,
both
visually
and proportionally,
to appear contemporane
ously
in various regions.
Instead, regional sculptural
styles
dominate at^ least
through
the firsthalf of the
sixth century.
If neither
sporadic study trips
to
Egypt by
Greek
sculptors
nor pattern
books (^) convince, how can^ we
explain
some
proportional similarity
between the five
kouroi
(see figs. 2, 3)
and the
Egyptian
canon? Let us
repeat
that cluster
analyses
do not indicate the
degree
of
similarityamong^ objects
clustered
together.
Let us
also remember that these kouroi did not all
group
with the
Egyptian
canon until far
along
in the
analy
and the Strangford
kouros consistently
form one group
for
all the stages represented
in the graph?
clusters down to
5 clusters (^) (Guralnick 1978, 467, fig. 5).^
The (^) Greeks, Turks,
and Italians only group^
with the Egyptian
canon in the cluster
analysis using
data set B and only
when the number of clusters
is reduced to 3 (Guralnick 1978,^ 466,^ fig. 4).
26 Guralnick (^) 1978, 470. Illustrations and extensive bibliog
raphy
for these kouroi can be found as follows. New York (Met
ropolitan
Museum of Art,
inv. no. 32.11.1,
ht. 1. m):
Richter
no. 1, figs. 25-32, 60-2;
Boardman 1978, (^) fig. 63;
Floren 1987, 252
n. 6; Stewart^ 1990, 108-9, 111-12, (^) figs. 49
55; Vorster^ 2002,120-22,304, (^) fig. 190a-e.^ Thera^ (Athens, Na
tional Archaeological
Museum, inv. no. 8, preserved ht. 1.
m):
Richter 1970,69-70,
no. 49, figs. 178-83; Boardman^ 1978,
fig. 101; Floren 1987, 177 n. 7, pi. 13.2;Kreikenbom 2002,
fig. 218a-d;^
Kaltsas 2002,41,
no. (^) 22. Melos (Ath
ens, National^ Archaeological Museum,^
inv. no. 1558,
ht. (^) 2.
m): Richter^ 1970,96-7,
no. 86, figs. 273-79; Boardman^ 1978,
fig.
102; Floren^ 1987,178 n.^ 5, pi. 13.3;^
Stewart 1990,119, (^) fig.
Karanastassis 2002,180-81,312, (^) fig. 253a, b;
Kaltsas 2002,
no. 48. Tenea (Munich, Staatliche^ Antikensammlungen
und Glyptothek,
inv. no.^ 168, ht. 1. m):
Richter 1970,84-5,
no. 73, figs. 245-50;
Boardman^ 1978, fig.
121; Stewart^ 1986;
Floren 1987,
n. 12, pis. 14.2,15.1; Karanastassis^ 2002,
85,313, (^) fig. 262a-d.^ Ptoon^12 (Athens, National^ Archaeologi
cal Museum, inv. no.^ 12, preserved ht. 1. m):
Richter 1970,
122-23, no.^ 145, figs.
425-29, (^) 437; Ducat^ 1971, 346-51, no.
197, pis. 112-14;
Boardman 1978, (^) fig. 179;
Floren 1987,
n.
Stewart 1990,126, (^) fig. 170; Maderna-Lauter 2002, 230-32,
319, (^) fig. 309a-d;
Kaltsas 2002,62,
no. 80.
For the sake of consistency,
dates of statues^ are^ taken
from Boardman (1978) whenever^ possible.
28 Data set A: Guralnick (^) 1978, 465, fig.
fig.
- With
data set B, 9 out of 17 kouroi, including
the New York and the
Sounion (^) kouroi, did cluster with the Egyptian
canon when
the number of clusters was reduced to 3 (Guralnick 1978,466,
fig. 4). 29 Guralnick 1978,471.
30 Guralnick 1978,469.
31 Guralnick (^) 1978,471.
32 Gal. DeplacitisHippocratisetPlatonisb;
Plin. iW34.55. Sim
ilarly, according
toVitruvius (De
arch. 7, pref. 12),
in the sixth
century B.C.E.,^
the architects Rhoikos and Theodoros wrote a
book about their Temple
of Hera on Samos,
and Chersiphron
and Metagenes
likewise wrote (^) about their Temple
of Artemis
at Ephesos.
J.B.
CARTER AND
LJ.
STEINBERG
[AJA
ses
(see
table
l),
which
suggests
that no
very strong
proportional similaritydoes,^
in
fact, exist among these
kouroi and the
Egyptian
canon.
Still, the combined
results of the (^) cluster and principal components^ analy
ses do
appear
to
identify
some
degree
of
proportional
similarity among
these
objects,
and there must be a
reason for this.
We cannot tellwhich
proportioned
dimensions in
the data sets caused the cluster and
principal
com
ponents analyses
to
group
these
objects together.
In
1985, Guralnick^ used^
a thirdmethod,
z-score
profile
analysis (fig.4)^ ,
to examine the
proportions
of kou
roi. What the z-scores offer, and^
the (^) other methods
do
not,
is a visualization of the
proportional
dimen
sions. Using
the (^) z-scores, we^ are^ able to suggest
what
factors caused the earlier analyses
to produce
the
groups they
did.
Z-scores show how an object
relates to^ the average
(mean)
of its class; the z-score^ is the distance of the
object
from the mean, measured in standard devia
tions, for each proportioned variable.^
In
figure 4, the
straight
horizontal (^) axis, centered at^ zero, represents
the
proportioned
dimensions of the
average
Greek
man,
and the vertical y-axis
of the chart is marked
in
plus
and minus standard deviations
(SD)
from this
average
man. If human males represent
a statistically
"normal" (^) distribution, as^ is assumed, then the propor
tions of 68% of all human males will be between +
and -1 SD from the
average,
95% will be between +
and -2 SD, and 99.7% will be between +3 and -3 SD.
Basically, then, proportions
that fall outside the area
of+1 to -1 SD
begin
to be unusual,
and a
proportion
that falls outside +3 to -3 SD can be
expected
to occur
only
once in 357 men. The mean and standard devia
tions for the average
man as computed by
Guralnick
are not
statistically comparable
to the
proportions
of
the kouroi (see below,
under "Z-Score Profiles").
How
ever, the^ average
man used by
Guralnick does^ provide
a fixed
paradigm against
which other
objects (Greek
statues (^) and the Egyptian canon)^
are compared,
and
this
provides
useful information about how the
objects
relate to one another.
The
proportioned
dimensions that are
being
com
pared
to the average
man (^) are located at fixed points
on the horizontal x-axis and labeled at the bottom of
the
graph (see fig. 4). Again,
Guralnick has consid
ered the measurements^ as^ ratios (each
dimension of
an
object
is divided
by
its
height
from the
top
of the
knee to the
top
of the
head).
For each
proportioned
dimension,
a
point
is charted at the
appropriate place
for that dimension on (^) the x-axis and at a point
on (^) the
vertical y-axis showing
its distance from the average
man. (^) The distance on the vertical axis ismeasured in
units that should
correspond
to the SD for that
pro
portional
dimension in the
population
of all men.
For
example,
in
figure 4,^
the
proportional
shoulder
width of theNew York kouros is charted at a little less
than +1^ SD from the average
man. Thus,
for that pro
portional
dimension, the New^ York^ kouros^ should
be within 68% of a normal
population
of men.^ The
charted points
are then connected, creating
the ir
regular zigzag profile.
The chart
(see fig. 4)
shows that two kouroi
(New
York and Ptoon
and the
Egyptian
canon
have,
proportionally,
shoulders that are a littlewider than
those of^ the average
man (^) and very
slim waists. Three
vertical dimensions?knee to navel,
knee to nipples,
and knee to sternum?in both kouroi and (^) the Egyp
tian canon (^) are, proportionally, quite
similar to^ those
of the
average
man and to one
another; all are
slightly
less than those of an^ average
man (^) and are within less
than 0.9 SD from one another. The head of the Ptoon
12 kouros is somewhat on the
large side, and
the head
of theNew York kouros is
enormously large.
The
similarity
in the
proportions
of the five kou
roi that Guralnick associated most
closely
with the
Egyptian
canon can be seen
clearly
in a column chart
(fig. 5). Again,
the
average
man is
represented by
the horizontal axis. All five statues and the
Egyptian
canon have
slightly
wide shoulders
(except
theMelos
kouros), (^) very
narrow waists,
narrow hips,
and vertical
dimensions from knee to^ sternum^ that are^ quite
close
to those of the
average
man. The head
heights
tend
to be greater
than the average
man's.
In (^) fact, as^ Guralnick's z-score^ profiles show,^
most
kouroi follow this
pattern; they
have shoulders thatare
proportionately quite
broad in relation to the waists
and
hips,
and
they
have vertical dimensions from knee
to sternum that are
proportionately quite
similar to
one another and to those of the average
man. (^) Since all
kouroi seem^ to have average (^) proportions
from knee to
sternum that are close to the proportions
of an aver
Using
data setA with 21
objects (not includingThera),
the 4 kouroi grouped
with the Egyptian
canon when the clus
ters were reduced to 8, about
two-thirds of the way (^) through
the analysis (Guralnick^ 1978,^ 465,^ fig. 3).
With data set^ A and
objects,
the 5 kouroi did not all
group
with the
Egyptian
canon (^) until the analysis
was (^) extended by extrapolation
to 5
clusters (Guralnick 1978,467, (^) fig. 5). Using
data set B with (^21)
objects (not includingThera),^
the4 kouroi
grouped
with the
Egyptian
canon when the clusters were (^) reduced from 21 to 11
(Guralnick 1978,466, (^) fig. 4).
34 Guralnick's z-score^ charts place
the anatomical dimen
sions on^ the vertical axis and the standard deviations on the
horizontal axis;
the orientation is reversed here.
Guralnick 1982,
n. 7; 1985,400.
J.B.
CARTER AND
L.J.
STEINBERG
[AJA
age man,^
it is primarily
the non^ average (^) proportions
that differentiate the kouroi and are most influential
in grouping
certain kouroi as proportionally
similar.
One important
non average
factor is the (^) difference
between the
proportional
width of the shoulders and
the
proportional
width of the waist relative to these
dimensions for the average
man. (^) In these five kouroi,
the shoulders are a littlebroader than
average (except
Melos),
while thewaists are a
good
deal more^ slender
than average.
Other kouroi show a^ strong proportional
exaggeration
of the width of the shoulders as well as
the slimness of the waist. The Sounion and Munich
kouroi, for
example,
are alike in
having very
broad
shoulders
and +2.
SD, respectively) and very
narrow (^) waists (-2.04 SD^ for^ both) ,36In
a group
of (^) late
kouroi
(Ptoon 20,Aristodikos, Strangford,
and theKri
tios
Boy),
a lesser but stillnotable difference between
shoulders and waist is realized
by
shoulders that are
proportionally
broader (^) than average
and waists that
are narrower (^) than average
but with less exaggeration
of either dimension
(table 2).
There was thus a
preferred
idealization of themale
physique
in the sixth
century
B.C.E. that favored a
greater-than-average proportional
difference between
thewidth of the shoulders and thewidth of thewaist
and close-to-average
vertical proportions
between
knees and sternum. In addition,
most (^) kouroi have pro
portionally large heads,
long
lower
legs,
and tall total
height.
Kouroi
may
be
relatively
slender or
relatively
thicksetand stillfit this
general description.
A
regional
preference
for stockier
proportions explains why,
for
example,
the Munich and Anavyssos
kouroi?which
are both Attic and both dated to the third
quarter
of
the sixth century
B.C.E.?are associated by
multivari
ate
analyses (see^
table
l).
This pattern, indeed,^
caused Guralnick to wonder
whether all kouroi
embody
a
single proportional
ideal.
She noted the same variations we have
just described,
namely,
that the same basic proportional
ideal occurs
both inmore^ slender kouroi and in broader kouroi
formuch of the sixth
century
and
again
inmore natu
ralistic kouroi (^) toward the end of the series.40 She con
cluded that "the proportional patterns
most likely
came from a
widely accepted approach
to idealization
which individual
sculptors
felt free to
modify
in
detail,
if not in basic lines."41 This seems entirely
reasonable.
Within this
generally preferred
idealization, however,
Guralnick
hypothesized
that
proportional
similarities
between
any
twokouroi are the result of the deliberate
use of the same proportional
scheme. She suggests,
for
example,
that the Sounion kouros from Attica
(ca.
B.C.E.)
and the Ptoon 20 kouros from
the
Apollo
Ptoon
sanctuary
at Thebes
(ca. 510-
B.C.E.) may have^
grouped together early
in cluster
analyses
because both conform to^ a^ (non-Egyptian)
canon "infrequently
used but long
known." We, (^) by
contrast, believe^ that
regional styles,operating
within
the generally preferred idealization,^
are a better ex
Measured from Guralnick 1985, 400, (^) fig. 1; 406, (^) fig.
The main proportional
difference between these two kouroi
is the width of the hips (Sounion:^
-0.75 (^) SD; Munich: +0.
SD). Sounion^ (Athens, National^ Archaeological Museum, inv.
no. 2720, ht.^ 3.05 m,
ca. (^) 590- B.C.E.):
Richter (^) 1970,42-4,
no. 2, figs. 33-9; Boardman^ 1978, fig. 64;
Floren^ 1987, 252
n.
9, pi. 20.1; Stewart^ 1990,111-12, figs. 44,
45; Kaltsas^ 2002, 39,
no. 17; Vorster
fig.
193a-e. Munich (Mu
nich, (^) Glyptothek,
inv. no. (^) 169, ht. (^) 2.08 m, ca. 540- B.C.E.):
Richter 1970,118,
no. 135, figs. 391-94;
Boardman 1978, (^) fig.
Floren 1987, 256
n. 22, pi. 20.4;
Karanastassis 2002,
77,312, (^) fig. 251a-d.
37 The information in table 2 ismeasured from Guralnick
1985, 407, (^) fig. 8;
dates are from Boardman 1978, (^) figs. 180,
182, 145, 147.^ Ptoon^20 (Athens, National^ Archaeological
Museum, inv. no. 20,
ht. 1.03 m, ca.^ 510- B.C.E.):
Richter
no. 155, figs. 450-57;
Ducat 1971, 355-62,
no. 202,
pis.
117-19; Boardman^ 1978, fig.
180; Floren^ 1987, 315
n. 41;
Stewart 1990, 126, (^) fig. 180;
Kaltsas 2002, 71-2,
no. 102;
Mad
erna-Lauter 2002, 232, 319, (^) fig.
310a-c. Attic style
of Ptoon
20: Boardman (^) 1978, 88; Stewart (^) 1990, 124; Maderna-Lauter
2002, 232. Aristodikos^ (Athens, National^ Archaeological
Mu
seum, inv. no. 3938, ht.^ 1.95 m,
ca. (^) 510- B.C.E.):
Richter
no. 165, (^) figs. 492, 493;
Boardman 1978, (^) fig. 145;
Floren 1987, 258
n. 31, (^) pi. 20.5;
Stewart 1990, 133, (^) fig. 218;
Kaltsas (^) 2002, 66, no.^ 94; Maderna-Lauter (^) 2002, 227-29, 319,
fig.
307a-e. Strangford (London,^
British Museum, inv. no.
B
475, ht.^ 1.01 m,^
ca. (^) 510- B.C.E.):
Richter (^) 1970, 136, no.
159, figs. 461-63; Boardman^ 1978, fig.
- Kritios Boy (Ath
ens, (^) Acropolis Museum, inv. no. 698, ht.^ 0.86, ca.^ 490-
B.C.E.):
Richter 1970, 149,^
no. 190, figs. 564-69;^
Boardman
fig.
147; Stewart^ 1990, 133-35, figs. 219,^ 220;^
Kaltsas
2002, 58, no.^ 69.
38 Of the 23 statues charted in z-scores (Guralnick 1985),
19 have heads that are proportionally greater
in height
than
the head of an average
man by
more than +1 SD. Of the 11
complete
kouroi charted in z-scores by
Guralnick (1985),
are proportionally
taller by
more (^) than +1 SD from baseline
to knee-top
than is an^ average man,^
and 9 are proportionally
taller in total height
than an^ average
man (though only
3 are
taller by
more (^) than + SD). 39 See also the z-scores^ for these two statues (Guralnick
1985, 406, (^) fig. 7).
Both statues (^) have wide shoulders,
narrow
waists, and^ wider-than-average hips.
Kroisos from Anavyssos
(Athens,
National Archaeological Museum,^
inv. no. 3851,
ht.
1.94 m, ca. 530 B.C.E.):
Richter 1970, 118-19,
no. 136, (^) figs.
395-98; Boardman^ 1978, (^) fig. 107; Floren^ 1987, 255
n. 21, pi.
20.3; Stewart^ 1990,122, (^) figs. 132,134; Kaltsas^ 2002,58,
no. 69;
Karanastassis 2002,177-79, 312,^ fig.
252a-d.
Guralnick 1985,404-7.
41 Guralnick 1985,407.
42 Guralnick (^) 1978,467, fig. 5; 469.
2010] KOUROI^
AND STATISTICS 113
Table 2. Proportioned
Dimensions of^ Four^ Late^ Archaic^ Kouroi^ in Standard^ Deviations^ from^
an Average
Human
Male
(adapted
fromGuralnick 1985, fig. 8).a
KourosWidth^ of^ Width^ of^
Height
from
Height
from
Height
from
Shoulders Waist
Knee-Top
to
Knee-Top
to
Knee-Top
to
Navel
Nipple
Sternum
Ptoon 20
(ca.
B.C.E.)
+2.08 -0.73 -0.44^ -0.09^ -0.
Strangford (ca.
B.C.E.)
Aristodikos
(ca.
B.C.E.)
Kritios
Boy (ca.
B.C.E.) +2.^
a Guralnick presents
this information as^ a^ z-score^ chart; the values shown here were^ obtained by measuring
the distances on
Guralnick's chart and converting
the distances to standard deviations. Each dimension was considered as^ a proportion
of the
statue's height
from the top
of the knee^ to the^ top
of (^) the head.
planation
for the
proportional
similarities that
may
exist among
kouroi than are^ standardized propor
tional
systems
used
sporadically
in different
places
at
different times.
It is
probably
the influence of
regional style
thatbest
accounts for the
seemingly
random collection of five
kouroi thatGuralnick considered closest to the
Egyp
tian canon^
(see figs. 2, 3).
These five are all^ slender
kouroi that share
idealizing proportions
of
modestly
wider-than-average shoulders,^ very^
narrow waists,
and
close-to-average heights
from sternum to knee-top.
All
five have been associated with a
style
of
sculpture
that
appears
to (^) have originated
on (^) Naxos and exercised
notable influence on other
Cycladic
islands and the
mainland. The New York kouros, which
belongs
to the
earliest
group
of kouroi known fromAttica,
is carved
fromNaxian marble; Attic
sculptors probably adopted
the kouros
type
fromNaxos and with ita Naxian
pref
erence for slender stature and linear surface pattern
ing.
The Thera kouros
(ca. 570-
B.C.E.)
is also
made of Naxian marble. Boardman noted similarities
between thehead of thiskouros, the head of theMelos
kouros, and
the head of a
probably
Naxian kore from
theAthenian
Acropolis (Athens,^ Acropolis Museum,
inv.no.
677); recently,
Kreikenbom has described the
Thera kouros as a somewhat inept
imitation of Naxian
and Parian sculpture by
a local Theran carver.44 The
Melos kouros (ca.
B.C.E.),
also of Naxian marble,
is
universally
considered to be the work of either a
Naxian sculptor
or a sculptor
under strong
Naxian
influence.45 The Tenea kouros (ca.
B.C.E.),
from
near
Corinth, is
sculpted
from Parian marble. While
there is general agreement
about the Cycladic
connec
tions of this statue,
scholars have tended to associate
itmore with Paros thanNaxos, in
part
because of the
relatively
smooth
carving
of the torso.
Finally,
Ptoon
(^12) (ca. 530-520 (^) B.C.E., of^ "island^ marble"), found
in the Ptoon
sanctuary
in Boeotia,
belongs
to what
Ridgway
has called the International
Style.47Ridgway
sees the
regional
schools of the earlier sixth
century
beginning
to
merge
in the decade 540-
B.C.E.;
she
believes that local
styles
cannot be
distinguished
after
- At the Ptoon sanctuary,
the local style
came (^) under
Cycladic
influence from Naxos and Paros in the de
cades between ca. 550 and 530 B.C.E.;
Naxian sculptors
may
have
migrated
to themainland when
Lygdamis
became tyrant
on (^) Naxos and confiscated unfinished
works for resale ca. 540 B.C.E. (Arist. [Oec] 2.2.2,lines
1346b9-13)
.48Parian^ and East Greek^ kouroi
generally
appear
heavier than their Naxian counterparts,
with
fluid
modeling
of the surface. Attic kouroi become
more thickset in the third
quarter
of the sixth
century
B.C.E., (^) perhaps
under Parian influence,
but they
con
tinue themodeled athleticmusculature ofAttic kouroi
from the second quarter
of the sixth century.
After ca.
B.C.E.,
some Ptoon statues begin
to resemble the
International Style
of contemporary
Attic kouroi. The
well-muscled
style
of Ptoon 12 thus blends
Cycladic
43Stewart 1990, 111; Ridgway 1993,88.
44 Naxian marble: Kaltsas (^) 2002, 41, no.^ 22; see^ also Board
man 1978,71;
Kreikenbom (^) 2002,148.
45Pedley 1976,35-6;^
Boardman 1978,71;
Floren 1987,178;
Stewart 1990,119; (^) Ridgway 1993,85; Karanastassis^ 2002,181.
46Stewart (^) 1986, 61 (Paros-trained sculptor);
Floren (^) 1987,
(a
Corinthian sculptor inspired by
both Parian and
Naxian kouroi);
Martini (^) 1990, 213 (more
Parian than Nax
ian) ;Sturgeon 2006,47 (closest
to sculptures
from Paros and
Attica).
47Ridgwayl993,80,84-5. 48 Stewart (^) 1986,119.
2010] KOUROI^
AND STATISTICS 115
Fig.
- Kouroi considered proportionally
similar by
Guralnick: left,
Florence kouros (also
known as the Milani kouros), (^) prov
enance uncertain, island marble,
ca. (^560) B.C.E. Florence, Florence^ Archaeological Museum^ (H. Koppermann;?^
DAI
Rome,
neg. 1962.0001);^ right,
Ptoon 10 kouros, from the Ptoon sanctuary
in Boeotia, Naxian marble,
ca. 550 B.C.E. Athens, National
Archaeological Museum,^
inv. no.^10 (G. Fafalis; Kaltsas (^) 2002, cat.^ no.^ 44).
York and Ptoon 12 kouroi to those of the Munich
kouros and the three-times-life-sized statue^ of Isches
from Samos
(fig. 7).
Made of local Samian marble and
dated ca.^ 590-580 (^) B.C.E., the Isches kouros provides
a
good example
of the
heavy physique
and softmod
eling
of East Greek kouroi.55We would
expect
Isches
to be more like the Parian/Attic
proportional type
than theNaxian/Attic. The z-scores of Isches
suggest
that this is the case^
(fig. 8);^
like theMunich kouros,
he has notably
wide (^) shoulders, narrow^ waist, and verti
cal dimensions from knee to sternum that are a little
greater
than those of an^ average
man. By
contrast,
the New York and Ptoon 12 kouroi are^
consistently
more narrow and not as tall (heads excluded)
in their
vertical dimensions.
Guralnick noted the same two basic proportional
types
and observed that the first
group may^
have "a
general proportional configuration
related to that of
the Egyptian
canon."57 This (^) characterization, however,
suggests
a causal relationship
that is not warranted. A
visual examination of all these statues shows immedi
ately
that the statues of theNaxian/Attic
group
have
parison
of the z-scores^ of Sounion and New York (^) (Guralnick
1985,401, (^) fig. 2) shows^ similar^ profiles,
but Sounion has over
all wider horizontal dimensions than New York.
55
Kyrieleis (1986,^ 38)^
first dated Isches to ca. 580-
B.C.E. but later (Kyrieleis
1996, (^) 57) revised^ the date^ to ca.
600-580 B.C.E. The statue is now^ in the Vathy
Museum on
Samos (ht.
as restored 4. m).
For bibliography,
see Kyrieleis
1986,35-41, (^) pis. 14-19; 1996; Floren^ 1987,
n. 31, pi. 30.5;
Kreikenbom (^) 2002,144,309, fig.
229a-c.
56Munich z-scores^ are measured from Guralnick (^) 1985,406,
fig. 7;
z-scores (^) of Isches are (^) taken from Guralnick 1996b, 521,
table 1.
"Guralnick (^) 1978,466.
J.B.
CARTER AND^
LJ.
STEINBERG
[AJA
Fig.
- Kouros dedicated^ by
Isches in the Sanctuary
of
Hera on^ Samos, Samian (^) marble, ca.^ 590-580 B.C.E.
(E. Gehnen;?^ DAI^ Athens, (^) neg. 1988/363).
relatively
slender
proportions,
while the statues of the
Parian/Attic group have^
relatively
more thickset
pro
portions.
These resemblances are^ very (^) general; they
probably
do reflect the influence of
regional styles,
but
they
lack the
specificity
that
might imply
an intendonal
use of standardized
proportional
schemes. The
Egyp
tian canon^ describes a^
relatively
slender
figure,
and
so it ismore like the slender
group
of kouroi, but this
does not mean^ that these kouroi were^ proportioned
according
to the
Egyptian
scheme.
According
to z-scores
(see fig.4),
the
Egyptian
canon
conforms to the same
preferred
idealization shown
by
kouroi: ithas proportionally wider-than-average
shoul
ders,
a very
narrow (^) waist, narrower-than-average hips,
close-to-average
vertical dimensions between knees
and (^) sternum, and taller-than-average
head, lower
legs,
and total
height.
Nine of the 13
proportions
of
the
Egyptian
canon charted
by
Guralnick fall within
68% of
average
men
(i.e.,
within?
SD).
The
Egyptian
proportions
that
depart
from the
majority
ofmen^ are
the
very
thin and slender waist
and -3.
SD),
the slender
hips (-1.45 SD),^
and the tall
height
of the
knees from the baseline
(+2.72 SD).
This is not an
unusual kind ofmasculine ideal.
Exaggerations
of these
proportions
are common to idealized male figures
of
many
eras and cultures, (^) including
modern western
culture. The triangular
torso of male figures
on Greek
geometric
vases is an extreme case of this stylization.
Less
obvious, perhaps,
is the
long
lower
leg,
but this
elongation,
too, was^ a^ frequent
characteristic in Greek
and Roman art^ and in classicizing figures
thereafter.
Greek
geometric
bronze
figurines
and
painted
male
figures generally
have
elongated legs,
and
frequently
the lower
leg
is
(contrary
to
nature) longer
than the
thigh.
Since these exaggerated
dimensions were at
home inGreek art before the firstkouroi, there isno
reason thatGreek
sculptors
need have^ imitated these
idealizing proportions
from
Egyptian
models.
Guralnick's studies also demonstrate how the pro
portions
of kouroi evolve toward greater
naturalism.
Ifwe look at the absolute distance
(in SD
from the
average male)
between the
proportional
widths of
shoulders and waist for all the kouroi included in the
z-score profiles,
we can see (^) how the relation between
proportional
shoulder and waist width
changes
over
time (table 3).
The sum
(i.e., the absolute
distance)
tends to diminish,
though unevenly,
over the course
of the sixth
century, regardless
of
regional preferenc
es for slender
figures (e.g., Melos,^
Ptoon
or for
stockier
figures (e.g., Kea, Aristodikos).^
As the sum
shrinks,
the statues become more like the average
man, since, by definition, all proportions
of the aver
age
man (^) have the value of zero standard deviations.
Even at the end of the series, however, the absolute
distance remains
unusually large, only
once
slipping
below two standard deviations (with theKea^ kouros).
Even the more^ naturalistic kouroi^ continue^ the^ gener
ally preferred
idealization.
GURALNICK'S METHODOLOGY^ FOR KOUROI
The Statistical
Techniques
Used inGuralnick 'sStudies
In the
previous
section, we
accepted
the
validity
of
Guralnick's published
results and examined her^ inter
pretations
of those results. We now^ look^ at her method
Measured from Guralnick (^) 1985,403, fig.
59Cf.Boardman (^) 2006,20.
,i Hollander (^) 1978,98-9.
61 For geometric
bronze male figures,
see, (^) e.g., Schweitzer
pis.
130, 131, 136-39, 164, 165, 182-84, 185.^ For^ geo
metric vases, see, e.g.,
Schweitzer 1971, (^) pis. 35,36,40,69,^
62The z-scores^ are measured from charts^ inGuralnick^ 1985;
date of the Isches kouros: Kyrieleis 1996,57;^
date of Paros kou
ros: Stewart 1990, (^) pis. 118,119;
date of Ptoon 10: Kaltsas (^) 2002,
no. 44; all other
dates: Boardman 1978.
J.B.
CARTER AND
LJ.
STEINBERG
Table 3. Sums of^ the Absolute^ Z-Scores^ of^ the
Proportioned
Widths of Shoulders
Second Canon and Greek Kouroi
(adapted
from Guralnick 1985, figs. 1, 3-8; 1996b,
[AJA
114
and Waists of the
Egyptian
table 1).
Object
Date Z-Score for Z-Score for
(B.C.E.)
Width of Shouldersa^ Width^ ofWaista
Sum ofAbsolute
Standard Deviations of
Shoulders and Waist
Egyptian
canon
New York
Sounion
Isches
Ram-bearer
Kleobis
Biton
Thera
Volomandra
Florence
Thebes 3
Melos
Paros
Keratea
Ptoon 10
Tenea
Munich
Anavyssos
Kea
Ptoon 12
Ptoon 20
Strangford
Aristodikos
Kritios
Boy
ca. 600-
ca. 590-
ca. (^) 590-
ca. (^580)
ca. 580
ca. (^580)
ca. (^) 570-
ca. 570-
ca. 560
ca. (^550)
ca. (^550)
ca. (^550)
ca. 550
ca. (^550)
ca. (^550)
ca. (^) 540-
ca. (^530)
ca. 530
ca. (^) 530-
ca. (^) 510-
ca. 510-
ca. (^) 510-
ca. (^) 490-
a The z-scores are expressed
in standard deviations for average
human males. For all kouroi^ except Isches,^
Guralnick presents
this information only
in z-score (^) charts; the values shown here were obtained by measuring
the distances on^ Guralnick's charts
and converting
the distances to standard deviations.
ters,but^ it
joins
cluster F with the
Anavyssos, Kea,
and
Aristodikos kouroi when^ there are
13 clusters. A first
response might
be to conclude thatMunich ismore
similar to Keratea and Paros^ than to Anavyssos, Kea,
and Aristodikos,
since itfirst
groups
in cluster E with
Keratea and Paros with a
larger
number of clusters
clusters)
and then returns toE with 12 and 11 clusters.
However, this is^
probably
not true, since theMunich
kouros groups
soon (^) and consistently
with the Anavys
sos kouros in the two other cluster analyses
and is close
to
Anavyssos
and Kea^ in the
principal components
charts. Thus,
one (^) should interpret
Guralnick's clusters
as
implying
that for a
predetermined
number of clus
ters, (^) groups
of objects
are most similar as aggregates;
one should not conclude that an individual
object
in
one
group
ismore^ similar to other individual
objects
in the same
group
than to
any
other
object
in another
group. (The^ exception
to thiswould be at the
early
stage
of
analysis
when clusters have
only
two
objects
in them. In this case, the
objects
that are most^ similar
to each other will most
likelypair together.)
Guralnick
might
have chosen^ another
similarity
measure to (^) cluster the objects
rather than minimiz
ing
the trace ofW. There were^ a number of different
algorithms
available in the 1970s and 1980s, and each
algorithm
used one of several different similarity
mea
2010]
KOUROI AND STATISTICS 119
sures and a
particular
method tominimize the simi
larity
measure (^) used. Studies have shown that analyses
of the same data set
by
different
algorithms
and dif
ferent similarity
measures produce
different results.
Given a particular
set of variables,
some (^) methods per
form better than others. For
example, minimizing
the
determinant ofW or
maximizing
the trace of BW
(where
B is the between-cluster covariance matrix)
has the advantage
of accounting
for covariance in the
data
(which very likely
existed inGuralnick's data
sets).
The
primary advantage
of the traceW measure^ is the
ease with which it can be
computed;
the
capacity
of a
computer
to perform
the algorithm
was an important
consideration in the 1970s
(and
is a far lesser concern
today)
.69For our^ purposes,
it is important
to note that
other
algorithms using
different
similarity
measures
or differentminimization methods
might
have
pro
duced different
groupings
than Guralnick's
analyses
did. Furthermore,
it is not
possible
to know a
priori
which method will work best for a
given
data
set, and,
when the classification of the
objects
isnot known be
forehand
(as
was the case with Guralnick's
studies),
it
would not be
possible
to determine which results are
better even^ after several methods had been used.
Another important
consideration is the robustness
of cluster
analysis
to the
particular
values in the data
set. For
example,
it iswell known that themethod is
sensitive to
outliers, the
presence
ofwhich can distort
the model results.71The
sample
of kouroi
certainly
includes outliers such as the
Dermys
and
Kitylos pair
and the Kleobis and Biton
pair.
The
Dermys
and
Kitylos pair
is so different from other kouroi that if it
were put
in a^ cluster with other kouroi,
the variance
of that cluster would increase
dramatically. Thus,^
the
program
avoids
putting
the
Dermys
and
Kitylos
outlier
into a cluster as
long
as itcan
(see table 1 [theDermys
and
Kitylos pair
formed itsown cluster until the num
ber of clusterswas^ reduced to
five]).
In order to
keep
Dermys
and
Kitylos isolated, however,^
the
program
is
forced to
group together
other
objects
that
may
not
actually
have a
high degree
of
similarity.
Cluster analysis
is also sensitive to (^) the exact (^) values of
data. Small changes
in the value of a variable can move
the data point
from one^ cluster to another.72 The re
sults of Guralnick's cluster analyses
thus depend
on (^) the
reliability
of the exact values of the data she has used
for the kouroi and the Egyptian
canon. (^) Guralnick first
measured kouroi in 1968 forher doctoral dissertation.
For this
project,
which did not use statistical
analyses,
she included
eight life-sized,completely preserved
kou
roi; of^ these, she^ herself^ measured^ six, and^ she^ used
previously published
dimensions for the other two.
Guralnick
adopted
themeasurements^ of these kouroi
from her dissertation for her statistical studies, adding
measurements of additional kouroi taken in 1974.
The tables in her dissertation that record the dimen
sions of the initial
eight
kouroi
report
the
"averages
of actual dimensions."75 This indicates thatGuralnick
took several measurements and then used the average
value for her
study.By
the central limit theorem, one
expects
that the greater
the number of measurements
taken and averaged,
the more accurately
and precisely
the actual dimension can be estimated. The amount of
precision
in the estimate ismeasured
by
the deviations
of the measurements from their mean. Guralnick does
not provide
these deviations or the number of measure
ments taken,
so we cannot (^) estimate this uncertainty;
but the fact that she used
averages implies
that there
was (^) measurement error of unknown magnitude
in the
data. While the amount of deviation in Guralnick's
data
might
have been
small, it isnevertheless^
likely
to
have affected the clustering.
The data used for the
Egyptian
canon
present
more
serious problems
with respect
to exact values because
Egyptian figures
do not conform
exactly
and consis
tently
to
surviving
or restored
grids.
Guralnick remarks
on this in her dissertation. There she
gives
a
range
of
values for the shoulder width
to 7
units)
and waist
width
to 3
units)
of the second canon.76 For the
cluster analyses
and the z-score profiles, however,^
Gur
alnick used a
single
set of values for the dimensions of
the
Egyptian
canon, which means^ that she^ must have
either used an average
value or^ selected one^ value
within the
range
shown
by Egyptian figures
on some
other basis. If somewhat different values for the
Egyp
tian canon^ had been used in the
analyses,
we would
68Friedman and Rubin (^) 1967; Everitt et al. 2001,65-7,94-9;
Baxter (^) 2003,103-4.
Seber 1984,
- It is possible,
for example,
that Gural
nick extrapolated
the division into five clusters in her cluster
analyses
with 28 objects
because this task exceeded the capac
ity
of the
computer (Guralnick1978,467,^ fig.5).
70Shennanl997,254.
71
Punj
and Stewart (^) 1983, 143-44; Baxter 1994, 165-66;
Baxter (^) 2003,102. Principal components analysis
is also sensi
tive to outliers (James
and McCulloch 1990,142;
Baxter 1994,
Morrison (^) 1990, 385: "small perturbations
in the data
might
lead to very
different clusters."
73 Guralnick 1970, 3, 40. Guralnick
measured Dermys
and
Kitylos,
and the Tenea, Volomandra, Melos, (^) Anavyssos, and
Munich kouroi. She used previously published
measurements
for the New York and Aristodikos kouroi.
Guralnick 1978,461.
75 Guralnick (^) 1970,108-24, tables 1-8.
76 Guralnick 1970, 41-2,127,
table 10. Guralnick also gives
a range
of 1.75 to 2.25 units for the height
of the face.
2010]
KOUROI AND STATISTICS 121
peared.
The resultwould be similar to a
bell-shaped
curve (^) from which the mean (^) and standard deviation for
each proportional
ratio could be computed.
However, Guralnick^ could^ not^ have^ plotted
a bell
shaped
curve in this
way
because theNATO
publica
tion does not
provide
these data for all the individual
men in the
study.Rather,
theNATO
publication
re
ports only
themean^ and standard deviation for each
dimension of the men. The authors were interested
in actual dimensions of real men, not proportional
re
lationships. By contrast, Guralnick's
project
involved
representations
of human figures,
some (^) life-sized and
some larger
or smaller than life-sized. She therefore
compared
kouroi
using proportional
ratios rather than
actual (^) dimensions, and this means^ that she needed the
mean (^) and standard deviation for each proportional
ratio of real (^) men, rather than for each dimension,
to
compare
with the kouroi.
A
hypothetical example may
be
helpful
here.
Sup
pose
we (^) wanted to compute
the "perfectly average"
university
nationwide with respect
to student-faculty
ratios. The correct
way
to do this is to find the ratio
for each school in the
nation,
sum the ratios
together,
and then divide
by
the number of schools to
get
the
average student-faculty
ratio for^ a U.S. university.
This
method is
equivalent
to
measuring
the NATO men,
expressing
the dimensions of each man^ as^ a^ ratio, and
then computing
the average
value of all the ratios.
In contrast, one could divide the total number of
students
attending
all U.S. universities
by
the number
of universities to get
the average
number of students
per
school
(call
this
S).
Then one could divide the total
number of
university faculty
in all U.S. universities
by
the number of universities to
get
the
average
number
of
faculty per^
school
(call
this
F). Finally,
one could
divide the
average
number of students
per
school
(S)
by
the
average
number of
facultyper
school
(F)
to find
the ratio of students to
faculty
across the
country (S/
F). This method, which^
is
equivalent
to themethod
that Guralnick (^) used, does not^ give
the average
student
faculty
ratio forU.S. universities but rather is
simply
a
ratio of two averages,
S and F.
The twomethods will not
yield
the same result.
For our
purposes,
the
important
difference is that the
second method does not^ produce
a (^) random variable
similar in kind to the ratio obtained from
averaging
the ratios of the individual schools
(which
were com
puted
from the number of students and faculty
at (^) each
individual
institution). Hence,
the value obtained
by
the firstmethod isnot
comparable
to the value com
puted by
the second method.
Guralnick does not
explain
themethod she used to
chart the z-scores of kouroi and korai in her articles
of 1982 and
1985; however,
in her more recent
study
of the colossal Isches kouros from Samos
(see fig. 7),
she does
provide enough
information to
permit
a
reconstruction of her procedure. Here,^
in addition
to a z-score profile
that compares (^) proportions
of the
Isches kouros with those of an^ average
Greek (^) man,
Guralnick includes a table
showing
the values used to
create the z-score profile;
the information is given
in
table 4. The columns show
(left
to
right)
the actual
dimensions
(in cm)
of the Isches kouros, the mean
and standard deviations fordimensions ofGreek men
as
reported by
theNATO
study,
and the z-scores for
the Isches kouros
using
the distance from
knee-top
to
head-top
as (^) the proportioning
base. The standard de
viations for Greek men are given
in centimeters,
indi
cating
the
spread
of values around themean^ value of
the dimensions of Greek men.^ However, Guralnick's
data do not include the measures of interest for com
paring
the
proportions
of Isches to those of realmen,
that is, themean and standard deviations of the ratios
between the dimensions and the
proportioning
base
(which,
as
ratios,would^ not be^
expressed
in
any
unit
of measurement).
Guralnick
apparently proceeds
as follows. Each di
mension of the Isches kouros isdivided
by
the
height
of the statue from
knee-top
to
head-top
(329.1 cm).
This ratio is then used to
compute
what the dimen
sions of Isches would be if the statue had the same
height
from knee-top
to head-top
as (^) the average
Greek
man.89 Then the difference between each converted
dimension of the kouros and the mean dimension
forGreek men is
expressed
in terms of the standard
deviation for that
(unproportioned)
dimension. Thus
the (^) ratio of (^) shoulder width to knee-top-to-head-top
height
for Isches is 131.9 cm/329.1 cm^
(see
table
Suppose
one (^) school has 300 students and 50 faculty
mem
bers,
another has 600 students and (^20) faculty,
another has 700
students and 100 faculty,
and another has 300 students and 40
faculty. Using
the first procedure,
the average
value of the stu
dent-teacher ratio is 12.6 students per faculty
member. Using
the alternative procedure,
the average
number of (^) students
per
school (475 students) is^ divided^ by the average number^ of
faculty per
school (52.5 (^) faculty),
and the overall ratio is com
puted
as 9.0 students per (^) faculty
member.
87Guralnick 1996b, 512, (^) fig.
(z-score (^) profile); 521, table^
(data
used for the profile).
88Herzbergetal.
1963, (^153) (shoulderwdth.), 154 (chest
wdth.), 156 (waistwdth.), 157 (hipwdth.), 128 (ht.
of ster
num), 129 (ht.
of nipples),
(ht.
of navel),
(ht.
of knee
top).
89E.g.,
letx
the shoulder width of Isches if the statue were
the same^ height
as an average
man. Then x/knee-top
to head
top
distance of the average
man
shoulder width of (^) Isches/
knee-top
to head-top
distance of Isches.
J.B.
CARTER AND^
L.J.
STEINBERG
[AJA
114
Table 4. Dimensions of Isches (^) Kouros, Average
Dimensions and Standard Deviations of Greek Men,
and Z-Scores
forDimensions of the Isches Kouros
(after
Guralnick
1996b,
tables 1, 2; fig. 1).
Variable Isches
(cm)
Average
Greek
Man
(cm)
Standard Deviation
of theVariable for
Greek Men
(cm)
Z-Score of Isches for
Variables
Proportioned
to Isches'
Height
from
Knee-Top
to Head-Top
Wdth. of shoulders 131.9 44.
Wdth. of chest 83.630.
Wdth. of waist 29.2771.
Wdth. of
hips
Ht. knee-top
to navel 149.4 51.
Ht.
knee-top
to
nipples
Ht.
knee-top
to sternum 243.7 88.
Ht. of head 66.9 22.
Ht.
knee-top
to
head-top
If the kouros had the same
height
from
knee-top
to
head-top
as the
average
Greek man^ (121.1 cm), then
its shoulder width would be 48.54 cm, or^ 3.85 cm
more than the shoulder width of the
average
Greek
(44.69 cm). (^) Expressed
as a z-score (i.e.,
in terms of
the standard deviation of 2.25 cm for the shoulder
width of Greek men),
the shoulder width of Isches is
(i.e., 3.85/2.25).
Guralnick uses^ the z-scores
computed
in this way
to create z-score profiles
of the
Isches kouros.
This is not^ a valid
procedure
because it does not
compare
the
proportional
value of Isches' shoulder
width relative to^ his
knee-top-to-head-top height
with
the computed
mean and SD of this proportional
value
for the
average
Greek. We would have to
begin by
com
puting
the
proportional
value of the shoulder width to
the
height
from
knee-top
to
head-top
for
every
Greek
man in the NATO
sample. But,
as noted
previously,
theNATO
study
does not
give
the individual dimen
sions of each man^ measured, only
the average
value
and standard deviation for each dimension. (^) Guralnick,
therefore,
could not have produced
a (^) random vari
able and an associated SD for the
proportional
values
of themen^ thatwould be
statistically comparable
to
the proportional
values of the statue. (^) Hence, we^ must
conclude that the z-score^ profiles
in which the dimen
sions of the Isches kouros are compared
to the average
Greek man have little or no statistical
meaning.
The
same must (^) be true (^) of the z-score profiles
in Guralnick's
earlier studies of kouroi and korai.
They
do not show
statisticallymeaningful comparisons
between the kou
roi
(or
the
Egyptian canon)^
and an
average man,^
and
the
profiles
would have different
shapes
if
they
were
charted with z-scores for proportional
values.
Measuring
the
Egyptian
Canon
Guralnick (^) seems, for^ the most^ part,
to (^) consider the
Egyptian
second canon^ as^ an^ invariable and compre
hensive proportional system.
The surviving
evidence
for the
Egyptian system,however,
indicates that it
was
neither as fixed nor as comprehensive
as her studies
imply.
Of the 11 anatomical dimensions in data setA, two
or three
appear
tobe defined
by grid
lines in the
Egyp
The table
published by
Guralnick (1996b, 521, table 1)
contains several errors. The height
from top
of knee to ster
num is given
as (^) 240.7 in tables 1, 3a, 3b, 4,^ 5,
and 6 but ap
pears
as (^) 243.7 cm in table 2. The value of 243. produces
the
z-scores of 0.24 and -0. given
in table 1. For height
from top
of knee to navel,
the top-of-head
to top-of-knee
z-score (^) is given
as -1.13 in the (^) table. This should be +1.13,
and it is charted as
a plus
value in the z-score^ chart^ created^ from^ this data^ (Gur
alnick (^) 1996b, 512, fig.
1). For^ depth of chest, the base^
to top
of-head z-score is given
as -0.49; this should^ be^ -1.49.^ For^ base
to navel, the base to top-of-head
z-score is given
as 0.00;
this
should be+1.33.
Guralnick (^) 1996b, 512-15, figs.
The Egyptian
canon is treated as a single
set of values in
the cluster analyses
and the z-score^ analyses. Eight examples
of the Egyptian
canon are (^) charted in the principal compo
nents analyses (Guralnick^
figs.
6, (^) 7), and^ the
article on Boeotian kouroi mentions two sets of values for the
Egyptian
canon included in cluster analyses,
as if they
were
two additional statues (Guralnick 1996a, 39).
