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Guralnick's Statistical Analysis of Greek Kouroi and the Egyptian Canon: A Critique, Lecture notes of Statistics

This document critically examines Guralnick's statistical studies on Greek kouroi and their relationship with the Egyptian canon. The analysis reveals limitations in her methods and procedures, casting doubt on her claims for Greek use of the Egyptian system for proportioning human figures. However, her observations about similarities among kouroi add support to conclusions formed by scholars on stylistic grounds.

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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: (1) the ap
proximate synchronism of the earliest Greek statues in
stone and the resumption of direct contacts between
Greece and Egypt around the
middle 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 that Greeks
learned sculptural techniques from the
Egyptians; if
so,
one could expect to
find
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.1 Guralnick concluded that "at
least through the third
quarter of the sixth century
Greek sculptors made conscious use of the contempo
rary
Egyptian 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
of Greek sculpture.3
However, 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 is well 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, they
do not in
fact demonstrate
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 of
Guralnick'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.4 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,
2000.
2Guralnick 1985,409.
3
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 (2007,274,283
n. 27) 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,
148) 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.
103
American
Journal of
Archaeology
114 (2010) 103-28
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe
pff
pf12
pf13
pf14
pf15
pf16
pf17
pf18
pf19
pf1a

Partial preview of the text

Download Guralnick's Statistical Analysis of Greek Kouroi and the Egyptian Canon: A Critique and more Lecture notes Statistics in PDF only on Docsity!

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.

  1. 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,

  1. (^) 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.

  1. 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.

  1. 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.

  1. 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.

  1. 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.

  1. 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

  1. 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.

  1. 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.

  1. 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,

  1. 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).