Harris 2003.4


84 85

If one measures the conventional mesiodistal and 
buccolingual crown diameters of the 32 permanent teeth 
in the human dentition, there are 64 variables, which is 
comparable to an extensive battery of craniometrics or 
anthropometrics (e.g., Davenport, 1927; Martin, 1928).  
One might suppose that there is a lot of statistical 
information—several axes of variation—in the odon-
tometrics based on tooth types, side, arcade, a tooth’s 
position in its morphogenetic field, sex, race, and so on.  
However, the morphological and statistical redundancy 
among tooth types has long been recognized, and this 
redundancy sharply diminishes the information con-
tent of batteries of crown dimensions.  Bateson (1894) 
included teeth in his anatomic examples of meristic 
series that included phalanges, vertebrae, and ribs.  This 
phenomenon of multiple analogous skeletodental units 
that develop clinally along a growth axis also is termed 
polyisomerism (Gregory, 1934).  The supposition is that 
the shared morphologies are controlled by common 
control mechanisms (genes, gene products), but verifi-
cation has only recently been provided (e.g., Yamaguchi, 
1997; Green, 2002).  Weiss (1990), Jernvall (2000; Jernvall 
and Jung, 2000; Jernvall and Thesleff, 2000), and others 
suggest that polyisomerism is a conservative, efficient 
mechanism for increasing (or, occasionally, decreasing) 
the anatomic units, which is more obvious phylogeneti-
cally, but occurs ontogenetically as well.  The “several” 

Where’s the Variation?  Variance Components in Tooth Sizes 
of the Permanent Dentition

Edward F. Harris*
Department of Orthodontics, The Health Science Center, University of Tennessee 38163

*Correspondence to:  Edward F. Harris, Department of Ortho-
dontics, The Health Science Center, University of Tennessee, 
Memphis 38163.  E-mail eharris@utmem.edu

A version of this paper was presented at the Annual Meet-
ing of the American Association of Physical Anthropologists, 
Tempe, AZ, 2003.

ABSTRACT     Studies have shown that there are only a 
few canonical axes of tooth size variation in the perma-
nent dentition.  Despite the numerous measurements 
that might be taken (e.g., crown length and breadth of 32 
teeth = 64 variables), most of the canonical structure is 
explained by 3 or 4 overarching axes of variation.  This 
study used maximum likelihood components of vari-
ance analysis to determine where the major sources of 
statistical variation are among the crown dimensions in 
the permanent dentition.  Mesiodistal and buccolingual 
crown dimensions were measured on all permanent 
teeth (excluding M3s and averaging sides) in 100 Ameri-
can whites and 100 American blacks, evenly divided by 
sex.  The SAS program varcomp estimated the sources of 
variation across 7 aspects of the dentition, namely race, 
sex, arcade, tooth (incisor, canine, premolar, molar), po-
sition (mesial, distal), dimension (MD, BL), and a residu-
al term.  Most variation is shared; residual variance was 
just 21.8% of the total.  Considering the six components 

of shared variance, the greatest (82.8%) was due to tooth 
type (I, C, P, M).  In contrast, only 4.9% was attributable 
to the black-white race difference, which confirms re-
sults of other biological data that the preponderance of 
variation is within groups, not among them.  More strik-
ing is the lack of variation between males and females 
(1.2%)—confirming the insensitivity of tooth crown 
dimensions for forensic purposes.  Very little shared 
variance (0.6%) was due to tooth position, indicating 
that the mesial “pole” tooth that is metrically and mor-
phologically more stable does not possess much more 
informational content statistically.  Whether the tooth 
was maxillary or mandibular accounted for 6.9%.  In a 
practical sense, the large variance due to tooth type im-
plies that dental anthropologists commonly will want to 
include variables from all tooth types (I, C, P, M) rather 
than multiple measurements within a tooth type, since 
tooth type is the canonical axis of variation.  Dental An-
thropology 2003;16(3):84-94.

canonical axes of variation expected from a battery of 
tooth dimensions do not actually occur because tooth 
crown dimensions are invariably positively intercorre-
lated (e.g., Moorrees and Reed, 1964; Potter et al., 1968; 
Henderson, 1975; Townsend, 1976; Harris and Bailit, 
1988).

Genetic covariance among continuous-scale variates 
like crown dimensions arises from pleiotropic effects of 
the contributing genes (e.g., Falconer, 1989).  Indeed, the 
principal theme of Butler’s seminal studies of morpho-
genetic fields (1939, 1956, 2001) is the developmental 
dependencies (covariance) of tooth morphologies and 
dimensions of teeth within the three major fields in 
mammals, namely incisors, canines, and postcanine 
tooth types.  The consequences for the dental anthro-
pologist are that much of the informational content of 
many tooth crown dimensions are statistically redun-
dant.  Measuring more teeth or measuring more dimen-
sions of the same teeth does not proportionately increase 



84 85

the researcher’s ability to discriminate between sexes, 
or populations, or races, or species.  Falk and Corruccini 
(1982) have shown this quite simply:  the discrimina-
tory power among groups was much better using cra-
niometric variables (with less covariance among traits; 
Solow, 1966) than an equivalent battery of tooth crown 
dimensions.

MATERIALS AND METHODS

The data analyzed here consist of maximum me-
siodistal (MD) and buccolingual (BL) tooth crown 
dimensions of all 14 permanent tooth types, omitting 
M3s.  Measurements were made on the full-mouth 
dental casts of 100 American whites and 100 American 
blacks using electronic-readout sliding calipers with the 
beaks machined to fit well into the embrasures between 
teeth.  Measurement technique followed that described 
by Selmer-Olson (1949).  There was an equal number of 
males and females in each race, and the subjects were 
contemporary American adolescents with all 28 teeth 
fully erupted without any restorations that would affect 
the measurements.  Teeth on just one side of the mouth 
were measured (either left or right, on an individual ba-
sis), but numerous studies have shown that the variance 
attributable to side is meager and due to just bilateral 
asymmetry plus technical error and may safely be ig-
nored without biasing the other effects (e.g., Lundström, 
1948; Potter and Nance, 1976).

Statistical analysis

It is implausible from what is known about odon-
tometrics (e.g., Kieser, 1990; Hillson, 1996) to suppose 
that either genetic or environmental variation in tooth 
size is distributed across the dentition in even a vaguely 
uniform manner.  Instead, some of the axes of variation 
will account for appreciably more than other sources of 
variation.  Six axes of variation were estimated in the 
present study.  Variation was compared by (1) race, (2) 
sex, (3) arcade (maxilla or mandible), (4) tooth type (in-
cisors, canines, premolars, and molars), (5) dimension 
(mesiodistal versus buccolingual crown diameters), and 
(6) position (the mesial or distal tooth within a morpho-
genetic field).

To find out how the variance in tooth size is appor-
tioned across these six axes, model II, maximum-likeli-
hood estimates of variance components were estimated 
(Hartley et al., 1978) using the SAS procedure varcomp 
(SAS, 1989).

Two “races” were contrasted, American blacks and 
whites, but the perspective is to view these as random 
samples from the “universe” of possible races (e.g., 
Coon, 1965).  Similarly, any number of crown dimen-
sions could be measured on a tooth (cf. Corruccini, 1977, 
1978; Black, 1979; FitzGerald and Hillson, 2002), the 
conventional two assessed here (i.e., the standard MD 
and BL dimensions) are best viewed as a sample of two 
picked from a population of dimensional possibilities.

RESULTS AND DISCUSSION

Multivariate analysis

Most of the total variance for odontometrics is shared 
(common) rather than unique variance.  This has long 
been recognized (e.g., Garn et al., 1965, 1968; Moorrees 
and Reed, 1964) insofar as all MD and BL crown dimen-
sions are positively intercorrelated with one another 
throughout the dentition.  This is true for the present 
data set (Table 1) where every one of the 378 pairwise 
comparisons is positively and significantly correlated at 
P < 0.001 (n = 200 for each comparison).  This means that 
“size” is a pervasive controlling factor throughout the 
dentition.  It also means that (1) tooth size can be pre-
dicted with some accuracy from other tooth sizes (e.g., 
Moyers, 1988; Tanaka and Johnston, 1974) but that (2) 
since all dimensions are intercorrelated, they all tend to 
estimate the same thing (namely “overall size”) rather 
than carrying unique, nonredundant information.  De-
velopmentally, these statistical intercorrelations appear 
to reflect the communalities of a few rather than many 
axes of ontogenetic control (Weiss, 1990; Salazar-Ciudad 
and Jernvall, 2002).

Statistical redundancy also has been illustrated in the 
several studies of human tooth size using factor analy-
sis (e.g., Potter et al., 1968; Harris and Bailit, 1988).  For 
the present data, there are just three orthogonal axes of 
shared variation among the 28 crown dimensions, with 
“overall size” accounting for most (83%) of this (Fig. 1).  
The other two axes of variation are (1) BL breadths of the 
anterior teeth contrasted with MD lengths of the cheek 
teeth (premolars and molars), accounting for 10% of the 
shared variation, and (2) MD lengths of the incisors con-
trasted with BL breadths of the posterior teeth (canines, 
premolars and molars) accounting for 7%.  Collectively, 
these three axes of shared variation (i.e., variation not 
unique to a single crown dimension) is 73% of the total 
variation.

PCA has been performed across a broad range 
of human samples, showing concordant results and, 
thus, the nature of the covariance matrices probably is 
essentially independent of the population under study.  
It is obvious that these three canonical axes of metric 
control of the dentition are far fewer than the 28 variables 
measured for the dentition, and this “reduction” is due 
to statistical (and developmental) redundancy among 
crown sizes.

Variance components

Results from the SAS program varcomp disclosed 
that, taking total variance as 100%, the shared variance 
accounted for by the six variables in the model was 
79.2% while the residual variance, unique to individual 
measurements accounted for the other 21.8% (Fig. 2).  
This is about a four-to-one ratio of explained to residual 
variances, suggesting that the six factors listed above do, 

VARIANCE COMPONENTS IN TOOTH SIZE



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86 87

Fig. 1. Results of principal components analysis on the 200 cases in the present study (28 crown dimensions).  Top:  
Distribution of eigenvalues showing how most of the variation is in the first canonical component and how quickly 
the subsequent values descend, so that just the first three are larger than 1.0.  The other three panels are graphs of 
the variables’ weights on each of the three principal components.

collectively, account for most of the variability in this 
data set in the statistical sense.  Variance components of 
the six factors tested here are expressed as percentages 
of the explained variance (Table 2).

Caveat

Partitioning total phenotypic variance into the 

relative fractions due to the six sources (listed above) is 
done to disclose differences in the relative contributions 
of these contributors to anatomic variation.  So, for ex-
ample, variations among the four tooth types (58.8% of 
total) is found to be enormously greater than variations 
between the MD and BL crown diameters at 2.5% (i.e., 
between the two conventional axes used to reflect size 

PC I

PC II

PC III

VARIANCE COMPONENTS IN TOOTH SIZE



88 89

TABLE 2. Estimates of the proportion of variance for each of 
the 7 parameters in the model

 Source Estimate Percentage

 Tooth Type 2.47247 58.76
 Arcade 0.20484 4.87
 Race 0.14707 3.49
 Dimension 0.10735 2.55
 Sex 0.03633 0.86
 Position 0.01656 0.39
 Residual 0.83461 19.84

 Total  100.00

Fig. 2. Pie chart showing the apportionment of tooth 
size variation based on the six variables in the model 
(see text for details).

variation).  Whether large or small, these components 
do not address whether there are statistically significant 
differences between groups within one of these six ca-
nonical dimensions.  For example, the smallest source of 
variation in the present analysis is “position”—whether 
a tooth is the mesial, stable tooth or the distal, vari-
able tooth within a morphogenetic field (I, P, M).  Even 
though position only accounts for 0.4% of the total vari-
ance, there still are highly significant statistical differ-
ences in mean size and in variance between mesial and 
distal teeth within a field (Kieser, 1990).  Consequently, 
these two issues (source of variation versus statistical 
significance) are unrelated issues.

Tooth type

By far, the largest variance component (82.8%) is 
tooth type, namely whether the tooth is an incisor, ca-
nine, premolar, or molar (Fig. 3).  This finding has an in-
tuitive appeal because heterodonty—the segmentation 
of the dentition into functionally specialized tooth types 
(incisors for nipping, canines for piercing, premolars for 
trituration, and molars for crushing)—is the fundamen-
tal arrangement of the primate dentition (Todd, 1918; 
Butler 1939, 1956; Swindler, 2002).  The other anatomic 
effects in the present analysis simply involve duplica-
tion within the fields:  duplication across the upper and 
lower arch producing structurally similar antagonists; 
duplication of a distal tooth creating the short meristic 
series that Weiss (1990), Jernvall (2000; Jernvall and Jung 
2000), and others point out is an efficient method of in-
creasing the number of structures, essentially by dupli-
cating existing ones.  The other sort of duplication (not 
included here) is tied to the ontogeny of bilateral sym-
metry, where left and right paired structures develop, 
apparently using the same genetic information, sym-
metrically across the midline.  It would seem, then, that 
the four morphogenetic fields (one for each tooth type) 
constitute the basic organizing theme—with most of the 
variation among fields—and that, within fields, teeth 
enumerated front-to-back (the “pole” and the “variable” 
tooth; Dahlberg, 1945, 1951), side-to-side (bilateral sym-
metry), and craniocaudally (creating analogous tooth 
morphologies in the two jaws) consume comparatively 

little variation.  In a practical sense, this large variance 
due to tooth type implies that dental anthropologists 
commonly will want to include variables from all tooth 
types (I, C, P, M) rather than multiple measurements 
within a tooth type, since tooth type is the canonical axis 
of variation.

Arcade

While it is a distant second in terms of absolute 
variance, arcade (Fig. 4) counts for the next-largest 
component of variance (6.9%), which is in concert with 
the results of factor analysis of dental metrics showing 
that, aside from an overall size effect, most factors or 
principal components (i.e., intercorrelated multivariable 
dimensions of teeth) typically are arcade-specific (e.g., 
Potter et al., 1968; Brown and Townsend, 1979).  Perhaps 
this has been shown most clearly by Potter et al. (1976) 
who characterized the few axes of genetic variation in 
the dentition.  One genetic factor was bilateral sym-

Fig. 3. Graph of mean tooth size by tooth type.

E.F. HARRIS



88 89VARIANCE COMPONENTS IN TOOTH SIZE

metry; every genetic factor identified for a dimension 
on one side included the antimeric dimension on the 
other.  Secondly, Potter disclosed a buccolingual crown 
size factor that extended throughout the maxillary (but 
not the mandibular) teeth.  Thirdly, a genetic factor in-
fluenced both MD and BL dimensions of the mandibular 
anterior teeth.  It is noteworthy that these genetic factors 
control regions of the dentition, not specific teeth.  Recent 
computer modeling (Salazar-Ciudad and Jernvall, 2002) 
shows comparable results, namely that controlling just 
a few parameters can account for both the ontogenetic 
and phylogenetic variations within and among tooth 
types, both metrically and morphologically.

In the present study, Figure 4 displays the arcade 
differences graphed across the 14 tooth crown dimen-
sions.

Race

The estimate of variance for “race” in this study 
might be criticized because only two groups were in-
cluded and because American blacks and whites have 
experienced several generations of low level gene flow, 
primarily from whites to blacks (literature reviewed in 
Pollitzer, 1999).

On the other hand, Subsaharan Africans and Ameri-

can whites are at either end of the contemporary spec-
trum of human tooth sizes (Harris and Rathbun, 1991), 
except of course for the megadont native Australians 
(e.g., Smith et al., 1981).  Odontometric studies of Ameri-
can blacks and whites routinely find that blacks possess 
significantly larger teeth (Richardson and Malhotra, 
1975; Macko et al., 1979; Vaughan and Harris, 1992).  In 
the future, it may be informative to increase the mix of 
ethnic samples in this assessment of the sources of tooth 
size variation.

The critical issue, however, is recognition of the small 
component of variance attributable to the black-white 
difference, estimated here at 3.5% (Fig. 5).  The minor 
contribution of “race” is no longer surprising (Lewon-
tin, 1972), but these data are confirmatory, using quite 
a different tissue system, that races have been defined 
historically using very superficial criteria, whereas the 
great preponderance of variation is among individuals 
within groups, not among them.

Dimension

Dimension of the tooth crown—whether the crown is 
measured mesiodistally or buccolingually—accounted 
for 3.6% of the total variance.  This is interesting because 
it shows that these geometrically orthogonal axes of a 

Fig. 4. Top:  Graphs of the mean crown sizes by tooth and arcade and plot of the maxillary-minus-mandibular size 
differences (bottom).



90 91

crown are largely coupled in terms of their ontogeny and 
genetic control.  If these two commonly-measured axes 
of crown size (MD and BL) were not strongly related, 
one would expect appreciably more variance to be due 
to this contrast of measured dimensions.  Researchers 
who have studied the genetic control of tooth size (e.g., 
Sofaer et al., 1971; Potter et al., 1976; Townsend and 
Brown, 1978) have commented on differences between 
MD and BL dimensions, but the results often are 
inconsistent among studies, suggesting that sampling 
fluctuations may be at work.

The suggestion has been advanced that MD dimen-
sions have lower heritabilities than BL dimensions be-
cause teeth compete for size of the dental lamina in the 
dental arch.  In contrast, BL dimensions do not.  This 
scenario seems to be insufficient as concerns a couple of 
developmental points.  Teeth do not develop from the 
dental lamina—like beads on a string—instead, they 
develop from projections of condensed mesenchyme 
(i.e., the presumptive dental papilla) that extend away 
from the presumptive occlusal plane, with considerable 
space between them (Arey, 1965; Slavkin, 1974; Ooë, 
1981).  The tooth buds develop in a three-dimensional 
array such that, while their bony crypts may overlap 
mesiodistally, they are offset mediolaterally and cranio-
caudally (van der Linden and Duterloo, 1976; Duterloo, 
1991).  Teeth do not compete for space until their fully 
formed crowns erupt into the oral cavity where under-
developed arch size may cause an arch-size to tooth–

size discrepancy (Little, 1975).  The high prevalence of 
crowding in contemporary westernized populations is 
a recent epidemiological problem that seems to be pre-
dominately acquired rather than inherited (Corruccini 
and Potter, 1980; Harris and Smith, 1980).

Sex

It is well documented that males have bigger teeth 
than females as statistical averages (e.g., Mijsberg, 1931; 
Gonda, 1959; Garn, 1966; Garn et al. 1964, 1967; Harris 
and Bailit, 1987), though the amount of sex difference 
is specific to a population, not a fixed effect (Hanihara, 
1978).  It is a bit surprising, then, that variance due to sex 
accounted for just 1.2% of the total variation in the pres-
ent study (Fig. 6).  On the other hand, humans are char-
acterized by their trivial sexual dimorphism in tooth 
size compared to the great apes (e.g., Harvey et al., 1978; 
Swindler, 2002).  Garn et al. (1967) showed that the ca-
nine was the most dimorphic tooth in humans, at 4-6% 
depending on the group studied, which pales against 
such nonhuman primates as Papio and Pan, where the 
canine is more than half again as large in males as in fe-
males.  The issue should also be considered that univari-
ate analysis tends to exaggerate sex differences because 
redundant male-female differences are included in each 
test (Potter, 1972).

Ditch and Rose (1972) used discriminant functions 
analysis to correctly determine sex in an average of 

Fig. 5. Plot of crown dimensions by race (top) and black-minus-white differences in mean size showing that black 
have larger means throughout the dentition (bottom).

E.F. HARRIS



90 91VARIANCE COMPONENTS IN TOOTH SIZE

93% of their cases (depending on the set of variables 
analyzed), and Garn and coworkers (1977, 1978) ar-
rived at similar success rates.  Brown and Townsend 
(1979) reported lower correct allocations (ca. 75% or 
less) using data from aboriginal Australians—the same 
as reported by Hanihara (1979)—indicating that the de-
gree of sexual dimorphism is not tied to the tooth sizes 
of a group per se.

In passing, researchers also have provided discrimi-
nant functions based on crown sizes of the primary 
teeth that correctly identify sex better than expected 
from chance (DeVito and Saunders, 1990; Tsutsumi et 
al., 1993) even though the primary teeth are much less 
dimorphic (Harris, 2001).

Position

Depending on their position within a morphogenetic 
field (I, P, M), teeth are labeled as “stable” or “variable” 
(Butler, 1939; Dahlberg, 1945).  This dichotomy refers to 
the metric and morphological variation exhibited by a 
tooth.  A stable, early-forming tooth is larger, possesses 
more and larger cusps and other crown features, and is 
less likely to be reduced in size or congenitally absent.  
These and other considerations led Dahlberg (1945, 
1951, 1986) and others to characterize the “fields” of the 
human dentition (Fig. 7).  Several studies have shown 

that the increased variability of distal “variable” teeth is 
due to diminished genetic control (e.g., Lundström, 1948; 
Alvesalo and Tigerstedt, 1974).

(As an aside, this study did not account for the ap-
parent field reversal, where LI1 is more variable than 
LI2, which Kjaer (1980) attributes to the weak vascular 
supply in the mandibular midline because of the sym-
physis menti.)

The present study shows that position is a compara-
tively small axis of variation, estimated at 0.4%, making 
it the most trivial of the factors studied in this model.  
This also emphasizes the caveat (above) that estimating 
the relative sources of variation in the dentition is a dif-
ferent issue than whether particular teeth exhibit statis-
tically significant differences.  A key metrical attribute of 
a pole tooth within a field is its relative metric stability 
(Townsend and Brown, 1981).  Coefficients of variation 
are graphed in Figure 8, where it is seen that it is not a 
foregone conclusion that the later-forming tooth pos-
sesses significantly great variance statistically.  For the 
six contrasts in Figure 8, just three achieved significance 
(α = 0.05 for one-tail tests).  Just the maxillary incisors (I2 
> I1) and the upper and lower molars (M2 > M1) exhibit 
significantly more variance in one tooth vis-à-vis the 
other.  In all these instances, the distal tooth is always 
the more variable tooth.

Fig. 6. Plot of tooth crown dimensions by sex (top) and plot of the male-minus-female differences (bottom) showing 
that, characteristically, males have larger mean crown dimensions.



92 93

OVERVIEW

What are the major axes of variation in the perma-
nent dentition in terms of tooth size?  Results of the 
present study show that the canonical axis is among 
tooth types, which accounts for more than half of the 
variation (59%).  There is a dramatic drop-off after tooth 
type is accounted for.  Arcade (4.9%), race (3.5%), and 
crown dimension (2.6%) have only minor but com-
paratively intermediate values.  Least influential are sex 
(0.9%) and tooth position within a field (0.4%).  None of 
these axes of variation hinges on any one tooth, and the 
fundamental lack of more and more-prominent axes of 
variation is assumed to be due to the strong, pervasive 
statistical and developmental correlations among crown 
dimensions.

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The annual meeting of the Canadian 
Association for Physical Anthropology will be 
held in Edmonton, Alberta, October 23-25 of 2003.  
Contributed papers and posters for a symposium on 
Dental Anthropology are welcome. 

For further information, contact Dr. Nancy 
Lovell, Department of Anthropology, University 
of Alberta, Edmonton, T6G 2H4 Canada.  E-mail:  
nancy.lovell@ualberta.ca

CAPA Meeting

E.F. HARRIS

Decoding Your Subscription
Want to know when your subscription to 

Dental Anthropology expires?  Membership in the 
Association and, thus, your subscription to Dental 
Anthropology is on an annual basis coinciding with 
the calendar year.  Have a look at the mailing label 
on the evelope that this issue arrived in, and you 
will see the year for which your dues have been 
paid.  The year is located in parentheses to the right 
of your name.  So, if the mailing label says “(2003)” 
you are paid to the end of this calendar year.

In order to extend your membership, fill-out the 
relevant portions of the enclosed form—remember 
to include appropriate payment—and mail it to the 
Secretary-Treasurer of the Association:

Dr. Heather H. Edgar
Maxwell Museum of Anthropology
MSC01 1050
1 University of New Mexico
Albuquerque, New Mexico 87131-0001 USA

telephone:  (505) 277-4415
e-mail:  hjhedgar@unm.edu