BULLETINS

OF THE

Zoological Society of San Diego

No. 18

1. TAIL-LENGTH DIFFERENCES IN SNAKES WITH NOTES ON
SEXUAL DIMORPHISM AND THE COEFFICIENT
OF DIVERGENCE.

2. A GRAPHIC METHOD OF SHOWING RELATIONSHIPS.

By L. M. KLAUBER

Consulting Curator of Reptiles, Zoological Society of San Diego

SAN DIEGO, CALIFORNIA
September 1, 1943

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ZOOLOGICAL SOCIETY OF SAN DIEGO

BOARD OF DIRECTORS
1943-1944

T. M. Russell, President
C. L. Cotant, First Vice-President
Gordon Gray, Second Vice-President
Fred Kunzel, Secretary Robert J. Sullivan, Treasurer

F. F. Annable Mrs. Robert P. Scripts

Dr. Thos. O. Burger L. M. Klauber

W. D. Crandall F. T. Olmstead

Milton G. Wegeforth

EXECUTIVE COMMITTEE
T. M. Russell, Chairman
C. L. Cotant
Gordon Gray
Fred Kunzel
Robert Sullivan
F. T. Olmstead

FINANCE COMMITTEE

C. F. Cotant, Chairman
F. F. Annable
F. T. Olmstead
Robert J. Sullivan

hospital and research

COMMITTEE

Rawson J. Pickard, M.D.,
Chairman
W. C. Crandall
Joshua F. Baily, Sc. D.

Howard A. Ball, M.D.

Thos. O. Burger, M.D.

Denis L. Fox, Ph.D.

George F. Kilgore, M.D.

Hall G. Holder, M.D.

Eaton M. MacKay, M.D.
Francis M. Smith, M.D.

Q. M. Stephen-Hassard, D.D.S.
Claude E. ZoBell, Ph.D.
Theodore D. Beckwith, Ph.D.
James D. Edgar, M.D.

Meyer Wiener, M.D.

A. M. Burlingame
C. B. Perkins
F. M. Klauber

LEGAL COMMITTEE
Fred Kunzel
Gordon Gray

building and maintenance

COMMITTEE

F. T. Olmstead, Chairman
George Ray
William P. Kemper
Robert J. Sullivan

EXHIBIT AND EQUIPMENT
COMMITTEE

F. F. Annable, Chairman
Dr. T. O. Burger
Gordon Gray
Mrs. Robert P. Scripps
F. M. Klauber
Earl Warren

GROUNDS COMMITTEE

Mrs. Robt. P. Scripps, Chairman
Mrs. Seth Anderton
Fred Kunzel
F. F. Annable
Allen Perry

EDUCATION COMMITTEE

W. C. Crandall, Chairman
James H. House
Mrs. W. H. Porterfield
W. E. Harper

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Robert J. Sullivan, Chairman
Mrs. Robert P. Scripps

C. F. Cotant

MARINE MUSEUM (STAR OF INDIA)
Capt. Robert Baker, Chairman
Fred Kunzel
Gerald MacMullf.n
Bates Harper

D. P. Marvin
W. C. Crandall

BULLETINS

OF THE

ZOOLOGICAL SOCIETY OF SAN DIEGO

No. 18

I. TAIL-LENGTH DIFFERENCES IN SNAKES,
WITH NOTES ON SEXUAL DIMORPHISM
AND THE COEFFICIENT OF DIVERGENCE.

II. A GRAPHIC METHOD OF SHOWING
RELATIONSHIPS.

BY L. M. Klauber

Consulting Curator of Reptiles,
Zoological Society of San Diego

SAN DIEGO, CALIFORNIA

SEPTEMBER 1, 1943

TABLE OF CONTENTS

Page

I. Tail-Length Differences in Snakes, with Notes on Sexual

Dimorphism and the Coefficient of Divergence 5

Introduction 5

Statistical Formulas 5

Spurious Correlation 9

The Coefficient of Divergence as a Measure of Sexual

Dimorphism in Scutellation 11

Methods and Precautions in Tail-length Studies 15

Variation of Tail-length within Fdomogeneous Populations 21

Correlation of Tail Length and Body Thickness 39

Difference Problems 40

Sexual Dimorphism 46

Rattlesnake Tail Proportionalities 51

Acknowledgments 5 6

Summary and Conclusions 57

Bibliography 5 9

II. A Graphic Method of Showing Relationships 61

Klauber: Tail-length Differences in Snakes

5

TAIL-LENGTH DIFFERENCES IN SNAKES,
WITH NOTES ON SEXUAL DIMORPHISM
AND THE COEFFICIENT OF DIVERGENCE

Introduction

In the course of certain studies of herpetological correlations the pro-
portionate tail lengths of snakes were investigated. This led to a detour
of such extent that it is deemed advisable, in the interest of clarity and
balanced treatment, to offer this discussion of tail length separately. At
the same time it appears opportune to give examples of the use of the
coefficient of divergence as a measure of sexual dimorphism or ontogenetic
differentiation.

The tail-length ratios of snakes often prove useful in diagnosis and
systematics. The proportion seems to be a rather stable character, so that
when differences between related forms do appear, they are likely to be
of importance. However, it is a character with respect to which the
determination of the significance of differences is not simple. In contrast
with characters of lepidosis, which are subject to individual and terri-
torial variations, and sometimes sexual dimorphism, the tail ratio has all
of these and usually an ontogenetic variation as well, that is, a change in
the ratio of the tail length to the length of body as a snake grows.

As is always the case when comparisons are to be made, we must make
sure that our samples — that is, the groups of specimens to be compared —
are homogeneous. Thus, if we are comparing two series with respect to a
character in which sexual dimorphism is present, and we fail to treat the
sexes separately, we may find an apparent difference which really results
from an accidental sexual unbalance in the samples, rather than a true
difference between the forms being compared. And so, as it will be shown
later that tail length is ontogenetically variable in most species of snakes,
complete homogeneity can only be secured if we limit our samples to
specimens of uniform age. The virtual impossibility of obtaining ade-
quate series under such a restriction renders it necessary to make special
statistical provisions for combining specimens of diverse ages. Failure to
take this ontogenetic variation into consideration in making species com-
parisons may lead to false conclusions. It is the purpose of this paper to
discuss methods of combining specimens, and to show the extent of
ontogenetic variation in several example species. Methods of evaluating
differences are developed. Sexual dimorphism is treated.

Statistical Formulas

Problems such as those of tail proportionality, its variation within a
homogeneous series of specimens, and differences between series, are sub-

6

Bulletin 18: Zoological Society of San Diego

ject to both analytic and graphic attack. If the results are to be trust-
worthy both methods are usually advisable, since they supplement each
other in affording an understanding of the nature of the variation involved.

Where, as in the present case, a preliminary survey of the most super-
ficial nature indicates that ontogenetic variation is probably present, the
problem becomes one of correlation — the correlation of tail length propor-
tionality with age. This is not to say that the coefficient of correlation
necessarily affords the best measure of the concomitant relationship; on
the contrary, in most morphological surveys of this kind, where the cor-
relation between a body part and the whole body, or between two body
parts is under investigation, the correlation is sure to be high. In such
cases the direction of the regression line, and the extent and nature of the
scatter of the individual specimens about that line, will be of greater in-
terest and importance than the numerical value of the correlation co-
efficient.

The methods of calculating coefficients of correlation, of determining
regression lines and errors of estimate, will be found fully detailed in every
text book of statistical methods, and therefore will not be discussed here,
ffowever, certain formulas involving the relationship between two parts
and a whole — the body length, tail length, and length over-all of a snake,
for example — are not so readily available, or, if given in a text, may be
in a form not directly applicable to the present problem. These formulas
are therefore given here, although it should be understood that they in-
volve no originality whatever.

The symbols I shall use are as follows: L, B, T, represent length over-all,
body, and tail lengths, respectively. Thus, in each specimen, L = B + T.

Ml, Mi-„ Mr are the sample means of the same quantities, and (Tl, <Tp„ Or
their standard deviations; while (fiuL, Omb and Out ate the corresponding
standard errors of the means. (Tt.b and (Tt.l are the standard errors of esti-
mate of T on B or L; rTB and rT l are the corresponding coefficients of
correlation. V with any suffix represents a coefficient of variation; N the
number of specimens comprising a sample.

T = a ~b bL and T — a' + b'B are the statements of the linear regression
equations which will be discussed, a and a' being the regression constants
and b and b' the regression coefficients.

CD is a statistic to which I refer as the coefficient of divergence, de-
fined as the difference between two means divided by half their sum, that is,

CD = (Mx - My) / y2 (Mx + MY) ,

or, if expressed as a percentage,

CD% = 200 (Mx - My) / (Mx + My).

(Ten is the standard error of CD.

Klauber: Tail-length Dieeerences in Snakes

7

In most taxonomic work we are accustomed to measuring the lengths
of snakes over-all (L) and the tail lengths (T). The length of the body
(B) is not generally recorded. It is possible, without the necessity of
making the subtraction for each specimen, to calculate the statistics of B,
if the statistics of L and T , including their correlation, are available.1

We have, first, the two fundamental equations giving the variances
(standard deviations squared) of sums and differences:

<7l2 = Or2 + <7r2 + 2 (7b (7t rTB

and

CfiT = (Tlj + <V - 2 (7L (7T rTL

It will be observed that each equation contains four quantities; if any
three be known, the other two in both equations can be readily com-
puted. The second equation can be revised to give the value of the vari-
ance of T instead of B by merely interchanging the T’s and B’s throughout.
Thus, any variance can be found if we have the other two and their
correlation.

Any correlation coefficient may be derived from another, and the
several standard deviations, by the equations:

rTL — (Cfp, J'tb h” (7T) / (7l

and

^tb= (aL rTL- aT)/

Or, the correlations can be found directly from the three variances, from
the following equations:

7'tl = (Ol~ + CTt2 - 0»b") / 2 (7b (7t

and

^tb — (CTl" — 6t2 ~ <V) / 2 (7b <7t

This form will sometimes be convenient when it is desired to avoid the
preparation of a correlation table. Again, the T’s and B’s can be inter-
changed in the first equation of each pair, if it be desired to ascertain
the value of Tbl instead of rTL.

Another formula useful in problems dealing with the tail-length ratio
is that which permits one to compute the mean of the individual ratios
T / L or T / B without actually making the division for each specimen.2
Occasionally approximate values of these ratios are derived by dividing
the mean of the tails by the mean of the total lengths, but a result so

1 While the six equations which follow are stated in terms of the particular parts B and
T, and the whole, L, they are, of course, equally suitable for any items comprising two
parts and their sum.

2 The formulas are from Dahlberg, pp. 94 and 200; also see Pearl, p. 370.

8

Bulletin 18: Zoological Society of San Diego

calculated may be somewhat in error. A more accurate value is given by
the formula

Mt

Mt/ l =

AtAl Al2

1 _ r,rL +

M'i’M Ml"

The standard error of this ratio is

Mt

At/ l —

Ml

At2 AtAl

2rTL +

Mt2 MtMl

If one prefers to work with T / B instead of T L, it is only necessary
to substitute B for L throughout both equations, for they are not re-
stricted to a part and a whole as were those previously given. If it is not
desired to calculate Ttl, the following substitution may be made in the
middle term of the right hand member of either equation:

Ttl At Al = (Al“ “h Ax" — Air) / 2

It has been shown in these equations how the standard deviation of the
body length (Ap,), the correlation between the body length and the
length of the tail (rTp,), and the mean ratio of the tail to body length,
may be computed, even though all of the original measurements may have
been recorded in terms of length over-all (L) and tail length(T), and this
without the necessity of calculating the value of the body length ( B )
of each individual specimen, by subtracting T from L. I should also
point out that the constants of the regression equation of T on B can be
obtained from these data without the necessity of setting up a correlation
table of T and B, for

b' — tTb At / AP>,

Mr — Ml — Mt
and a' = Mx — b'M p

or the form b' — (AL2 — AT2) / 2Ap2 - /2 may be used. These equations
involve only factors derived directly from the original statistics of T and L,
or which can be determined by the use of the equations already given.

If the relationship between T and L is linear, that between T and B
will likewise be linear. Further, if there is a constant term in the T on L
equation, there must also be one in the T on B equation, provided the
correlation is high. The latter statement is equivalent to saying that if
the ratio of the tail length to length over-all changes as a snake grows,
then the ratio of the tail length to body length does likewise.

Klauber: Tail-length Differences in Snakes

9

If the correlation Ttl is quite high, approximations to the regression
constants of the T on B equation may be had directly from the following:

a' = a / ( 1 - b)
and b' — b / ( 1 — b)

With respect to the coefficient of divergence, its use will be enhanced
if we know its standard error. In problems involving large samples, when
determining the significance of the difference between two means, Mx - My,
it is satisfactory to compute the ratio of this difference to its standard
error (<Tmxj i- *Tmy2) and then find the resulting significance in a table
of areas under the normal curve. Thus, to determine the significance of
such a difference, when stated in terms of the coefficient of divergence, it
should be satisfactory to divide ( (7m x2 <7My2) 1/2 by /z (Mx + My) which
will thus leave the significance of the difference unchanged. Hence one
might consider ((JMx2 °mt2) 54 / Zz (Mx + My) to be an approximate
value of the standard error of CD. However, I have calculated the standard
error to be as follows:

(Tod — (My“ (TMx h Mx^CTmy2) ^ / (Mx My) 2

Another expression for the same value is

(Ten

( <TmX “1“ ^MT2 ) 1/2

Zz (Mx + My)

1 - 2

(Mx - My) (Omx2 — ^my")

(Mx + My)(0Mx2 + (Tmy2)

(Mx - My)2
(Mx + My)2

In this second form the relationship with the approximate value first
given above may be seen. Since the middle term under the second radical
is slightly greater than the third, this presumably2 more accurate value of
the standard error is somewhat lower than the approximation and will
therefore give a slightly higher significance. Thus, the approximation is
on the safe side.

Spurious Correlation

While the correlation coefficient rTn is useful in computation, it should
not be taken as an indication of the relationship between T and L, since it
involves a form of spurious correlation. For it is obvious that if a part be
larger than normal, the larger part plus a normal remainder will be
larger than normal, and hence the total will also be larger than normal.
This type of spurious correlation between a part and a whole sometimes
vitiates the values of the correlation coefficient which have been presented

3 I used the word "presumably” advisedly. I am not as sure of this derivation as I should
like to be. It was determined by substituting, in the formula for the standard error of
an index, the standard errors of two uncorrelated variables Mx and My. However,
Mx — My and Mx T My are correlated and recognition was given this fact. I had
expected to find the standard error somewhat higher than the approximation, instead of
lower as it worked out.

10

Bulletin 18: Zoological Society of San Diego

in biological studies. It is of little importance when the part under con-
sideration is relatively small, as, for example, in the case of snake heads;
however, it cannot be neglected in discussing snake tails (except in the
case of short-tailed forms like most of the rattlers), particularly in limited-
age groups, where the correlation will be lower than in a sample repre-
senting all ages.

The trend and extent of spurious correlation may be judged from the
fourth equation on page 7. It can be seen that, if the tail is very short
in comparison with the body, CTL / Cr approaches 1, and (Jt Or approaches
zero, so that Ttb is almost equal to Ttl* This is a verification of the state-
ment with respect to the head lengths of snakes.

The following examples wdl serve to illustrate some quantitative effects
of spurious correlation. In 267 adult female specimens of the Platteville
series4 of C. v. viridh, the coefficient of correlation between head and length
over-all is 0.906. This involves some spurious correlation, but it is almost
negligible, for the correlation between head and body is found, by the
use of the equations set forth above, to be 0.899. Similarly, the correla-
tion between the tail length and the length over-all in 102 female Platte-
ville juvenile C. v. viridh is 0.702; the corresponding coefficient for tail
and body is 0.6 5 5. Here the difference is somewhat greater, not only be-
cause the tail is proportionately longer than the head, and has a higher
variance, but because the initial correlation (0.702) is lower. For the
extent of the spurious correlation is affected by the relative sizes of the
standard deviations of the parts, and the gross correlation between either
part and the whole. Assuming substantially equal coefficients of varia-
tion of the parts, the extent of the spurious correlation is greater, the
greater a part may be proportional to the whole; but is less the higher
the true correlation. Thus, in 2 8 female specimens (all ages) of Masticophis
flagellum piceus, from Cape San Lucas, the correlation of the tail to
length over-all is 0.990, and the tail to body correlation is reduced only
to 0.987; and this, notwithstanding the proportionately long tail as com-
pared to the tail of the rattlesnakes, because the true correlation is very
high anyway. The adults of the same series, with a tail to length over-all
correlation of 0.924, have a tail to body correlation 0.869, showing a
greater difference because of the lower true correlation. In 142 specimens
of Pituophis c. annectens from western San Diego County, the correla-
tion between the blotches on the body and the entire number from head
to tip of tail is 0.962; but if the spurious correlation thus involved be re-
moved, the remaining correlation between the blotches on the body and
those on the tad is reduced to 0.637. Here it will be noted the spurious
correlation has been given in terms of the larger part; that is, the pro-

4 This and other series which have been employed in previous studies, will be found
described in Occ. Pap. No. 1, San Diego Soc. Nat. Hist., p. 2, 193 6. Similar territorial
designations for other collections of specimens will be used in this discussion.

Klauber: Tail-length Differences in Snakes

11

portion of the part to the whole is greater than in the other examples.
Hence, the spurious difference is greater, even though the unadjusted or
crude correlation is relatively high.

The Coefficient of Divergence as a Measure of

Sexual Dimorphism in Scutellation

The coefficient of divergence is a useful statistic for evaluating dif-
ferences, since it places them on a common basis of comparison, independent
of the unit used in measuring any particular character, or the numerical
values involved. For example, we may find that the females of one snake
species average 10 more ventrals than the males, while those of a second
species have an average sexual difference of only 5. But the true relative
importance of the sexual dimorphism may not be as 10 to 5, for a single
ventral may be of greater importance in the one case than the other. If
the males in the first species average 200 ventrals and those of the second
100, then the females would have 5 per cent more in both cases, and the
extent of the sexual dimorphism might thus be considered equal in the
two cases. The percentage difference in the two cases may indeed be used
as a measure of a difference of this type. However, I consider the coef-
ficient of divergence, which is the ratio of the difference between the
means to the average of the means, instead of the ratio of the difference
to only one of the measurements, to be a better balanced statistic. It is
equally useful in comparing the extent of the differences shown by two
separate characters within a single species, as, for example, sexual di-
morphism in ventral and subcaudal scales, or characters not measurable
in the same units, such as ventral scales and body blotches.

On page 9 I have given an equation for determining the standard
error of this coefficient. Where the numbers of specimens in the samples
to be compared are relatively large, I think a statement of the coefficient,
plus and minus three times its standard error, will afford a good indica-
tion of the range within which the population coefficient of divergence
would probably fall. I realize that this is not at present recognized as an
approved method of stating such confidence limits.

Before applying the coefficient of divergence to the measure of sexual
dimorphism in tail proportions, I wish to illustrate its use in the measure
of simpler differences, that is, differences in characters not ontogenetically
variable. I shall first employ it as a measure of sexual dimorphism in
scutellation in several homogeneous series of rattlesnakes. As is always the
case in such studies, homogeneity is necessary to avoid pseudo-differences
resulting from an unbalanced territorial representation in the two samples
being compared. For example, if we take widespread samples of C. cinereous
and a considerable majority of the males happen to come from Arizona,
while most of the females are from Texas, an erroneously low figure for
the sexual dimorphism would result, since both sexes have more ventrals

12

Bulletin 18: Zoological Society of San Diego

in Arizona than Texas. On the other hand, if most of the males happen
to be from Texas and the females from farther west, the result would
be too high.

TABLE 1

Extent of Sexual Dimorphism in Efomogeneous Series of Rattlesnakes
In Terms of the Coefficient of Divergence in Per Cent.

Species

Series

Ventrals

Subcaudals

Tail Rings

C. basiliscus

Mexico

- 2.61

-+-

0.41

C. enyo

S. Lower California

- 3.33

0.53

C. m. molossus ....

Arizona

- 2.51

F=

0.52

C. m. nigrescens.

Mexico

- 2.17

W

0.35

C. adamanteus ....

Florida

- 3.64

f=

0.38

C. cinereous

Arizona

- 1.46

zq

0.27

C. cinereous

San Patricio

- 0.95

F-

0.29

C. tortugensis

Tortuga Island

- 0.97

F11

0.35

C. lucasensis

. Cape San Lucas

- 2.11

qz

0.24

C. ruber

San Diego County

- 1.61

A1

0.24

C. scut ul at us

Arizona

- 1.41

F1

0.28

C. v. viridis

Platteville

- 3.80

F3

0.12

C. v. viridis

Pierre

- 4.03

0.14

C. v. nun this ....

..Winslow

- 2.98

F1

0.36

C. v. iutosus

...Utah

- 2.62

0.36

C. v. oreganus ....

. Pateros

- 2.65

zq

0.15

C. v. oreganus. ..

..San Diego County

- 2.70

qz

0.15

C. m. pyrrhus ....

..San Diego County

- 0.62

zq

0.35

C. cerastes

Colorado Desert

- 2.42

F=

0.27

C. /. klauberi

..Arizona

- 0.81

F1

0.42

C. t. pricei

..Arizona

- 3.06

F1

0.55

21.03 qp 1.54

26.07 zq 1.69
16.13 zq 1.47
20.66 qz 1.28

20.00 qz 1.63

22.57 qz 0.97
26.96 zq 1.06

25.3 5 qz 2.54

21.57 qz 0.67
19.39 zq 1.11
26.22 qz 0.98
25.12 zq 0.45
26.16 zq 0.47
29.26 qz 1.32
20.29 qz 1.25
19.82 zq 0.56

21.01 qz 0.60
21.60 zq 1.57
24.28 qq 0.77
21.43 qz 1.18
15.45 zq 1.54

25.91 qz 3.19

28.58 qz 6.20

32.28 zq 2.67
36.64 qz 5.12
20.37 qz 2.18

31.03 zq 2.14

20.08 zq 6.06
19.51 zq 1.45

22.20 zq 2.20
31.40 qz 2.18
26.49 zq 1.36
29.83 zq 1.08

24.20 zq 2.44
23.61 zq 2.52
20.67 zq 1.14

26.08 zq 1.52
23.56 zq 3.21
24.55 qz 2.05
12.32 zq 4.50
13.48 zq 4.50

* Absent, or not clear enough to count. A negative value of the coefficient means that the females exceed
the males. The use of the cp sign throughout this paper indicates that the following figure is the standard,
rather than the probable, error.

Table 1 sets forth the sexual dimorphism of several series of rattlesnakes
in terms of the coefficient of divergence and its standard error, in three
characters in which the sex differences are known to be considerable, that
is, ventrals, subcaudals, and tail rings. Table 2 presents similar data on
four characters in which the sexes are usually assumed to be the same, these
being scale rows, supralabials, infralabials, and body blotches. In cases
where the males are higher, the coefficient is considered to be positive,
while it is negative where the females are greater. In the second table, in
order to simplify the statement of the coefficient, I have merely indicated
by asterisks whether the sexual difference is significant, instead of giving
the standard error.

Klauber: Tail-length Differences in Snakes

13

TABLE 2

Extent of Sexual Dimorphism in Homogeneous Series of Rattlesnakes
In Terms of the Coefficient of Divergence in Per Cent.

Scale

Supra-

Infra-

Body

Species

Series

Rows

labials

labials

Blotches

C. cinereous

Arizona

1.06*

- 0.42

1.20*

0.34

C. lucasensis

Cape San Lucas

- 1.05*

- 1.03*

0.63

1.54

C. ruber

San Diego County

- 0.23

- 1.43**

- 0.40

1.82*

C. scutulatus

Arizona

- 0.26

0.02

1.3 8**

0.85

C. v. viridis

Platteville

0.53*

1.64**

0.68*

- 1.62*

C. v. viridis

Pierre

- 0.11

1.12**

1.04**

- 0.11

C. v. oreganus

San Diego County

- 0.51

1.03**

0.28

- 1.02

C. cerastes

Colorado Desert

- 0.83

- 0.37

- 2.02*

- 0.39

Significant (5 per cent level or below)

highly significant (1 per cent level or below).

It will be observed that the sexual dimorphism in the ventrals is always
negative — that is, the females exceed the males- — -the extent varying
from somewhat above l/i per cent to 4 per cent. The divergence in both
subcaudals and tail rings is positive and much higher, running from 16
to over 30 per cent. There is evident a considerable degree of generic uni-
formity; and some of the related subspecies or species have fairly con-
sistent divergences. All coefficients are highly significant, that is, the ratio
of each difference to its standard error is above 2.5 8 (the 1 per cent
significance level) except the ventral differences in pyrrhus and klauberi,
which do not quite reach the 5 per cent level.

The sexual dimorphism shown by the other four characters presented
in Table 2 is much less consistent. In only half the cases tested is the
sexual dimorphism significant; in 5 instances the result is highly sig-
nificant. It is probable that there is a slight tendency of the females to
have more scale rows than the males, such being the case in six out of
eight samples. While not evident in the rattlesnakes, marked sexual dim-
orphism in scale rows is by no means unknown among snakes. For example,
in the sea snakes Lapemis hardwickii from Manila Bay the coefficient of
divergence in scale rows at mid-body, deduced from 2 3 males and 31
females, is 12.6 per cent and is highly significant. Do Amaral1' lists the
mid-body counts of 94 males and 116 females of Bothrops alternata, from
which the coefficient of divergence is found to be 7.7 per cent, the females,
as usual, having a higher number of rows. Miss Cochran has shown1 that,

5 The counts were made by the late Dr. J. C. Thompson and were received through the
courtesy of Mr. J. R. Slevin.

6 Mem. Inst. Butantan, Vol. 8, p. 178, 1934.

T Bull. 117, U.S.N.M., p. 330, 1941.

14

Bulletin 18: Zoological Society of San Diego

in the species of the genus Uromacer found in Hispaniola, the males al-
most universally have 1 1 scale rows at the base of the tail and the females
13. This indicates a coefficient of divergence of nearly 17 per cenit.

But in the rattlers these differences do not exceed one percent. Table 2
shows the sexual differences in labials and body blotches to be quite vari-
able, both in nature and extent, the highest being just above 2 per cent in
cerastes infralabials.

Of course there is nothing novel in the statement that male rattle-
snakes have fewer ventrals and more subcaudals than females; these facts
have been evident since the earliest studies of the genus were made. These
data are only presented to show how the coefficient of divergence can
be used as a measure of the extent of these differences; or to express them
in numerical terms, so that relative differences between species can be
evaluated. Thus we can say that adamanteus shows a higher sexual
dimorphism in ventral scales than cinereous , and that pricei is unusually
low (compared to most other rattlesnakes) in the extent of the sexual
dimorphism in subcaudal scales. Further we are able to say that in
adamanteus sexual dimorphism is more evident in subcaudals than in vent-
rals, and is higher in tail rings than either ventrals or subcaudals.

TABLE 3

Extent of Sexual Dimorphism in Homogeneous Series of Snakes
In Terms of the Coefficient of Divergence in Per Cent.

Species Series or Locality

Adelphicos q. sargii Volcan Zuml

Geo phis nasalis Volcan Zunil

As pi dura guentheri Ceylon

Uromacer catesbeyi Hispaniola

Diadophis a. similis San Diego Bay Area

Pbyllorhynchus d. perkinsi Borego Area

Arizona e. occidentals Desert San Diego Co.

Pituophis c. annectens San Diego County

Lampropeltis g. calif orniae San Diego County

Sonora m. linearis Laguna Island

Sonora o. annulata San Diego County

Tbamnophis o. ordinoides W. Oregon

Tbamnophis hammondii San Diego County

Tbamnophis radix Cook County, 111.

Hypsiglena ocbrorbynchus San Diego County

Micrurus n. nigrocinctus Panama and C. Z.

M'crurus n. divaricatus Honduras

Ventrals

- 7.99

qz

0.42

- 2.61

zp

0.28

- 4.52

A1

0.38

5.52

A1

0.67

- 5.57

qz

0.25

- 7.58

zp

0.19

- 5.15

qz

0.51

- 2.14

A1

0.26

- 0.79

A :

0.36

- 6.73

A 1

0.38

- 6.05

A ;

0.31

1.06

=P

0.35

4.17

A-

0.20

4.01

zp

0.36

- 3.22

qz

0.58

- 7.79

A z

0.38

- 6.24

A1

0.41

Subcaudals

13.40 qz 1.61
20.92 zp 0.75
20.51 qp 1.18

1.03 qp 0.47
9.28 zp 0.68

25.41 zp 0.60
8.90 zp 0.90
9.30 zp 0.69
7.80 zp 0.87

12.37 qp 1.00
12.49 zp 0.81
12.48 qp 0.64
10.94 zp 0.32
12.24 qp 0.71
15.10 zp 1.29

28.42 zp 0.95
27.89 zp 1.13

Klauber: Tail-length Differences in Snakes

15

Table 3 presents data on the ventral and subcaudal sexual divergence
of a number of snakes other than rattlers. Again we note the superiority
of the females in ventrals, except in certain genera such as Uromacer and
Thamnophis ; also that divergence is usually greater in subcaudals than
ventrals; and that sexual dimorphism in subcaudals tends to be greater
in short-tailed than in long-tailed species.

TABLE 4

Extent of Racial Dimorphism

in Homogeneous Series

of Crotalus viridis

In Terms of the Coefficient of Divergence in

Per Cent.

Difference

Difference

between

between

Platteville

Platteville

and

and

New Mexico

Nuntius

Character

Specimens

Specimens

Scale rows

3 . 5 0 *

8.81**

Ventrals, males

2.55* *

4.36**

Ventrals, females

2.92**

5.07**

Subcaudals, males

1.98*

5.02**

Subcaudals, females

1.96

9.00**

Supralabials

-0.55

0.00

Infralabials

0.25

3.31**

Body blotches

5.04**

1.88*

Tail rings, males

2.26

2.81*

Tail rings, females

3.79

3.96

* Significant (5 per cent level); ** highly significant (1 per cent level).

In Table 4 the coefficient of divergence has been employed as a measure
of territorial variations or clines. Here are shown, first, the differences be-
tween the Platteville series of Crotalus v. viridis from northeastern Colo-
rado and the same subspecies in New Mexico; and secondly the differences
between the Platteville specimens and C. v. nuntius from northeastern
Arizona, which intergrades with C. v. viridis along the Arizona-New
Mexico border. It will be observed that the more important characters —
scale rows, ventrals, and subcaudals — show an increasing difference with
wider territorial separation; but this trend is not manifest in all characters.

Methods and Precautions in Tail-Length Studies

If a morphological character such as tail-length proportionality is to
be thoroughly investigated, a graphical approach is to be strongly recom-
mended, even though the final determination of the regression equation
be by analytical methods. By graphical is meant the plotting of each
specimen on rectangular co-ordinate paper with lengths over-all (or body
lengths) along the horizontal scale and tail lengths on the vertical (Fig. 1).

16

Bulletin 18: Zoological Society of San Diego

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120 160 200 240 280 320 360 400 440 480

LENGTH OVER-ALL MM.

Klauber: Tail-length Differences in Snakes

17

The scales should be so arranged that the origin is in evidence; that is,
even if no snakes of the species being investigated are below 300 mm. in
length, do not begin the bottom scale at 300, but at zero. By so doing it
will be possible to check the regression formula by noting the points of
intersection with the axes. In plotting specimens the use of different sym-
bols for the sexes is to be recommended; if specimens from different dis-
tricts are to be compared, colors may be employed to advantage. However,
if the fit of regression lines to the primary data is to be studied, it will not
be found advisable to plot too many series on a single sheet.

This graphical method of attack is virtually imperative, even though
followed by a complete correlation study, since it will give a necessary
picture of the general relationship. It will indicate at once whether suf-
ficient curvature is in evidence in a trend line to warrant a test for
linearity. It will give an idea of the extent of sexual dimorphism, and also
of territorial differences. While not determining the significance of the
constant term ( a ) in the regression equation, it will at least indicate its
relative importance and whether it is at all safe to consider the tail-length
ratio ontogenetically constant. It will afford a good idea of the nature
and extent of the scatter of the individual specimens about the regression
line. Aberrant specimens will stand out conspicuously; these may then be
given further study to find whether they may not result from inaccurate
sexing, incomplete tails, inaccurate measurements, mistakes in recording,
or other errors, as will not infrequently be found to be the case. And even
though these non-conformists prove to be true aberrants, it may be de-
sirable to give consideration to their elimination from the data used in
the regression computation. While this may seem a questionable or non-
scientific procedure, it is often advisable; for a single badly distorted or
freakish specimen can considerably affect the determinations, particularly
if the sample be relatively small; and it is to be remembered that what we
seek is the mode (using the word in its non-technical sense) of the normal
population, uncomplicated by freaks.

The graphical set-up is also useful in checking the accuracy of the con-
stants of the regression equation, as determined by the subsequent cal-
culations.

If it be desired to depend entirely on a graphic determination of the
regression line, a procedure requiring only a minor fraction of the time
needed for an analytical study, the use of a black thread, rather than a
straightedge, is to be recommended. One end of the thread should be
fastened to the back edge of the table or drawing board; the other end,
to which a small weight is attached, is allowed to dangle over the near
edge. Then the sheet containing the plot of the specimens may be manipu-
lated under the thread until what seems to be the best fit is obtained. The
co-ordinates (Tj and L1; T2 and L2) of any two points on the line (pref-

18 Bulletin 18: Zoological Society of San Diego

erably near opposite ends) are recorded. The required equation T -a + bL
may then be determined from the following:

a= (riL2-T2L1)/(L2-Ll) and b= (T2 - T,) / (L, - Lx)

Values of B may be substituted for values of L if one prefers working
with body length instead of length over-all. The calculations will be
simplified if one of the points be selected on an axis, so that either T1 or Ll
become zero.

With some practice, particularly if the dispersion about the regression
line is moderate in extent, and there is a fairly even ontogenetic distri-
bution of the specimens, a quite accurate determination of the regression
line can be obtained by this simple method. In placing the line, it is well
to remember to balance vertical deviations on each side of the line against
each other, rather than deviations perpendicular to the line. If this be not
done, a tendency will be noted whereby the graphically determined line
will have a slightly steeper slope than a line found by the least-squares or
analytical method.

A still more accurate placing of the regression line may be secured by
totaling the lengths over-all (or body lengths) and the lengths of the
tails, and dividing by the number of specimens, thus securing the values of
Ml and Mt. These values give the co-ordinates of a point through which
the regression line should be made to pass. Or the juvenile and adult means
may be determined separately and the line passed through the two points
thus located.

In an analytical determination, in computing the values of Mt and Mb,
and the standard deviations, no specimen should be included in calculating
one statistic unless it be available for all. For this reason, even though
values of B, rather than L, may be used, no specimen with an incomplete
tail should be tabulated.

The determination of boundary lines between the sexes will be found
discussed on page 5 6.

Incomplete tails are particularly prevalent in attenuated forms; this
is complicated in some species by the absence of a characteristic terminal
cone,s which makes it almost impossible to determine whether the tail
really is complete. Such is the case, for example, in Masticophis flagellum.
There is some evidence from examinations of broods that certain species
of snakes normally lose the tips of their tails shortly after birth, possibly
in the course of shedding. Also, it seems as if some snakes are able to
regenerate a tail tip in a manner which makes it quite difficult to ascer-
tain whether the tail is complete.

s In some forms the presence of a prominent terminal cone, containing a rather char-
acteristic longitudinal crease, makes the determination quite definite.

Klauber: Tail-length Dielerences in Snakes

19

Errors in sexing should be minimized; obviously they will tend to re-
duce the true extent of sexual dimorphism. They are likely to occur most
frequently among the juveniles, yet the juveniles cannot be omitted if
accurate regression lines are to be obtained.

A graphic set-up of the ventral-subcaudal relationship will often sub-
stantiate tail-length discrepancies in certain specimens — that is, the same
specimens will be found to deviate from the mode in both plots, thus
clearly indicating those which should be the subject of a re-examination.
While it should not be forgotten that there may be cross-correlation in
these characters, so that aberrance in one may be coincident with ab-
normality in the other, this type of double check will frequently disclose
errors of sexing, measurement, or recording.

As we may well suspect the existence of ecological variations, our
samples should be limited, as far as possible, to collections from quite re-
stricted localities. However, from the standpoint of accuracy of determi-
nation, meaning the adherence of the sample statistics to the true popu-
lation parameters, an increase in the number of specimens contained in
a sample is decidedly beneficial. Therefore, if there is no evidence of
territorial variation, it may sometimes be advantageous to employ a
larger sample from a less restricted locality.

No definite rule can be laid down with respect to the adequacy of
samples for the determination of tail length proportionality, except this:
the larger the better, if homogeneity be not sacrificed. One can hardly
expect to secure dependable results with less than twenty-five full-tailed
specimens of each sex. The accuracy will also depend to an important
degree on the ontogenetic scatter of the sample; it is essential that both
juveniles and adults be represented in the sample, if an ontogenetically com-
plete and accurate regression line is expected. These terminal individuals
tend to pull the line into its proper position; limited-length groups will
seldom produce the same, or even a closely approaching equation. The
reason for this is not necessarily a general curvature of the regression
line, under which circumstances the first degree regression line of a
limited-age group would tend to produce a tangent to the complete-age
line. Rather it is the greater effect that deviations of individuals from the
normal have upon the determination of the regression line when only a
limited-age group is included in the study.

There is, in fact, one condition in the method of analysis that to some
extent renders a graphical determination superior, or at least makes it
important for purposes of comparison. This is the fact that deviations
from the regression line (as will subsequently be shown) are approxi-
mately proportionate to tail length. Thus adult abnormalities — that is,
abnormal deviations from a truly representative regression line — affect
the determination of the line to a much greater degree than proportionate
deviations in juveniles. Sometimes a single aberrant adult will throw the

20

Bulletin 18: Zoological Society of San Diego

line so far out of position that it will clearly not be truly representative
of the species, particularly in the juvenile range; this is especially true if
there be a single very large adult which is in any way abnormal. Under
such circumstances a specimen of this kind should be omitted. In some series
where the number of juveniles is inadequate to balance the adults the
most satisfactory solution is to duplicate the entry for each juvenile.

This relative ineffectiveness of the juveniles might be compensated for
by using the logarithms of the measurements instead of the lengths them-
selves. However, if the linear relationship includes a constant term, the
line is no longer straight when plotted on a logarithmic basis, which
complicates the calculations. For this reason I prefer to use the simpler
method, but with a close scrutiny of the graphic situation to eliminate
any juvenile inadequacies or over-effects of aberrant adults. It should be
understood that these modifications are only made when an available
series is inadequate in numbers or unbalanced in ontogenetic distribution.

There is also the matter of accuracy and uniformity of measurement.
The tail length should be measured to the center, rather than the lower
edge, of the anal plate. Attempts to record stiff and contorted specimens
should be avoided. Where possible, specimens should be measured when
freshly killed, since the data so derived will represent a true natural rela-
tionship; furthermore, unhardened specimens may be stretched out along
a ruler and thus more accurately measured. However, one is usually de-
pendent on preserved specimens which have already set; in these there
will be some differential shrinkage between body and tail, resulting from
preservation. This is not likely to prove of importance in taxonomic work
unless two series to be compared have been preserved by different methods.

The following data will serve to illustrate the relative shrinkage resulting
from preservation in alcohol:

Shrinkage
in per cent

Specimens

Body

Tail

Masticophis f. piceus

22

2.76

0.34

Pbyllorbyncbus d. perkinsi

18

3.15

0.43

Pit uop bis c. annectens

28

2.09

0.22

It will be observed that the tail suffers almost no shrinkage; it is not
unusual for specimens to show a slight increase in length after preserva-
tion. It is probable that, on the average, the body shrinkage will not
usually exceed 3 per cent, although it will be found to be higher than
this figure in young specimens.

Klauber: Tail-length Dieeerences in Snakes

21

Variation of Tail Length Within Homogeneous Populations

Having discussed some of the methods and precautions involved in the
utilization of tail length proportionality in taxonomic studies, I shall
now illustrate their use with results secured from a variety of species. As is
so often the case in evaluating a diagnostic character it is desirable first
to ascertain the extent and the nature of the variation of such a character
within homogeneous groups, before attempting to discover the degrees
of difference between groups or the extent of sexual dimorphism.

Assuming the availability of homogeneous series of snakes, the principal
questions to be answered respecting the tail-length ratios are: (1) is the
relationship linear, or substantially so? (2) does the ratio of the tail
length to total length remain constant, or vary with age? (3) in what
manner does the dispersion of individuals about the species regression line
change with age?

To answer these questions, 48 species and subspecies of snakes were
investigated, the resulting statistics being set forth in Tables 5 to 9, in-
clusive. However, before the determinations are discussed, it is necessary
to supply some descriptive information on the several samples, since they
vary considerably in adequacy, and the geographical data cannot be
given in the tables, although the numbers of available specimens will be
found there.

Lichanura roseofusca roseofusca Cope.

A series fairly well distributed ontogenetically; the specimens are from
extreme northern Lower California, and San Diego and southern River-
side counties, in California. Both coastal and desert specimens are included,
since there seem to be no important tail-ratio differences. I was rather
surprised to find that the scarred and blunt tails, so frequent in the
adults of this species, which I had always supposed were mutilated and
truncated, are presumably complete, since they fall along a cortsistent
regression line. Evidently, the bluntness must result from internal causes,
possibly from the swelling of the anal scent glands, which are very large
and odorous in this species.

TABLE 5

Relationship of Tail Length to Length Over-all

Species or

Number of
Specimens

Coefficient of
Correlation, rTL

Regression
Coefficient, b

Regression
Constant, a

Standard Errors

Ob

On

Standard Error
of Estimate

Mean Coefficient
of Variation, %

Subspecies

M

F

M

F

M

F

M

F

M

F

M

F

M

F

M

F

Lichamira r. roseofusca

43

57

.983

.992

.166

.143

- 14.32

- 6.15

.0049

.0026

2.94

1.61

6.22

3.75

7.83

4.81

Adelphicos q. sargii

67

62

.984

.971

.162

.121

- 2.98

-0.93

.0036

.0039

0.88

0.90

1.56

1.80

5.56

6.84

Geophis brachycephala

25

24

.997

.990

.204

.163

-4.26

- 0.28

.0031

.0050

0.70

1.24

1.10

1.36

2.83

3.52

Geophis nasalis

115

87

.980

.976

.199

.147

- 6.37

-2.21

.0038

.0036

0.81

0.76

1.96

1.44

5.53

5.10

Diadophis a. similis

... 133

134

.978

.981

.212

.179

- 5.50

- 3.59

.0039

.0031

1.10

0.95

2.48

3.08

4.64

6.22

Diadophis p. arnyi I

71

40

.948

.973

.192

.159

- 0.28

- 0.40

.0077

.0061

1.88

1.47

3.28

2.76

7.24

7.6 5

Diadophis p. arnyi II

247

183

.983

.974

.210

.164

- 4.87

- 2.00

.0025

.0028

0.57

0.61

2.27

2.36

5.48

7.32

Boaedon l. lineatus I

23

33

.974

.986

.181

.130

- 7.76

-4.36

.0091

.0039

5.59

2.87

7.07

3.95

7.12

4.49

Boaedon I. lineatus II

21

12

.994

.998

.206

.145

- 10.60

- 3.13

.0050

.0028

2.25

1.50

2.23

1.97

2.81

2.87

Masticophis f. piceus

25

34

.986

.991

.270

.268

-4.69

1.21

.0095

.0065

8.38

6.97

17.86

13.43

8.57

4.95

Masticophis lateralis

23

24

.996

.992

.315

.304

-9.11

- 5.40

.0058

.0082

5.20

7.66

7.45

10.02

2.85

3.76

Salvador a g. virgultea

23

23

.996

.995

.237

.230

- 2.38

-2.01

.0047

.0049

3.34

3.18

4.72

4.29

2.98

3.02

Phyllorhyn chns b. brown i

25

12

.960

.966

.161

.084

- 7.60

-1.06

.0097

.0072

2.87

2.19

2.54

1.79

6.52

7.50

Phyllohrynchus d. nubilus

8

4

.996

.999

.177

.080

- 11.62

0.11

.0062

.0023

1.85

0.66

1.09

0.39

2.75

1.79

Phyllorhynchus d. perkinsi I

214

148

.973

.976

.179

.106

- 12.01

- 3.41

.0029

.0020

1.02

0.67

3.58

2.15

7.32

6.75

Phyllorhynchus d. perkinsi II

34

16

.991

.957

.182

.107

- 12.30

-2.48

.0043

.0087

1.60

2.63

2.25

2.72

4.26

9.44

Arizona e. occidentals I

50

25

.993

.998

.132

.131

1.85

- 5.53

.0023

.0016

1.62

1.28

3.05

1.90

3.36

2.34

Arizona e. occidental's II

46

12

.990

.995

.143

.133

- 2.63

- 3.40

.0031

.0041

2.02

2.05

3.36

2.45

3.79

4.23

Pituophis c. annectens

155

144

.993

.991

.165

.148

3.91

4.61

.0016

.0017

1.37

1.20

7.48

6.52

5.58

6.53

Pituophis c. deserticola I

31

23

.994

.995

.127

.120

8.50

5.31

.0026

.0027

2.23

2.39

4.94

4.44

4.59

4.31

Pituophis c. deserticola II

23

15

.990

.994

.148

.144

0.57

- 7.35

.0047

.0044

4.50

3.86

6.68

4.87

4.95

4.37

Lampropeltis g. calif orniae

278

249

.977

.984

.131

.128

3.62

1.73

.0017

.0015

1.05

0.62

8.04

6.41

10.84

8.78

Khinocheilus l. lecontei I

30

23

.991

.994

.136

.126

3.01

1.77

.0034

.0031

2.10

1.56

3.77

2.74

4.58

4.56

Rhinocheilus l. lecontei II

35

10

.988

.992

.139

.136

1.79

- 1.99

.0038

.0062

2.51

3.43

3.68

3.27

4.05

4.70

Khinocheilus l. clams

29

13

.987

.983

.146

.145

- 1.77

-8.10

.0046

.0081

3.04

4.87

3.62

3.59

3.95

4.68

Toluca l. lineata I

20

29

.995

.982

.212

.148

- 2.22

0.00

.0050

.0056

0.99

1.08

1.07

1.41

2.78

5.06

Toluca l. lineata II

39

28

.990

.981

.220

.168

-4.71

- 3.10

.005 1

.0065

1.04

1.30

1.82

1.92

4.78

6.58

Toluca l. varians

71

69

.993

.979

.221

.161

- 3.74

- 0.14

.0031

.0041

0.62

0.92

1.71

2.13

4.53

6.22

Sonora m. linearis

43

57

.996

.990

.219

.181

-2.74

0.00

.0030

.0034

0.87

0.74

1.15

1.39

2.64

3.66

Sonora o. occipitalis

39

13

.971

.969

.195

.184

- 2.82

- 5.61

.0079

.0141

2.24

3.78

1.95

2.43

3.72

5.66

Sonora o. annulata

200

119

.974

.978

.204

.178

- 2.92

- 2.49

.0034

.0035

0.94

0.94

2.68

2.39

5.04

5.21

Chilomeniscus s. stramineus

22

22

.957

.954

.137

.122

1.83

0.17

.0108

.0086

2.01

1.71

1.13

1.36

4.20

5.66

Thamnophis hammondii

168

153

.994

.995

.246

.220

- 0.29

3.56

.0021

.0019

0.87

0.81

4.29

4.45

4.69

5.00

Thamnophis o. ordinoides

74

84

.991

.983

.274

.233

- 6.86

- 2.67

.0044

.0048

1.56

1.74

2.71

4.37

3.09

5.63

Thamnophis o. biscutatus

107

95

.992

.993

.254

.232

3.97

4.32

.0031

.0028

1.52

1.31

6.28

5.67

5.35

5.51

Conopsis nasus

61

70

.982

.981

.152

.117

- 0.17

0.45

.0038

.0028

0.92

0.68

1.79

1.36

5.06

4.92

Hypsiglena ochrorhynchus

55

30

.971

.986

.187

.152

- 4.64

- 1.19

.0064

.0048

1.88

1.34

2.52

1.82

5.06

4.57

T rimorpbodon vandenburghi

22

16

.974

.993

.163

.135

- 0.05

3.76

.0085

.0043

4.81

2.64

4.44

4.17

4.90

5.29

Tantilla eiseni

18

15

.995

.992

.263

.2 51

-4.53

- 8.34

.0067

.0090

1.90

2.02

1.67

2.52

2.44

5.63

Elapsoidea niger

1 5

29

.967

.974

.086

.082

- 3.67

- 7.43

.0062

.0037

2.34

1.48

1.56

1.43

5.56

5.63

Micrurus a. mayensis

14

16

.977

.976

.161

.115

-7.28

- 1.13

.0104

.0077

6.76

4.67

3.57

2.87

3.69

4.24

Micrurus n. nigrocinctus

39

32

.994

.991

.148

.099

- 3.99

- 0.87

.0028

.0025

1.28

1.45

2.88

3.27

4.51

6.35

Micrurus n. divaricatus

38

28

.996

.995

.162

.105

- 6.53

- 0.43

.0023

.0021

1.14

1.25

3.06

2.62

4.70

4.65

Agkistrodon m. mokeson

169

131

.973

.976

.122

.120

10.69

7.71

.0022

.0024

1.28

1.24

5.45

4.30

7.12

6.34

T rimeresurus gramineus

24

16

.994

.997

.219

.164

- 12.98

0.52

.0053

.0033

2.84

1.40

2.34

1.56

2.29

2.30

Trimeresurus elegans

17

12

.990

.997

.188

.156

- 6.55

3.78

.0070

.0035

5.30

2.16

5.90

2.69

4.54

2.71

Bothrops insularis

92

94

.958

.956

.145

.122

2.78

4.34

.0053

.0039

3.68

2.85

3.96

4.88

3.87

5.31

Atheris squamigera

22

25

.965

.977

.195

.154

- 13.67

- 5.23

.0126

.0070

6.00

3.57

4.72

3.21

6.09

4.43

TABLE 6

Relationship of Tail Length to Body Length

Species or

Nun

ber of

Coefficient of

Reg

ession

Regression

Standard Errors

Standard Error

Mean Coefficient

Subspecies

Specimens

Correlation, rTIt

Coefficient, b

Constant, a

<V

fl

„■

of Estimate

of Variation, %

M

F

M

F

M

F

M

F

M

F

M

F

M

F

M

F

Lichanura r. roseofusca

43

57

.975

.990

.197

.167

- 16.32

-6.93

.0070

.0032

2.94

1.61

7.44

4.38

9.37

5.61

Adel phi cos q. sargii

67

62

.978

.962

.192

.136

- 3.33

- 0.79

.005 1

.0050

0.86

1.03

1.85

2.05

6.60

7.78

Geo phis brachycephala

25

24

.996

.985

.256

.193

- 5.28

- 0.14

.0054

.0072

0.98

1.48

1.38

1.62

3.56

4.20

Geo phis nasal is

115

87

.969

.966

.246

.171

-7.42

-2.27

.0059

.0050

1.05

0.91

2.44

1.71

6.90

6.04

Diadophis a. similis

133

134

.964

.972

.265

.215

-6.05

- 3.82

.0063

.0045

1.45

1.17

3.14

3.74

5.88

7.57

Diadophis p. arnyi I

71

40

.920

.962

.230

.186

1.15

- 0.01

.0118

.0086

2.32

1.75

4.05

3.27

8.93

9.09

Diadophis p. arnyi II

247

183

.972

.963

.262

.193

- 5.60

- 1.95

.0040

.0040

0.74

0.74

2.87

2.82

6.93

8.74

Boaedon l. li neat us I

23

33

.967

.984

.219

.149

- 8.65

-4.77

.0126

.0049

6.40

2.52

7.98

4.29

8.04

4.87

Boaedon 1. lineatus 11

21

12

.991

.998

.259

.170

- 13.01

- 3.62

.0080

.0038

2.92

1.76

2.81

2.31

3.53

3.36

Masticophis f. piceus

25

34

.973

.983

.364

.362

- 3.31

4.19

.0178

.0121

11.51

9.48

24.42

18.32

11.71

6.75

Masticophis lateralis

23

24

.993

.984

.457

.432

- 12.00

- 5.00

.0084

.0170

5.27

1 1.06

10.89

14.36

4.17

5.39

Salvadora g. virgultea

23

23

.993

.992

.310

.298

-2.55

- 2.09

.0125

.0124

6.83

6.23

9.59

8.38

6.05

5.89

Phyllorhynchus b. brou/ni

25

12

.943

.959

.188

.091

- 8.14

-0.91

.0138

.0085

3.51

2.40

3.02

1.96

7.75

8.18

Phyllorhyncbus d. nubiius

8

4

.995

.999

.215

.087

- 14.04

0.12

.0092

.0027

2.35

0.72

1.32

0.43

3.35

1.95

Phyllorhynchus d. perkinsi I

214

148

.960

.970

.214

.118

- 13.68

- 3.28

.0043

.0025

1.29

0.76

4.35

2.41

8.91

7.54

Phyllorhynchus d. perkinsi II

34

16

.987

.946

.222

.119

- 14.71

- 2.38

.0065

.0109

2.03

2.97

2.75

3.04

5.20

10.56

Arizona e. occidentals I

50

25

.990

.998

.151

.151

2.37

- 5.75

.0031

.0021

1.86

1.31

3.50

2.19

3.87

2.69

Arizona e. occidentalis II

46

12

.986

.994

.167

.153

- 5.75

- 2.70

.0042

.0055

2.37

2.37

3.92

2.82

4.43

4.87

Pitiiophis c. annectens

155

144

.990

.988

.197

.174

5.10

5.76

.0022

.0023

1.63

1.41

8.95

7.66

6.67

7.66

Pituophis c. dcserticola I

31

23

.992

.993

.145

.136

9.93

6.19

.0034

.0035

2.53

2.71

5.66

5.08

5.26

4.93

Pituophis c. deserticola II

23

15

.986

.992

.172

.168

1.25

- 8.31

.0064

.0060

5.27

4.54

7.83

5.69

5.80

5.10

Lampropcltis g. calif orniae

278

249

.969

.979

.149

.145

4.74

2.38

.0023

.0019

1.19

1.05

9.26

7.3 5

12.48

10.07

Rhinocheilus 1. lecontei I

30

23

.988

.992

.157

.144

3.73

2.15

.0045

.0041

2.41

1.75

4.36

3.13

5.29

5.21

Rhinocheilus 1. lecontei II

35

10

.984

.989

.161

.157

2.48

-2.09

.0051

.0083

2.90

3.98

4.27

3.78

4.70

5.44

Rhinocheilus /. clarus

29

13

.982

.977

.170

.168

- 1.S6

- 8.89

.0063

.0111

3.57

5.77

4.24

4.20

4.62

5.46

Toluca 1. lineata I

20

29

.992

.975

.268

.173

- 2.68

0.22

.0050

.0056

1.27

1.26

1.36

1.65

3.53

5.93

Toluca 1. lineata II

39

28

.984

.973

.280

.201

- 5.73

- 3.42

.0084

.0094

1.37

1.59

2.34

2.31

6.12

7.91

Toluca 1. varians

71

69

.989

.970

.282

.190

- 4.59

0.18

.0031

.0041

0.81

1.09

2.19

2.54

5.81

7.41

Sonora m. linearis

43

57

.994

.985

.280

.220

- 3.38

0.20

.0079

.0051

1.37

0.90

2.37

1.70

5.43

4.47

Sonora o. occipitalis

39

13

.955

.954

.238

.222

- 2.50

- 5.99

.0079

.0141

2.81

4.73

2.42

2.98

4.62

6.93

Sonora o. annul at a

200

119

.959

.967

.252

.214

- 2.69

- 2.44

.0053

.0052

1.20

1.21

3.36

2.91

6.32

6.33

Chilome nis cus s. stramineus

22

22

.941

.940

.156

.136

2.54

0.57

.0125

.0111

1.98

1.94

1.31

1.55

4.86

6.43

Thamnophis hammondii

168

153

.989

.991

.324

.280

0.12

4.89

.0038

.0031

1.15

1.03

5.69

5.69

6.21

6.40

T ham no phis o. ordinoides

74

84

.982

.968

.374

.299

- 8.50

- 2.35

.0084

.0085

2.20

2.37

3.73

5.93

4.25

7.65

Thamnophis o. biscut atus

107

95

.986

.989

.338

.300

6.14

6.13

.0056

.0047

2.02

1.69

8.42

7.37

7.18

7.16

Conopsis nasus

61

70

.975

.975

.178

.132

0.08

0.66

.0053

.0036

1.09

0.76

2.11

1.54

5.97

5.57

Hypsiglena ochrorhynchus

55

30

.955

.981

.227

.178

- 4.75

- 1.17

.0096

.0067

2.35

1.59

3.09

2.14

6.21

5.38

T rimorphodon vandenburghi

22

16

.963

.990

.193

.156

1.08

4.56

.0121

.0058

5.74

3.19

5.30

4.82

5.86

6.12

T ant ilia eiseni

18

15

.991

.985

.355

.332

- 5.77

- 10.71

.0123

.0160

2.63

2.81

2.27

3.36

3.31

7.52

Elapsoidea niger

15

29

.961

.969

.093

.089

- 3.79

-7.92

.0075

.0043

2.59

1.63

1.71

1.56

6.08

6.13

Micrurus a. mayensis

14

16

.967

.970

.190

.130

- 7.58

- 0.78

.0144

.0087

7.96

4.68

4.15

3.24

4.30

4.80

Micrurus n. nigrocinctus

39

32

.992

.988

.173

.110

-4.50

- 0.84

.0036

.0031

1.52

1.61

3.37

3.63

5.29

7.05

Micrurus n. divaricatus

38

28

.995

.993

.193

.117

-7.57

- 0.40

.0023

.0021

1.38

1.40

3.66

2.94

5.61

5.22

Agkistrodon m. mokeson

169

131

.965

.969

.137

.136

12.74

9.30

.0029

.0031

1.42

1.39

6.20

4.88

7.12

6.34

T rimeresurus gramineus

24

16

.990

.996

.280

.196

- 16.07

0.71

.0087

.0047

3.75

1.67

3.00

1.87

2.93

2.76

Trimeresurus elegans

17

12

.984

.996

.231

.184

-7.25

4.59

.0106

.0049

6.59

2.70

7.26

3.19

5.59

3.21

Bo/hrops insularis

92

94

.943

.943

.167

.137

4.99

6.21

.0062

.0050

3.66

3.21

4.63

5.53

4.53

6.02

Atheris squamigera

22

25

.945

.968

.236

.180

- 14.86

- 5.39

.0194

.0092

7.68

4.03

5.84

3.58

7.54

4.94

TABLE 7

Mean and Limiting Lengths of Specimens Comprising the Samples

Mean Length

Mean

Body

Mean Tail

Standard Deviations

Range of Length Over-all*

Over-all

Length

Length

0L

O

B

(Tp

Minimum

Maximum

Species or Subspecies

M

F

M

F

M

F

M

F

M

F

M

F

M

F

M

F

Lichanura r. rose of use a

5 66.2 1

586.19

486.79

508.2 5

79.42

77.95

195.63

210.52

163.36

180.35

32.96

30.43

276-33

284-35

840-110

951-133

Adelphicos q. sargii

1 9 1 . S2

226.11

163.45

199.77

28.07

26.34

53.37

59.71

44.74

52.54

8.79

7.42

90-13

98-12

268-44

335-41

Geophis brachyccphala

211.52

239.58

172.64

200.92

38.88

38.67

71.79

56.24

57.15

47.12

14.68

9.24

114-20

127-20

342-66

340-52

Geo phis nasalis

209.37

207.03

174.01

178.77

35.37

28.26

48.32

43.32

38.73

36.99

9.83

6.54

106-16

112-14

292-54

283-42

Diadophis a. similis

277.88

296.81

224.54

247.39

53.34

49.43

54.61

86.82

43.11

71.38

11.82

15.81

135-24

121-19

390-79

507-88

Di ado phis p. arnyi I

239.56

229.83

192.21

193.80

45.35

36.03

50.62

72.33

41.02

60.93

10.25

11.78

121-24

112-18

322-59

330-51

Diadophis p. arnyi II

220.37

209.64

179.00

177.34

41.37

32.30

57.40

62.39

45.41

S2.32

12.25

10.47

112-20

115-17

315-56

361-57

Board oit /. lineatus I

590.43

712.12

491.26

624.18

99.17

87.94

165.20

178.44

135.32

155.32

30.71

23.45

227-34

252-25

820-140

944-107

Boaedon 1. lineatus II

436.76

493.50

3 57.19

424.83

79.57

68.67

99.15

213.50

78.71

182.45

20.59

31.12

222-37

299-39

557-101

938-133

Masticophis f. piceus

790.40

1009.71

581.92

738.21

208.48

271.50

382.05

359.71

279.56

263.75

104.51

97.19

302-75

306-73

1481-395

1531-384

Masticophis lateralis

.. 8 58.87

894.38

597.65

627.96

261.22

266.42

275.74

253.3 5

189.09

176.62

87.09

77.62

343-99

406-117

1218-380

1385-417

Salvador a g. vir gill tea

678.78

625.70

520.21

483.52

158.57

142.17

214.24

187.83

163.51

144.60

51.00

43.49

315-72

280-60

1027-244

892-200

Phyllorhynchus b. browni

290.08

296.33

251.08

272.41

39.00

23.92

53.36

75.60

44.86

69.25

8.93

6.60

166-18

180-13

362-49

384-32

Phyllohrynchus d. nubilus

288.50

272.50

249.00

250.50

39.50

22.00

65.82

100.02

54.17

91.99

11.71

8.04

194-22

173-14

343-50

408-33

Phyllorhynchus d. perkinsi I

340.34

329.91

291.52

298.01

48.82

31.89

84.20

90.09

69.25

80.55

15.47

9.81

173-19

171-16

495-75

486-44

Phyllorhynchus d. perkinsi II

356.79

292.00

304.03

263.19

52.76

28.81

90.02

80.74

73.64

72.14

16.57

9.04

168-16

190-19

510-82

435-50

Arizona e. occidentalis I

673.98

662.56

583.42

581.16

90.56

81.40

187.52

243.15

162.86

211.24

24.86

31.97

262-34

213-24

1061-133

1063-133

Arizona e. occidentalis II

63 5.94

462.42

547.44

404.50

88.50

57.92

162.81

179.03

139.52

155.32

23.57

23.85

275-36

245-34

863-124

796-103

Pituophis c. annectens

789.66

642.21

655.63

542.29

134.03

99.92

386.06

325.93

322.53

277.64

64.05

48.80

322-56

322-48

1844-284

1504-210

Pituophis c. deserticola I

783.06

815.26

675.45

712.26

107.61

103.00

346.15

3 52.12

302.38

309.96

44.08

42.42

415-60

439-57

1495-189

1566-197

Pituophis c. deserticola II

910.48

826.20

775.52

714.53

134.96

111.67

303.92

296.50

259.14

253.83

45.33

42.97

420-60

417-51

1364-187

1194-165

Lat?ipropeltis g. calif or niae ...

540.5 1

5 59.10

466.33

486.08

74.18

73.01

281.16

276.36

244.59

241.21

37.57

35.81

205-29

210-25

1301-161

1233-143

Khinocheilus 1. lecontei I

583.1 3

462.57

500.83

402.52

82.30

60.04

206.43

187.30

178.40

163.73

28.31

23.75

202-27

231-31

842-111

740-97

Rhinocheilus l. lecontei II

639.94

526.40

549.09

456.80

90.86

69.60

166.54

176.30

143.40

152.35

23.46

24.17

280-37

273-33

890-121

749-96

Khinocheilus l. clarus

641.79

586.00

5 50.03

509.23

91.76

76.77

148.19

127.88

126.65

108.41

21.89

18.84

361-48

343-43

843-120

787-103

Toluca 1. lineata I

192.20

187.90

153.65

160.04

38.55

27.86

49.32

47.92

38.87

40.84

10.52

7.24

77-14

81-11

248-45

262-39

Toluca l. lineata II

194.59

191.82

156.44

162.61

38.15

29.21

57.87

56.99

45.16

47.43

12.87

9.78

93-16

90-12

293-59

263-42

Toluca l. varians

187.62

213.62

149.90

179.36

37.72

34.26

65.67

62.84

51.19

52.76

14.61

10.34

96-18

102-16

335-67

331-50

Sottora m. linearis

211.01

209.41

167.43

171.42

43.58

37.99

59.14

54.10

11.80

11.36

13.03

9.91

115-22

126-22

328-70

307-55

Sonora o. occipitalis

283.03

263.3 1

230.72

220.38

52.31

42.92

40.23

49.82

32.45

40.70

8.07

9.47

126-22

168-27

330-62

312-52

Sonora o. annulata

274.10

271.81

221.00

225.86

51.10

45.95

56.23

62.32

44.82

51.27

11.80

11.36

131-23

130-23

368-76

382-63

Chilomeniscus s. stramineus

183.95

196.09

156.95

172.05

27.00

24.05

26.51

34.57

22.91

30.39

3.79

4.41

111-15

96-12

230-31

244-29

Thamnophis hammondii

373.79

388.85

282.26

299.90

91.52

88.94

155.17

193.75

117.13

151.27

38.35

42.78

156-38

145-35

729-176

989-219

T hamnophis o. ordinoides

345.04

345.21

2 57.37

267.62

87.68

77.60

71.73

99.34

52.15

76.36

19.84

23.50

184-43

158-36

527-134

603-141

Thamnophis o. biscutatus

.... 445.39

425.13

328.18

322.21

117.21

102.92

194.67

211.24

145.31

162.34

49.89

49.32

208-58

205-50

797-210

922-207

Conopsis nasus

234.17

230.93

198.76

203.37

35.41

27.56

60.79

57.57

51.58

50.83

9.41

6.89

100-14

105-13

334-49

344-38

Hypsiglena ochrorhynchus

290.3 1

270.23

240.53

230.43

49.78

39.80

53.79

70.21

43.78

59.58

10.39

10.80

168-27

157-21

391-67

375-52

Trimorphodon vandenburghi

5 5 5.23

5 56.00

464.64

477.13

90.59

78.88

1 14.21

249.10

95.67

215.49

19.14

33.89

264-45

234-35

738-114

1054-137

T ant ilia eiseni

277.94

211.53

209.39

166.87

68.56

44.67

60.69

74.83

44.77

56.13

16.04

18.91

130-28

142-25

375-92

373-82

Elapsoidea niger

369.47

398.62

341.40

373.24

28.07

25.38

66.83

73.69

61.11

67.64

5.93

6.22

264-19

281-16

471-38

569-39

M icrurus a. mayensis

644.64

595.2 5

548.07

527.69

96.57

67.56

95.42

108.29

80.11

95.84

15.74

12.80

443-64

478-55

743-108

777-87

Micrurus n. nigrocinctus

458.1 3

527.91

394.41

476.38

63.72

51.53

179.16

233.76

152.72

210.58

26.62

23.42

218-29

220-21

745-113

890-88

M icrurus n. divaricatus

442.76

541.04

377.61

484.75

65.16

56.29

217.82

238.76

182.58

213.75

35.40

25.16

196-26

236-23

935-150

970-101

Agkistrodon m. mokeson

541.31

498.20

464.76

430.40

76.54

67.80

188.64

158.51

165.79

139.47

23.59

19.57

201-40

180-32

915-110

785-95

T rimer esurus grarnineus

525.21

410.13

422.92

342.44

102.29

67.69

92.07

123.50

71.90

103.29

20.34

20.28

310-55

247-40

660-132

608-102

Trimeresurus elegant

725.06

613.42

595.12

514.00

129.94

99.42

209.98

232.43

170.55

196.21

39.94

36.33

430-75

387-61

1160-215

1151-184

Bo t hr ops insularis

68 5.37

718.22

5 83.17

626.30

102.20

91.93

91.15

129.44

78.03

113.75

13.80

16.51

440-63

380-48

850-124

1000-118

At her is squamigera

468.32

504.08

390.86

431.64

77.46

72.44

81.49

93.99

65.78

79.57

16.44

14.82

318-48

320-45

595-103

660-100

The figures following hyphens are tail lengths.

Klauber: Tail-length Dieeerences in Snakes

23

TABLE 8

Assumed Standard Body Lengths
For Computation of Proportionalities and Sexual Dimorphism

Juvenile

Lichanura r. roseofusca 300

Adelphicos q. sargii 100

Geophis brachycephala 100

Geo phis nasalis 100

Diadophis a. similis 110

Di ado phis p. arnyi 1 110

Diadophis p. arnyi II 110

Boaedon l. lineatus I 220

Boaedon l. lineatus II 200

Masticophis f. piceus 2 50

Masticophis lateralis 250

Salvador a g. virgultea 2 50

Phyllorhynchus b. broivni. .. 160

Phyllorhynchus d. nubilus ... 160

Phyllorhynchus d. per kin si I 160
Phyllorhynchus d. perkinsi II 160

Arizona e. occidentals 1 250

Arizona e. occidentals II 2 50

Pituophis c. annectens 3 30

Pituophis c. deserticola 1 330

Pituophis c. deserticola II 3 30

Lampropeltis g. calif orniae. .. 2 50

Rhinocheilus l. lecontei 1 200

Rhinocheilus l. lecontei II 200

Rhinocheilus l. clams 200

Toluca l. lineata I 70

Toluca l. lineata II 80

Toluca l. varians 90

Sonora m. linearis 100

Sonora o. occipitalis 110

Sonora o. annulata 110

Chilomeniscus s. stramineus.. 100

Thamnophis hammondii 120

T ha m n ophis o. ord in oid es 110

Thamnophis o. biscut at us 200

Con op sis nasus 100

Hypsiglena ochrorhynchus . ... 140

Trimorphodon vandenburghi 220
T ant ilia eiseni 100

Sex Reaching
Largest Size

Median

Adults

500

700

155

220

190

280

155

220

205

300

180

250

180

250

420

620

325

450

675

1100

550

850

475

700

230

300

230

300

280

400

280

400

525

800

525

800

715

1100

715

1100

715

1100

62 5

1000

450

700

450

700

450

700

135

200

150

220

170

250

175

250

185

260

185

260

150

200

285

450

205

300

400

600

200

300

220

300

410

600

175

250

24

Bulletin 18: Zoological Society of San Diego

TABLE 8
(continued)

Sex Reaching
Largest Size

Juvenile

Median

Adults

Elapsoidea niger

... 240

340

440

Micrurus a. mayensis

... 200

400

600

Micrurus n. nigrocinctus ...

... 200

400

600

Micrurus n. divaricatus

... 200

400

600

A°kistrodon m . mokeson

... 200

400

600

Trimeresurus gramineus

... 200

350

500

Trimcresurus departs

350

675

1000

Bothrops insular is

... 300

500

700

At her is squamigera

... 270

385

500

Adelphicos qradrivirgatus sargii (Fischer)

This is a fine series in the collection of the California Academy of Sciences,
from Finca El Cipres, Volcan Zunil, Guatemala. As noted by Mr. J. R.
Slevin,0 all specimens were dug out of piles of debris in a single locality;
this is, therefore, an ideal series from the standpoint of territorial homo-
geneity. The size distribution is good. The nomenclature is that of Dr.
H. M. Smith.10

Geo phis brachyce phala (Cope)

A series of moderate size from Boquete, Panama, in the collection of the
California Academy.11 Although territorially well concentrated, the series
of both sexes are somewhat lacking in adolescents.

Geophis nasalis (Cope)

A beautiful series taken by J. R. Slevin in cafetal debris at Finca El
Cipres, Volcan Zunil, Guatemala.12 All ages are represented. The nomen-
clature is that of Smith.13

Diadophis amabilis similis Blanchard.

A series from cismontane San Diego County, California, mostly in my
own collection. The age distribution is good.

Diadophis punctatus arnyi (Kennicott)

I. A series in my collection from southern Kansas and northern
Oklahoma.

9 Proc. Cal. Acad. Sci., Ser. 4, Vol. 23, No. 26, p. 403, 1939.

10 Proc. Rochester Acad. Sci., Vol. 8, p. 192, 1942.

11 Proc. Cal. Acad. Sci., Ser. 4, Vol. 23, No. 32, p. 474, 1942.

12 Proc. Cal. Acad. Sci., Ser. 4, Vol. 23, No. 26, p. 404, 1939.

13 Smithsonian Misc. Coll., Vol. 99, No. 19, p. 4, 1941.

TABLE 9
Tail

Ratio of Tail

Ratio of Tail

Length to Length

Length to Length

Over-

-all at

of Body at

Mean

Length

Mean Length

Species or Subspecies

M

F

M

F

Lie ban ura r. rose of use a

140

.133

.163

.153

Ad el phi cos q. sargii

147

.116

.172

.132

Geo phis brachycephala

184

.161

.225

.192

Geo phis nasalis

169

.136

.203

.158

Di ado phis a. si mil is

192

.167

.238

.200

Di ado phis p. arnyi I

191

.157

.236

.186

Di ado phis p. arnyi II

188

.154

.231

.182

Boaedon l. lincatus I

168

.123

.202

.141

Board on l. lincatus II

182

.139

.223

.162

M asticophis f. piceus

264

.269

.358

.368

Masticophis lateralis

304

.298

.437

.424

Salv ad ora g. virgultca

234

.227

.305

.294

Phyllorhynchus b. brou/ni

134

.081

.155

.087

Phyllorhynchus d. nubilus

.137

.081

.159

.088

Pbyllorhynchus d. per kin si I

143

.099

.167

.107

Phyllorhynchus d. perkinsi II

148

.099

.174

.109

Arizona e. occidental's I

134

.123

.155

.140

Arizona e. occidentals II

139

.125

.162

.143

Pit no phis c. annectens

170

.156

.204

.184

Pit uo phis c. deserticola I

137

.126

.159

.145

Pituophis c. deserticola II

148

.135

.174

.156

Lam pro pelt is g. calif or niae

137

.131

.159

.150

Rhinocheilus l. lecontei I

141

.130

.164

.149

Rhinocheilus l. lecontei II

.142

.132

.165

.152

Rhinocheilus l. clarus

143

.131

.167

.151

Toluca l. lineata I

201

.148

.251

.174

Toluca l. lineata II

196

.152

.244

.180

Toluca l. varians

201

.160

.252

.191

Sonora m. linearis

207

.181

.260

.222

Sonora o. occipitalis

185

.163

.227

.195

Sonora o. annulata

194

.169

.204

.203

Chilomeniscus s. stramineus

147

.123

.172

.140

Thamnophis hammondii

249

.229

.324

.297

T hamnophis o. ordinoides

254

.225

.341

.290

Thamnophis o. biscut at us

263

.242

.357

.319

Conopsis nasus

151

.119

.177

.136

Hypsiglena ochrorhynchus

171

.147

.207

.173

Trimorphodon vandenburghi

163

.142

.195

.165

T ant ill a eiseni

247

.211

.327

.268

Elapsoidea niger

077

.064

.082

.068

M icrurus a. mayensis

150

.114

.176

.128

Micrurus n. nigrocinctus

139

.098

.162

.108

Micrurus n. divaricatus

147

.104

.173

.116

Agkistrodon m. mokeson

141

.136

.165

.158

Trimeresurus gramineus

195

.165

.242

.198

T rimeresurus elegans

179

.162

.218

.193

Bothrops insular is

149

.128

.175

.147

Atheris squamigera

165

.144

.198

.168

Ratios

Ratios of Tail Length to Body Length at

Males

Standard Body Length

S

Females

Coefficient of
Ontogenetical
Divergence*
Male Female

Juv.

Median

Adult

Juv.

Median

Adult

.142

.164

.173

.144

.153

.157

19.7

8.6

.159

.171

.177

.128

.131

.132

10.7

3.1

.203

.222

.237

.192

.192

.193

15.5

0.5

.172

.193

.212

.148

.152

.160

20.8

7.8

.210

.235

.244

.180

.197

.202

15.0

11.5

.240

.236

.235

.186

.186

.186

- 2.1

0.0

.211

.231

.240

.175

.182

.185

12.9

5.6

.180

.199

.206

.127

.137

.141

13.5

10.5

.194

.219

.230

.152

.159

.162

17.0

6.4

.351

.358

.361

.379

.368

.366

2.8

- 3.5

.409

.43 5

.443

.412

.423

.426

8.0

3.3

.300

.304

.306

.290

.294

.295

2.0

1.7

.137

.155

.161

.085

.087

.088

16.1

3.5

.127

.154

.168

.088

.088

.088

27.8

0.0

.129

.166

.180

.098

.106

.110

33.0

11.5

.130

.169

.185

.104

.110

.113

34.9

8.3

.161

.156

.154

.128

.140

.143

-4.4

11.1

.156

.161

.163

.137

.145

.148

4.4

7.7

.212

.204

.201

.191

.182

.179

- 5.3

- 6.5

.175

.159

.154

.155

.145

.142

- 12.8

- 8.8

.176

.174

.174

.143

.156

.160

- 1.1

11.2

.168

.156

.154

.155

.149

.148

- 8.7

-4.6

.176

.165

.162

.155

.149

.147

- 8.3

- 5.3

.173

.166

.164

.146

.152

.154

- 5.3

5.3

.162

.166

.167

.124

.148

.156

3.0

22.9

.230

.248

.255

.176

.174

.174

10.3

- 1.1

.209

.242

.254

.158

.178

.185

19.4

15.7

.231

.255

.264

.192

.191

.191

13.3

- 0.5

.248

.261

.267

.222

.222

.221

7.4

- 0.5

.215

.224

.228

.167

.190

.199

5.8

17.5

.228

.238

.242

.192

.201

.205

6.0

6.6

.181

.173

.169

.142

.140

.139

- 6.9

- 2.1

.325

.324

.324

.321

.297

.291

- 0.3

-9.8

.288

.332

.345

.277

.287

.291

18.1

4.9

.369

.354

.349

.331

.316

.311

- 5.6

- 6.2

.179

.178

.178

.139

.136

.134

- 0.6

- 3.7

.193

.205

.211

.169

.172

.174

8.9

2.9

.198

.195

.194

.176

.167

.163

- 2.1

- 7.7

.297

.322

.332

.225

.271

.289

11.1

24.9

.078

.082

.085

.056

.066

.071

8.6

23.6

.152

.171

.177

.126

.128

.128

15.2

1.6

.150

.162

.165

.106

.108

.109

9.5

2.8

.155

.174

.180

.115

.116

.116

14.9

0.9

.201

.169

.159

.182

.159

.151

-23.3

- 18.6

.200

.234

.248

.199

.198

.197

21.4

- 1.0

.210

.220

.223

.198

.191

.189

6.0

-4.6

.183

.177

.174

.158

.149

.146

- 5.0

- 7.9

.181

.198

.206

.160

.166

.170

12.9

6.1

* A positive figure indicates a higher tail-length ratio in the adults than the juveniles.

Klauber: Tail-length Differences in Snakes

25

II. A second series from Tulsa, Oklahoma, and surrounding counties, the
measurements having been made by the late Dr. Frank N. Blanchard, and
sent me through the courtesy of Dr. H. K. Gloyd. I have kept the two
series separate for comparative purposes. A few aberrant specimens were
omitted in making the computations on Series II.

Boaedon lineatus lincatus Dumeril & Bibron.

I. A small series from Sipi and Butandiga, Uganda; and Kaimosi,
Kenya Colony, Africa.

II. A small series from Lamu Island, in the Coast Province of Kenya.
While these series are rather inadequate in numbers, they are included to
illustrate territorial differences within a subspecies. The measurements
upon which the calculations are based were received through the courtesy
of Mr. Arthur Loveridge.

Masficophis flagellum piceus (Cope)

A small but ontogenetically well-distributed series from San Diego
County. Although a common snake, comparatively few of this species
have complete tails, and even some of those which have been included in
this study are not certainly complete.

Masticophis lateralis (Flallowell)

A small series from western San Diego County. Individuals of this species
with complete tails are relatively rare, although the snakes themselves are
quite common in this territory.

Salt' ad ora graham iae virgultea Bogert.

A small series from western San Diego County.

Phyllorhynchus broivni brcnvni Stejneger.

All available specimens were included; most of them are from the im-
mediate vicinity of Tucson, Arizona. While inadequate in numbers (par-
ticularly the females) the data have been included for comparison with
decurtatus. A single male juvenile has been triplicated.

Phyllorhynchus decurtatus nubilus Klauber.

A quite inadequate series from the vicinity of Tucson, included for
purposes of comparison with the previous subspecies.

Phyllorhynchus decurtatus perkinsi Klauber.

I. A large series, mostly in my own collection, from desert San Diego
County.

II. A small series from desert Riverside County, California, included
for comparative purposes.

26

Bulletin 18: Zoological Society of San Diego

Arizona elegans occidentalis Blanchard.

I. A series from desert San Diego County; there being few juvenile
males, the two smallest were duplicated.

II. A series from the Mohave-Victorville area of the Mohave Desert.
The females are quite inadequate in number.

Pituophis catenifer annectens (Baird and Girard)

A large series restricted to western San Diego County. The juveniles are
better represented than the adults; among the latter the dispersion is
quite high.

Pituophis catenifer desert icola Stejneger.

I. A small series from Imperial County, California. One large aberrant
female has been omitted.

II. A small series from the Mohave-Victorville section of the Mohave
Desert. One aberrant male has been omitted.

Lampropeltis getulus californiae (Blainville)

A large series from San Diego County, including many juveniles born
in captivity. All pattern phases have been included. This species shows
high dispersion among the adults.

Rhinocheilus lecontei lecontei Baird and Girard.

I. A series from coastal San Diego County.

II. A series from the Mohave-Victorville section of the Mohave Desert.
The female representation is inadequate.

Rhinocheilus lecontei clarus Klauber.

A series from eastern San Diego County; only 13 females are available.

Toluca lineata lineata Kennicott.

I. A series from the vicinity of Guerrero, Hidalgo, Mexico, the meas-
urements of which were received through the courtesy of Dr. Hobart M.
Smith. An aberrant male was excluded.

II. A second series, also from Dr. Smith, collected near Tezuitlan,
Puebla, and adjacent Veracruz.

Toluca lineata varians (Bocourt)

A large, well-distributed series from Acultzingo, Veracruz, Mexico; the
measurements are from Dr. Smith.

Sonora miniata linearis Stickel.

A good series from an island in Colorado River at Laguna Dam, close
to the Imperial County, California, shore. This tiny island is only a few

Klauber: Tail-length Dieeerences in Snakes

27

hundred feet long and this series is therefore the most concentrated, terri-
torially, of any available. Most of the specimens were collected by Wal-
lace F. Wood and Joseph R. Slevin, and are in the CAS collection. The
others are in the MVZ and LMK collections. Both sexes are well dis-
tributed ontogenetically.

Sonora occipitalis occipitalis (Hallowed)

A small series from the Mohave-Victorville area of the Mohave Desert.
The females are few in number, and the juveniles of both sexes inade-
quately represented.

Sonora occipitalis annulata (Baird)

A good series from eastern San Diego and Imperial counties in California.

Chilomeniscus stramineus stramineus Cope.

A series reported on by Dr. J. M. Linsdale, from Eureka, in the Cape
region of Lower California.14 Re-measurements entailed some minor changes
and the elimination of several specimens because of incomplete tails.

Thamnopbis hammondii (Kennicott)

A large and ontogenetically well-distributed series from San Diego
County.

Thamnopbis ordinoides ordinoides (Baird and Girard)

A series from western Oregon, mostly from the vicinity of Portland.
Ontogenetically, the distribution is good, although large adults are not
plentiful.

Thamnopbis ordinoides biscut at us (Cope)

A large series from the vicinity of Klamath Falls, Oregon, contained in
the collections of the California Academy of Sciences and the Museum
of Vertebrate Zoology. The measurements were given me by Dr. Henry
S. Fitch.

Conopsis nasus Gunther.

A good series in the Museum of Comparative Zoology, the measure-
ments having been received through the courtesy of Drs. Edward H.
Taylor and Hobart M. Smith. The specimens were collected at Alvarez,
San Luis Potosi, Mexico. As both sexes are weak in juveniles, the avail-
able specimens have been duplicated.

Hypsiglena ochrorhynchus Cope.

A fair series from San Diego County. Specimens from both sides of the
mountains are included, as no differences in tail proportionality are apparent.

14 Copeia 1936, p. 232.

2 8 Bulletin 18: Zoological Society of San Diego

Trimorphodon vandenburghi Klauber.

A small series from both sides of the mountains in San Diego County.

Tantilla eiseni Stejneger.

A small series from the coastal side of the mountains in San Diego
County. Desert specimens had to be omitted, as they have proportionally
shorter tails.

Elapsoidea niger Gunther.

A series from Nyange, Amani, and Bumbuli, Tanganyika, Africa. The
measurements were kindly furnished me by Mr. Arthur Loveridge.

Micrurus affinis mayensis Schmidt.

A small but ontogenetically well-distributed series from Yucatan,
Mexico. The measurements of this, and the two following series of the
same genus, were received through the courtesy of Mr. Karl P. Schmidt.

Micrurus nigrocinctus nigrocinctus (Girard)

A series from Panama and the Canal Zone. One aberrant female has
been omitted. The specimens are well distributed ontogenetically, but may
not be territorially consistent.

Micrurus nigrocinctus divaricatus (Hallowed)

A well-balanced series from Honduras.

Agkistrodon mokcson mokeson (Daudin)

A large series from eastern Kansas, the measurements of which were
received through the courtesy of Dr. Howard K. Gloyd. Several aberrant
specimens have been omitted. The ontogenetic distribution is good; the
dispersion is relatively high.

Trimeresurus gramineus (Shaw)

A small series from Formosa contained in the collection of the Cali-
fornia Academy of Sciences.

T rimeresurus elegans (Gray)

A series from Ishigaki shima, Loo Choo Islands, Japan, also in the
California Academy of Sciences collection.

Botbrops insular is (Amaral)

A series taken from data published by do Amaral.15 Some obviously
aberrant specimens were omitted; and, as there is a lack of juvenile males,
5 have been duplicated. All are from Queimada Grande Island, off the

15 Mem. Inst. Butantan, Vol. 1, No. 1, p. 62, 1921.

Klauber: Tail-length Differences in Snakes

29

coast of Sao Paulo, Brazil, this being the only place where these snakes
occur. Thus the series is territorially homogeneous.

Atheris squamigera (Hallowell)

A small series from Kaimosi, Nyanza Province, Kenya Colony, the meas-
urements of which were courteously sent me by Mr. Arthur Loveridge.
Owing to the lack of juvenile males, two were duplicated. One aberrant
adult male was omitted.

I return now to the three questions which are among the primary pur-
poses of this investigation. With regard to the first, that is, the linearity
of regression, it is to be regretted that publication limitations allow the
inclusion of only one of the scatter diagrams which have been prepared
for each of the forms investigated (Fig. 1). All show substantially
straight-line relationships, as is verified by the uniformly high values of
the coefficients of correlation shown in Tables 5 and 6. In two or three
series there seems to be evidence of an upward bending in the regression
line of the males and a possible drooping in the females; but in any case
this is so slight as to be below the level of significance in a linearity test.
In a few forms there are indications of a slight droop in the regression
line of the males, when the largest specimens are reached, as if the tail
did not quite keep pace with the continued, but slow, growth of the body.
But it can be fairly said, with respect to all of the species investigated,
that, from a practical standpoint, the relationship between the length of
tail and length over-all (or tail and body) is linear and may be treated
as such in all difference computations. It is probably significant to ob-
serve that the greater the number of specimens in a series, the less any tend-
ency to curvature is apparent.

I do not say that it is impossible to secure rather good fits with equa-
tions of the type T = cLk, which have been shown by Huxley and others
to be well suited to many problems of relative growth; rather, it is a fact
that straight-line equations of the form T = a + bL fit so well, and are so
much easier to use in problems of taxonomic differences or sexual dim-
orphism, that I think they are to be preferred in practice, where their use
is not inconsistent with the actual dispersions. I shall, however, in a later
table, present some data on the constants c and k in the parabolic growth
curves of several typical species.

As to the second question: Does the ratio of the tail length to total
length remain constant, or vary with age? The answers are to be found
in the values of a or a' (particularly the latter) in Tables 5 and 6. For,
if the proportion does remain constant, then a and a' should equal zero.
But we find that a and a' have relatively high values in many series; and,
comparing these values with their standard errors — admittedly not a
highly accurate procedure in this situation — half of them are found to
be significant (the ratio being greater than 1.96) or highly significant

30

Bulletin 18: Zoological Society of San Diego

(ratio greater than 2.5 8). We have the following summary of the results
derived from Table 6.”'

Character of a Males Females

Positive but not significant 9 12

Positive and significant 2

Positive and highly significant 4 4

Sub-total 1 3 18

Negative but not significant 9 18

Negative and significant 8 5

Negative and highly significant 18 7

Sub- total 3 5 3 0

Grand total 48 48

We see, first, that more than half the males (3 0) have a value of a' dif-
fering significantly from zero, and more than one-third the females (18),
as well. More than two-thirds of the males and nearly two-thirds of the
females have negative values of a'. These proportions are too high to be
due to chance. I think we may conclude that many species of snakes do
change their tail proportionalities as they grow; that, of these, a majority
have relatively longer tails as they age ( a ' negative), although a few have
tails which become shorter, Pituophis and Lampropeltis being examples.
In snakes in which the tails grow relatively faster than the bodies, the
males are likely to exceed the females in disproportionate growth. This
will be discussed further in connection with sexual dimorphism. Alto-
gether, admitting that small and insignificant values of a' have little im-
portance, and may result from the chance composition of the samples, we
still find a considerable number of subspecies in which the change of tail-
length proportionality with age is beyond question. Others, although
failing in the test for mathematical significance, suggest, by the evidence
of the graphical set-ups, that additional specimens would not be likely to
reduce a' to zero.

There is found to be no correlation between the value of a' and adult
tail-proportionality, using the appropriate figures in Tables 6 and 8 as
criteria; rather, the change in proportionality — its direction and degree —
seems to be a generic and even a species character without any widely
applicable uniform rule of variation.

Species with negative constants which are outstandingly high are
L. r. roseofusca, G. nasal is, D. a. similis, B. 1. Uneatus , Phyllorhynclms

1,1 To avoid the effects of spurious correlation, I have placed more emphasis on the
T to B relationship in Table 6, than the more customary T to L relationship of Table 5.

Klauber: Tail-length Differences in Snakes

31

(especially the males), T. eiseni, T. gramineus males, and A. squamigera.
Those with high positive values are Pituophis, L. g. calif orniae, A. m.
mokeson, and B. insularis. These latter are all short-tailed snakes, yet a
decrease in tail proportionality with growth is not a universal characteristic
of short-tailed snakes, as is shown by the presence of Lichanura and
Phyllorhynchus in the other category.

The standard errors of b and b' given in Tables 5 and 6 are not set
forth to indicate the significances of these statistics, since no such proof
is required; a lack of significance would be equivalent to saying that the
tail lengths of snakes are independent of body size, an obvious fallacy. But
these standard errors are useful in giving a rough indication of the extent
to which the population values of b and b' may differ from the values
calculated from the available series. Confidence limits of the coefficients
of correlation are omitted as having little interest in the present study.

The standard errors of estimate supply information on the extent of the
dispersion of the specimens (in mm.) about the regression lines. The two
final columns in Tables 5 and 6 present these figures reduced to percentages
of the mean tail lengths; in this form they offer a more useful basis for
comparison between species. The further use of this coefficient of varia-
tion, as an estimate of the dispersion at any age, will be referred to hereafter.

Table 7 gives the means and standard deviations of the measurements of
the specimens upon which these studies have been based. The minimum
and maximum limiting lengths listed in the final four columns are to
be viewed in the light of the numbers of specimens available, as set forth
in the first two columns in Tables 5 and 6. It is obvious that the larger
the sample — if there has been no conscious selection of specimens — the
closer will be the approach of the limiting specimens to true minima and
maxima of the wild population. Practical conditions often prevent the
collection of large specimens of the larger species, for which reason series
of the smaller species of snakes are more likely to represent true field
distributions. However, in some of the series treated it has been evident
that the field men considered the juveniles too small to be worth-while.

In Table 8 I have indicated certain standard sizes at three ages, at which
body proportionalities for comparative purposes are computed, as pre-
sented in Table 9. It is necessary to fix such standards if intersex and
intersubspecific comparisons are to be made, since otherwise the averages
Mu would not be the same for the groups compared. It will be noted
that I have again used the body length rather than length over-all as the
basic criterion; it seems to me that this constitutes a more pertinent and
fairer method. For example, if we are comparing the tail lengths of a
male and female, and we use snakes of the same length over-all, we are
unduly penalizing the male, since we are really taking a smaller snake to
compare with a larger, if, as we should, we view the tail as an appendage,
rather than a section of the body of the animal itself.

32

Bulletin 18: Zoological Society of San Diego

The juvenile and adult standard sizes in Table 8 are not extremes, but
rather are presumed to be averages of the young at birth, and large, but
not record-breaking, adults. Further, the standard adult size represents
large adults of the smaller of the sexes (in cases where one attains a con-
spicuously greater length than the other) , since it was desired to limit
calculations to lengths actually reached by both sexes, and thus avoid
fictitious extrapolations. I have indicated in the same table by asterisks
the cases in which one sex reaches a size notably larger than the other.
The median length is not a true median derived from the sample, but is
merely a half-way point between the other two terminal standards.

Table 9 presents the tail-length ratios at the mean lengths of the samples
(first 4 columns), and also the proportionalities, calculated from the re-
gression lines, at the selected standard body lengths. I should call attention
particularly to the changes in proportionalities from the juvenile to the
adult stages shown in the two final columns.1. This is a reiteration of the
statement that these ontogenetic variations are of importance in many
genera; and, if comparisons of tail ratios are to be made between species,
limited age groups must be used, or analytical methods of compensation
should be employed, for some of these ontogenetical differences exceed
20 per cent. It will be noted that, in considerably more than half the
series, the change in the male proportionality exceeds the female.

Should any doubt remain, upon the part of herpetologists who do not
favor analytical methods, concerning the reality of these ontogenetic
changes in tail ratio, I may cite, as a conspicuous example, the tail-to-body
ratios of ten juvenile and ten adult male P. d. perkinsi selected at random
from among the San Diego County specimens in my collection:

Juvenile

Adult

.168

.187

.172

.178

.185

.179

.174

.170

.182

.165

.176

.133

.138

.135

.131

.130

.142

.141

.139

.139

.135

mean .13 6

11 This difference has been given in terms of what may be called the "coefficient of onto-
genetic divergence”: (P- -PR) / (Pj + Pa) where Pj and Pa are the juvenile and
adult proportionalities.

Klauber: Tail-length Differences in Snakes

33

It will be observed that the juvenile with the highest ratio (.142) is
considerably below the adult with the lowest (.165). It will be seen how
questionable is the usual procedure of employing average tail-length
ratios, in discussing species differences, without taking into consideration
this extensive ontogenetic variation; for such an average figure is not so
much a species characteristic as it is the result of the ontogenetic compo-
sition of the particular sample at hand.

I should call attention to the fact that a and a' are expressed in milli-
meters and therefore the values of these constants are not comparable
between species; that is, the value of a does not in itself constitute a meas-
ure of ontogenetic variability, since a particular value would be of more
importance in a small species than a large. In some ways the expression
of the regression equations in the forms showing directly the nature of
the variability in the tail-length ratios

T/L = a/LJrb and TfB-a'/B-Pb'

shows the ontogenetic change most clearly, since a/L and a' /B are inde-
pendent of the unit of measurement.

As I review the equations derived for these 48 species, and consider the
inadequacies of several of the series, there are some concerning which
changes may be prophesied when more material shall have become available.

In A. q. sargii I should expect a slightly steeper slope in the females;
that is, a' will be increased negatively, and b' will also be higher. In
female G. brachycephala a' will, I think, eventually be found to approxi-
mate — 2. In D. a. arnyi, Series I, both sexes will develop larger negative
values of a' . In M. /. picens the female value of a' will tend to decrease.
When larger series of Phyllorhynchus are available, all will more closely
approach the regression line of San Diego County perkinsi. In A. e. Occi-
dent alis males, Series I, with a more adequate series of juveniles, I should
expect a' to become negative. In P. c. deserticola females, Series II, the
high negative value of a ' is probably accidental, and not truly representa-
tive. In Conopsis nasus a more adequate juvenile representation would prob-
ably increase the negative value of a' in the males.

In addition to the 48 series of which both analytic and graphic investi-
gations were made, there are available 27 additional series which were not
considered sufficiently extensive, or territorially compact, to justify an
analytic study. However, a simple graphic determination of the regression
lines of T on B has been made, with the results as set forth in Table 10.
These may be compared with related species in Table 6. Again we note
that negative values of a' predominate, particularly among the males.

In Table 1 1 statistics similarly derived are given for a number of sub-
species of garter snakes. For the basic measurements of most of these I am
greatly indebted to Dr. Henry S. Fitch. For comparative purposes the

TABLE 10

Relationship of Tail Length to Body Length — Supplementary Series

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Except Laguna Island.

Klauber: Tail-length Differences in Snakes 3 5

data on three series, T. o. biscut atus, T. hammondii, and T. o. ordinoides
(vicinity of Portland), are carried forward from Tables 6 and 9. A
considerable consistency in adult proportionalities will be found evident
in the last two columns of Table 11, both between territorial groups of
the same race, and between races. It is of interest to note these adult
tail-length proportionalities in the light of the relationships deduced by
Dr. Fitch in his monograph.18

On page 8 I have given an equation showing how the mean of the
tail ratios Mt/l (or Mt/b), which involves many individual computa-
tions, may be derived from the mean of the tails divided by the
mean of the lengths over-all, or body lengths (MT/ML or MT/MP>) from
other statistics readily available, provided the coefficients of correlation
have been calculated. In order to ascertain the importance of the differences
between these statistics, I have made four example computations with the
following results, which are given in terms of the multiplier to be applied
to Mt Mr in order to derive Mt/l:

Multiplier

Species Male Female

P. d. perkinsi (S. D. Co.) 0.9849 0.9926

S. o. annulata 0.9977 0.9969

It will be observed that in these four examples, where the samples are
fairly large and the correlation high, the mean proportion differs from the
proportion of the means by a maximum of 1.5 per cent. The latter pro-
portion therefore may well serve as an approximation. It is to be remem-
bered that neither of these statistics is particularly useful, since both depend
too much on the ontogenetic distribution of the samples. Should a single
statement of tail proportionality be desired to represent a species in mak-
ing interspecific comparisons, and it is not possible to restrict the state-
ment to some particular age (juvenile or adult, for example) then I should
recommend the use of the proportion at the median length, as determined
from the regression equation. But for most taxonomic studies adult pro-
portionalities are to be preferred, as will be discussed hereafter.

With respect to the figures given in Tables 5 to 9, it should be stated
that all of the derived statistics were calculated to a further degree of
accuracy than shown, usually to 6 figures after the decimal point. This was
done so that the derived statistics would not lose in accuracy through
rounding off; however, these additional figures are not entered in the
tables, as they would lead only to confusion and difficulty in making
comparisons, and would give an entirely unwarranted impression with re-
spect to the accuracy of the results. This matter of dropping figures is

18 A Biogeographical Study of the Ordinoides Artenkreis of Garter Snakes (Genus
Thamnophis) . Univ. Calif. Pubs, in Zook, Vol. 44, pp. 1-1 5 0, 1940.

TABLE 11

■M

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Klauber: Tail-length Difeerences in Snakes

37

mentioned only to explain seeming inaccuracies which may be found if
an attempt be made to check some of the derived statistics.

Parabolic equations of the form Y = cXk are frequently found to fit
quite closely the growth of body parts.19 They are also to be preferred on
certain theoretical bases. However, in the present study, as was the case
with rattlesnake heads,1'" I do not find that the tail follows a curve of the
form T = cLk any closer than a straight line of the form T = a + bL; in
fact, in most cases the straight line gives the better fit. And the straight
line is so much easier to use, in evaluating sexual dimorphism or species
differences, that, if a fit is at all close, it is definitely to be preferred to
the more complicated equation.21 In the present instance the high values
of the correlation coefficients (Tables 5 and 6) are in themselves a proof
of the closeness of fit of the straight lines. The parabola seems to me to
have only one recommendation in the present case; since the deviations
from the regression line are measured logarithmically, the juveniles are
given a proper weight as compared with the adults. In the present study,
as the straight line is to be preferred both on the score of closeness of fit
and simplicity, I shall go no further into the parabolas than to list in
Table 12 the constants c and k in the equation T — cLk for those seventeen
of the series previously discussed, which contain the most specimens. This
will serve to illustrate the magnitude of these constants in example series.

As to the third problem, that is, the nature of the dispersion of the
individual specimens about the regression line, here again we are some-
what dependent upon the graphic studies which cannot be presented. As
was the case with head-length studies in the rattlesnakes,22 it is clear that
the dispersion is substantially proportional to the tail length. Thus the
scatter widens toward the larger sizes, the points representing the individual
specimens filling an angular space about the regression line (Fig 1). On
the assumption that the dispersion is proportionate to tail length, the co-
efficient of variation would remain constant from the juvenile to the
adult state, instead of there being constancy in the standard deviation

19 Julian Huxley: Problems of Relative Growth, New York, 1932, p. 4; D’Arcy Went-
worth Thompson: On Growth and Form, Cambridge, 1942, p. 20 5.

20 Occ. Papers S. D. Soc. Nat. Hist., No. 4, p. 7, 193 8.

21 Of course, the two curves give the same line if a — o, for then k — \, and b — c.
Therefore, it would be expected that the parabolic expression would give a good fit
where a is small, even though the relationship be truly linear. I have mentioned the fact
that in one or two cases some evidence of an upward curve is present, while a few others
seem to droop, especially when the larger specimens are reached. These curves indicate
that the equation T = cLk might be appropriate, an upward curve indicating a value
of k greater than unity, while in a dropping curve k is less than one.

22 Occ. Papers, S. D. Soc. Nat. Hist., No. 4, pp. 1 1-17, 193 8. As other references to this
rattlesnake head-length study are made hereafter, to avoid confusion I should call
attention to the fact that in the former paper I used a as the regression coefficient and
b as the regression constant (H = aL + b) ; while in the present paper, following a
more general custom, I have reversed the symbolism (T = a + bL) .

38

Bulletin 18: Zoological Society of San Diego

TABLE 12

Constants of Tail-length Equations of the Form T = cLk

Males Females

Species

k

c

k

c

Lichanura r. roseofusca

1.221

.0344

1.071

.085 1

Adel phi cos q. sargii

1.113

.0802

1.132

.0582

Geo phis nasal is

1.179

.063 5

1.123

.0692

Di ado phis a. similis

1.149

.0845

1.059

.1205

Di ado phis p. arnyi II

1.213

.0594

1.058

.1125

Phyllorhynchus d. perkinsi I

1.227

.0383

1.147

.0417

Pituophis c. annectens

1.000

.1710

0.965

.1963

Lam propeltis g. calif orniae

0.954

.1866

1.007

.1211

Toluca 1. varians

1.176

.0800

1.096

.0964

Sonora m. linearis

1.127

.1047

1.007

.1734

Sonora o. annulata

1.101

.1130

1.045

.1334

T ham n o phis ha m m ondii

1.003

.2415

0.933

.3404

Thamnophis o. ordinoides

1.068

.1718

1.048

.1722

Thamnophis o. biscut atus

1.003

.2564

0.969

.2904

Conopsis nasus

1.056

.1119

1.014

.1125

A gkistrodcm m. mokeson

0.839

.3926

0.859

.3311

Bo t hr ops insular is

1.006

.1429

1.003

.1253

about the regression line (the

standard error

of estimate) .

, as is

the case

in most correlation problems. This is equivalent to saying that, in these
snakes, any deviation in the tail length of an individual from the mean
at his age is maintained through life without change in proportionality; in
other words, a snake which starts life with a tail 10 per cent longer than
average will continue to have a tail 10 per cent longer as it grows. The
following statistics of a few restricted-length groups of juveniles and
adults will show that this relationship is substantially true.33 The groups
selected are from among those treated in Tables 5 to 9.

Comparative Standard Errors of Estimate

Males Females

Species

Juvenile

Adult

Juvenile

Adult

Phyllorhynchus d. perkinsi,

I

5.77

4.63

6.06

5.88

Pituophis c. annectens ....

6.89

7.12

6.66

5.58

Sonora m. linearis

1.64

1.93

1.85

1.88

23 Loc. cit. pp. 11-17. See the more

elaborate discussion

of this

relationship in

the head

lengths of rattlesnakes, wherein

the

same approximate

constancy in the coefficient of

variation was found.

Klauber: Tail-length Dileerences in Snakes 3 9

In half these cases the adult variation is lower than the juvenile; in half
the contrary is true. This substantiates the results derived in the studies
of rattlesnake head dimensions (/or. c/7. p. 17), where much larger series
were available. I think, based on such figures, and the graphical studies
of every species, that we may assume a constant percentage deviation
about the regression line. This will greatly expedite studies of species dif-
ferences and sexual dimorphism, as will be subsequently discussed.

Correlation of Tail Length and Body Thickness

The fact that slim snakes have long tails, and stout snakes short, is
evident from observation. An endeavor was made to reduce the relation-
ship to figures. The results are rather inadequate, because of the difficulty
of measuring body diameters accurately, particularly after preservation.
Although the expected relationship is found to exist, the correlation is
not especially close.

From 8 to 20 specimens of each of 1 5

species of snakes were measured

at mid-body. The average values of the

body width,

or thickness, as a

percentage of body length, (a quantity

which might well be called the

index of attenuation of a snake) were

ascertained.

These figures were

correlated with the tail-to-body ratios, also expressed

as percentages. In

determining a representative figure for

the latter,

the proportions at

median body length were used, the sexes being averaged. The resulting
data follow:

Ratios in Per

Cent

Series

W/B

T/B

/.. r. roseofusca

... 3.04

15.8

A. q. s argi'i

... 2.90

15.1

G. brachycephala

... 3.22

20.7

G. nasalis

... 3.5 5

17.2

D. a. similis

... 2.22

21.6

M. /. piceus

... 1.89

36.3

M. lateralis

... 1.78

42.9

P. d. per kin si

... 2.72

13.6

A. e. Occident alis

... 2.24

14.8

P. c. an nr etc ns

... 2.45

19.3

R. 1. lecontei

... 2.44

15.7

S. m. linearis

... 2.36

24.1

S. o. annulata

... 2.69

21.9

T. vandenburghi

... 2.02

18.1

T. eiseni

... 2.19

29.6

The correlation between these pairs of figures was calculated and found
to be -0.558. This is significant (P = 0.029). The regression equation is
Y - 3.3 - 0.036 X, where X is the tail-to-body ratio and Y is the width-

40

Bulletin 18: Zoological Society of San Diego

to-body ratio, both figures being expressed as percentages. The adherence
of the several species to the regression line is not particularly close.

T. vandenburghi has a short tail for so slim a snake; while the two
Geophis have overly long tails for their relatively stout bodies. The parabola
Y = 5.68 - 0.199 X + 0.002 5 8 X2 fits the points somewhat better than
the straight line, but the deviations of some of the species remain con-
siderable.

Difference Problems

Having developed certain generalities with respect to the nature of the
variation of tail lengths in snakes, it becomes desirable to show how these
affect the calculations in difference problems. For the methods thus far
developed only have value if it can be shown that their use will facilitate
and render more accurate the evaluation of differences. As compared to
problems involving differences in lepidosis, the characters of which are
presumed to remain unchanged throughout life, we have here the complica-
tion of a character subject to ontogenetic change, both in proportionality
and dispersion.

It is possible, by the method of the analysis of covariance, to determine
the significance of the difference between the values from which two
regression lines have been evolved. In the present case, however, I do
not think we are interested in the average, or over-all, differences through-
out life; rather, we are usually concerned with the difference at some
particular time of life, often the adult stage,24 or at some particular body
size. If we are to deal with tail-length directly, rather than with pro-
portionality, we must have some method whereby the data of all specimens
can be transformed to usefulness at the size at which the computation is
to be made. This can be easily done if we assume linearity of regression
and constancy of the coefficient of variation, both of which suppositions
have been found substantially true in all the species investigated. One
method is to determine a probable tail length for each specimen at some
assumed standard body length, and thus have available an array of tail
lengths which represent a hypothetical assemblage of snakes of that
standard and uniform body size. However, this rather laborious process,
involving a separate calculation for each specimen, which was used in the
investigation of rattlesnake head problems ( loc . cit. p. 22), can be much
simplified, and the same result attained, by a modified method. We first
determine the standard error of estimate of T on L (or B) . This is taken
as the standard deviation of the tail lengths of snakes at length Ml (or Mb) .
Dividing this standard error by the mean tail length Mt gives the coefficient
of variation, which is presumed to remain constant at all ages. From this,
in turn, the standard deviation of the tail lengths at any assumed body
length may be readily computed. For example, let it be desired to determine

24 See diagram loc. cit., p. 21.

Klauber: Tail-length Difeerences in Snakes

41

the probable dispersion in the tail length of snakes of a certain species at
any arbitrary body length ffi. We first derive from the entire sample the
regression equation T = a' + b'B, the standard error of estimate Gt.b, and
the mean length of tail Mx. The value of V is found from V = Ot.b/Mt.
The mean tail length at body length 6, is found from T1 = a ' + b'B{, and
the desired dispersion is found from GTl = VTX = Gt.b (<*' + b'Bx) /Mq\
Having done the same with the other species or group with which a com-
parison is desired, the two tail-length arrays can be compared by the
method of determining the significance of the difference between two
means, for we have available all the required statistics: the two means,
the standard deviations of the arrays, and the number of specimens in
each sample. Of course, the same standard body length Bx must be used
for both species. The full number of specimens in each sample may be
used as Nx and as all specimens have entered into the derivation of

the regression line and the dispersion about that line. Either of two methods
of testing the significance of the difference between means may be em-
ployed.26

For example, we shall test the difference between the tail lengths in
adult Pituophis c. deserticola as found in Imperial County and in the
Mohave-Victorville area of California. It first becomes necessary to select
a standard body length at which the difference will be calculated. This
should usually be an adult length, since, as will be shown in studies of
sexual dimorphism, differences in tail length tend to reach their maxima
in the adult stage.

We take our standard lengths of Pituophis at a body length of 1100 mm.
(Table 8.) Using the regression formulas already derived (Table 6), we
find the following mean tail lengths of snakes of this size:

Area Males Females

Imperial County 169.01 15 5.70

Mohave-Victorville 190.90 176.39

The corresponding standard deviations of the tail length at this standard
body length, as derived from the equations above given and using the
values of V, the mean coefficient of variation, set forth in the two final
columns of Table 6 are:

Area Males Females

Imperial County 8.89 7.67

Mohave-Victorville 11.08 8.99

Applying the null method for determining the significance of the dif-
ference between two means, we find that both sexes show highly sig-

25 This use of the full values N\ and N-2, is a somewhat questionable procedure if the
ontogenetic distribution of the sample is poor.

26 Kenney, Part II, p. 140; Simpson and Roe, p. 192.

42

Bulletin 18: Zoological Society of San Diego

TABLE 13

Adult Tail-length Differences
(Expressed in Terms of the
Coefficient of Divergence in Per Cent)

Males

Females

(Geo phis brachycephala from Panama and

(Geo phis nasalis from Guatemala

8.94* *

1 8 . 0 9

(Diadophis a. similis from San Diego and
( Diadophis p. arnyi II from vie. of Tulsa

0.33

7.48 !i

(Diadophis p. arnyi I from S. Kansas and N. Oklahoma and
(Diadophis p. arnyi II from vie. of Tulsa

- 2.3 3*

0.32

(Boaedon /. lineatus I from Supi, Butandiga and Kaimosi and
(Boaedon /. lineatus II from Lamu Island

- 14.19**

- 15.78*

(Phyllorhynchus b. browni vie. Tucson and
( Phyllorhynchus d. nub ills vie. Tucson

-4.64

0.42

(Phyllorhynchus d. perkinsi I San Diego County and
(Phyllorhynchis d. perkinsi II Riverside County

- 2.72

- 2.47

(Arizona e. Occident alis I San Diego County ond

(Arizona e. Occident alis II Mohave Desert

- 5.74**

- 2.76:

(Pituophis c. annectens San Diego County and
( Pit no phis c. deserticola I Imperial Countv

26.86**

23.28*

(Pituophis c. deserticola II Mohave Desert and

(Pituophis c. deserticola I Imperial County

12.16**

1 2.45 :

(Rhinocheilus l. lecontei I San Diego County and
(Rhinocheilus l. lecontei II Mohave Desert

- 1.40

- 4.70:

(Rhinocheilus l. lecontei I San Diego County and
(Rhinocheilus l. clarus San Diego County

- 3.16*

- 5.70:

(Toluca l. lineata I Hidalgo and

(Toluca /. lineata II Puebla & Veracruz

1.42

- 5.62 =

(Toluca l. lineata II Puebla & Veracruz and

(Toluca l. i aria ns Veracruz

-2.61*

- 3.02

(Sonora o. occipitalis Mohave Desert and

(Sonora o. annulata San Diego and Imperial counties

- 6.04**

- 2.93

(Thamnophis o. ordinoides vie. Portland and

(Thamnophis o. biscut at us vie. Klamath Falls

- 3.72**

— 9.66:

(Micrurus n. nigrocinctus vie. Panama and

(Micrurus n. divaricatus Honduras

OO

l

1

ON

OO

OO

* Significant (P below 5 per cent); ** highly significant (P below one per cent). If the
value given is positive, the first named form has the longer tail; if negative, the second
is the longer-tailed.

Klauber: Tail-length Differences in Snakes

43

nificant territorial differences; that is, the Mohave- Victorville snakes are
definitely longer-tailed than those from Imperial County, the differences
not being attributable to the chance composition of the samples. The
racial dimorphism, or divergence, measured in terms of the coefficient of
divergence, is 12.16 per cent, in the males, and 12.45 per cent in the females.

Using the same method, all the pairs listed in Table 13 were investi-
gated, taking the pertinent statistics from Tables 5 to 9. The results are
given in terms of the adult coefficient of racial divergence, those showing
significant differences (P less than 5 per cent) being starred, while those
which are highly significant (P less than one per cent) are double starred.

In six of these pairs the standard adult sizes of the two components
differ somewhat (third column of Table 8); these are G. brachycephala—
nasalis, D. a. similis-p. arnyi II, B. 1. lineatus lineatus II, T. I. lineata I—
lineata II, T. 1. lineata ll—varians, and T. o. or dinoides— biscutatus . In two
cases, Boaedon and T bam no phis, the differences are relatively great. The
question naturally arises whether the method of evaluating differences
which I have used is a proper one in such situations, for by this method
one really compares the tail proportionality of an adolescent or young
adult of the larger form with a true adult of the smaller. This somewhat
involved problem has been discussed before, in the investigation of the
head sizes of rattlesnakes.^' In tail length investigations I should recommend
that, if the standard adult body length of the larger form exceeds the
smaller by more than 5 0 per cent, then a modified method should be em-
ployed as follows: Make the comparisons at the standard adult length of
the smaller form as before, but instead of calculating the tail length of
the larger from its regression line, calculate it from its own proportion at
standard adult body length. For example, take the comparison between
male T. o. ordinoides from the vicinity of Portland and T. o. biscutatus
from the vicinity of Klamath Falls. The standard adult length of the
former is taken at 3 00 mm. and of the latter, a much larger snake, at
600 mm. We find that ordinoides at 3 00 mm. has a tail length averag-
ing 103.61 mm., while biscutatus at the same length has an average tail
length of 107.54 mm. This difference is found to be significant; how-
ever, as above mentioned, we have really compared an adult ordinoides
with an adolescent biscutatus. Now, biscutatus at its adult length of
600 mm. has a tail ratio of 0.349. Therefore, at 3 00 mm. a hypothetical
adult biscutatus would have a tail length of 104.70 mm. The difference,
formerly 3.93 mm., is now reduced to 1.09 mm. and this is not significant.
A similar treatment of the Boaedon problem and female Thamnophis,
however, does not change the former conclusions; all differences remain
significant. This is a somewhat modified method of making a correction
for differences in ultimate length from that utilized in the head-length

27 Occ. Papers, S. D. Soc. Nat. Hist., No. 4, p. 34. The problem is of particular interest
in comparing the body proportions of stunted races with the parent forms from which
they were derived (see p. 29).

44

Bulletin 18: Zoological Society of San Diego

study.2S It may be pointed out that if our object is solely the demonstra-
tion of specific or subspecific differences, the mere fact that these special
provisions for taking care of differences in ultimate length are necessary,
are, in themselves, evidence of interspecific or subspecific differences.

The results of these example difference investigations represent too
many entirely unconnected taxonomic problems to warrant discussion; it
could not be expected that they would suggest any general relationships.
They are, after all, offered only to illustrate a method of test; to show
how similar, but more unified, problems might be handled in a mono-
graphic treatment of a genus. It will be observed that two species (or
subspecies) may differ significantly in the tail proportionality of one sex,
while none is evident in the other. One surprising result is the high dif-
ference between Pituophis c. annectens and P. c. deserticola, and the some-
what reduced, but still considerable difference, between the desert gopher
snakes of Imperial County and those of the Mohave Desert. This should
naturally suggest a further investigation of these two populations, now
usually considered to belong to a single subspecies, deserticola.

I have mentioned before how necessary it is, if an investigation of tail
length is to be thorough, that the specimens be plotted so that the aberrants
may be carefully resurveyed. By this means many cases of inaccurate
measurements, incorrect sexing, incomplete tails, misidentifications, and
other mistakes (besides true aberrants) will be disclosed. Such was the
case in the series used in the present investigation, for a number of errors
of all these classes were found and corrected; had they not been discov-
ered they would have affected the calculated proportionalities considerably.

Following this method through, it is feasible, if the analytical data be
available, to determine the probability that a particular specimen belongs
in a specific category. For example, let us consider the Lampropeltis zonata-
multicincta problem.29 For the purposes of this example I am going to
assume a territorial homogeneity in the tail length of this species which
may or may not be justified; we do not know the type locality of
Blainville’s doubtful specimen and therefore I shall use all available mountain
king snakes for comparison. Blainville mentions particularly the sharp tail
of his specimen, from which it may be assumed that the tail was complete.

From a study of 44 males and 5 3 females, we find that the tail length
equation of the males is T = —3.3 T 0.188B, and of the females
T = 2.3 + 0.171ZF Blainville’s specimen had a body length of 3 60 mm.
(Burts translation). At this body length the average male would have
a tail length of 64.3 8 mm., while the average female would have a

28 Loc. cit., p. 34.

Charles E. Burt: The Nomenclature of Western Coral King Snakes, Lampropel tis zonata
Versus L. multicincta, Copeia No. 2 of 1936, p. 94. For a history of the case see the
bibliography in Burt’s paper together with James L. Peters, Copeia, No. 2 of 1938, p. 93.

Klauber: Tail-length Dieeerences in Snakes

45

tail length of 63.86 mm. Our analysis of the original data produces the
following values of V: males, 7.86 per cent; females, 10.19 per cent.
From these figures we deduce the following standard deviations in the
tails of specimens having the average tail lengths given above: males.
5.06 mm.; females, 6.51 mm. Blainville’s specimen had a tail length
of 5 5 mm.; thus differing from the average male snake of this body
size by 9.3 8 mm., and from the average female by 8.86 mm. Dividing
these values by the respective standard deviations, we have these re-
sultant values of t: males, 1.8 5; females, 1.3 6. From a /-table we find
that about 7.2 per cent of males, and 18.1 per cent of females would
show as great a deviation from the respective means as does this speci-
men; half of these would be this much smaller than average, the other
half larger. Thus about 3.6 per cent of male snakes and 9 per cent of
females of this species would be expected to have a tail as short as
Blainville’s specimen. Our conclusion is that, while Blainville’s type
had a rather short tail for a California coral king snake, it is by no
means impossible that it may have represented that species, as far as
the tail-length criterion is concerned. If it was a coral king snake, the
odds favor its having been a female.

It may prove interesting to investigate the other species to which
Blainville’s type specimen might have belonged, using the criteria of
pattern, length over-all, and tail proportionality. The ringed or half-
ringed snakes of the area from which Blainville’s other types came
are the following:

Lam pro pelt is getulus calif or niae (ringed phase)

Lam propel tis multicincta
Rhino cloeilus lecontei lecontei
Rhinocheilus lecontei clarus
Sonora occipitalis occipitalis
Sonora occipitalis annulata
Sonora semianmilata semiannulata
Chilo m e nis c ns cin ctus

The last two are quite ruled out by considerations of size, as indeed
are the other two Sonoras as well, yet I shall apply the tail-length test
to the latter. Proceeding with the method previously illustrated in the
case of the coral king snake, multicincta, and repeating the figures
derived for that species, we have the following values of the propor-
tion of specimens of each species that would show as great a difference
from the average as does Blainville’s type, and in the same direction:

46

Bulletin 18: Zoological Society of San Diego

Proportion,

Per Cent

Male

Female

L. g. californiae

32.0

47.0

L. multicincta

3.6

9.0

R. 1. lecontei

5.5

3.6

R. 1. clarus

5.2

12.5

S. o. occipitalis

0.01-

0.02

S. o. annulata

0.01-

0.01

It is evident that the Sonoras do not fit; not only are they too small,
but their tails are proportionately much longer than the Blainville type.
The two subspecies of Rhinocheilus are possibilities, but do not fit the
pattern description, since the rings of Rhinocheilus are not complete
ventrally. As far as tail length is concerned, Blainville’s type is nearer
L. g. californiae than L. multicincta ; in fact, it fits unusually well. Male
L. g. californiae with a body length of 360 mm. would have an average
tail length of 5 8.3 8 mm., while females would average 54.58. These are
very close to the 5 5 mm. of the Blainville type. But the pattern described
by Blainville does not fit the ringed phase of californiae, particularly the
2 half rings on the head. Young specimens of californiae are very dark
brown, but Blainville would hardly have referred to them as black. Thus
the coral king snake remains the best possibility, although clearly the
identification is quite uncertain. The tail-length survey surely throws addi-
tional doubt on the allocation of Blainville’s snake to L. multicincta ; and
zonata, if it is to be considered valid, must be based on Lockington rather
than Blainville. Hence the decision becomes one of the interpretion of the
Code of Nomenclature, as discussed by Linsdale, Burt, and Peters.

In a problem of this character, if one can make a graphic determina-
tion of the regression lines, it would probably be satisfactory to assume
V at 8 per cent, judging by the values of V found in the 48 species
analytically investigated and set forth in the last two columns of Table 6.
Lam pro pelt is seems to be a genus with rather higher values of V than
most snakes.

Sexual Dimorphism

The coefficient of divergence, and the method, hitherto developed, of
deriving hypothetical tail-length arrays at any standard body size, afford
useful statistics for investigating and stating the extent and significance 30
of sexual dimorphism in tail length. The series listed in Tables 5 to 9 have
therefore been surveyed, with the results set forth in Table 14. While

30 The term as here used does not connote a determination of the reasons for sexual
dimorphism and its interspecific variations; here significance is used in its mathematical
sense and refers to the determination whether any intersex difference found might result
from the chance composition of the available series of specimens.

Klauber: Tail-length Dieeerences in Snakes

47

adult proportions are to be considered more important than juvenile, the
latter have been included to show ontogenetic trends in this characteristic.
Again the comparisons have been made on a basis of equal body lengths B,
rather than lengths over-all, L, since this seems a more logical procedure.
As previously mentioned, the standard adult lengths represent large, but
by no means exceptional adults, as determined from the material at hand.
If one sex reaches a distinctly greater length than the other, the standard
body length is based on the smaller, so that in no case are tail lengths de-
duced for body sizes greater than those attained in nature.

TABLE 14

Sexual Dimorphism in Tail Length of Analytic Series
(Expressed as a Percentage in Terms of the
Coefficient of Divergence)

Juvenile

Adult

Lichanura r. roseofusca

-1.1

9.8

Adelphicos q. sargii

21.6

30.0

Geo phis brachycephala

5.7

20.6

Geo phis nasal is

14.8

27.8

Di ado phis a. si mil is

14.9

18.7

Di ad op his p. arnyi I

25.6

23.1

Diadophis p. arnyi II

18.7

25.7

Boaedon l. li neat us I

34.8

39.2

Boaedon l. lineal us II

24.3

34.7

Mas tico phis f. pi ecus

-7.7

-1.4

Mast i co phis lateralis

-0.8

3.8

Salvador a g. virgultca

3.2

3.6

Phyllorhynchus b. broivni

46.3

58.3

Phyllorhynchus d. nubilus

36.4

62.9

Phyllorhynchus d. perkinsi I

27.8

48.5

Phyllorhynchus d. perkinsi II

22.6

48.8

Arizona e. occidentalis I

22.6

6.9

Arizona e. occidentalis II

12.6

9.9

Pit no phis c. annectens

10.4

11.8

Pit uo phis c. deserticola I

12.2

8.2

Pit uo phis c. deserticola II

21.0

7.9

Lampropeltis g. calif or niae

8.1

3.9

Rhinocheilus l. lecontei I

12.7

9.9

Rhinocheilus l. lecontei II

16.8

6.6

Rhinocheilus I. clarus

26.7

7.4

Toluca l. lineata I

26.7

37.8

Toluca l. lineata II

27.7

31.5

Toluca l. varians

18.5

32.2

Sonora m. linearis

10.3

18.7

Sonora o. occipitalis

24.8

13.6

48

Bulletin 18: Zoological Society of San Diego

TABLE 14
(Continued)

Sexual Dimorphism in Tail Length of Analytic Series
(Expressed as a Percentage in Terms of the
Coefficient of Divergence)

Juvenile

Adult

Sonora o. annul at a

17.1

16.7

Chilomeniscus s. stramineus

24.2

19.0

T ha mno phis ham mondii

1.2

10.7

Thamnophis o. ordinoides

6.7

17.1

Thamnophis o. hiscutatus

10.9

11.6

Conopsis nasus

25.0

27.8

Hypsiglena ochrorhyn clous

12.9

19.2

Trimorphodon vandenburgloi

11.3

17.4

T ant ilia eiseni

27.8

13.8

Elapsoidea niger

31.9

17.3

Micrurus a. mayensis

19.2

32.2

Micrurxis n. nigrocinctus

34.9

41.6

Micrurus n. divaricatus

29.7

43.2

Aokistrodon m. moke son

9.7

4.6

T rimeresurus gramineus

0.2

22.8

T rimeresurus elegans

6.0

16.6

Bo t hr ops insular is

15.1

17.6

At her is squamigera

12.2

19.6

(Negative values indicate a female tail length greater than the male.)

Sexual dimorphism in the tail length of snakes is so well known that it
requires no detailed study to demonstrate its existence. Thus, among the
48 series which were investigated analytically, all but two have a highly
significant adult sexual dimorphism, with P less than 0.01; in fact, in all
but 4 cases P is less than 0.0001, and this notwithstanding the fact that
several of the series contain relatively few specimens.

The four exceptions are: First, M. /. piceus, which is the only subspecies
with a female tail exceeding the male, although the difference is of doubt-
ful significance (P = 0.075); secondly, S. g. virgultea (P = 0.056) this
being the only other series in which the sexual difference is below the
usually accepted level of significance (P = 0.05); M. lateralis (P = 0.009),
and R. 1. lecontei (P = 0.0007) . The two last have a highly significant sexual
dimorphism and are mentioned only because they do not reach P = 0.0001.

In Tables 15 and 16 I have presented the coefficients of sexual di-
morphism of the series which were investigated only graphically. Neces-
sarily these coefficients are somewhat less accurate than those derived
analytically.

Klauber: Tail-length Dillerences in Snakes

49

TABLE 15

Sexual Dimorphism in Tail Length of Graphic Series
(Expressed as a Percentage in Terms of the
Coefficient of Divergence)

Juvenile

Adult

C. bottac

6.1

33.5

G. semidoliatus

16.9

27.2

N. pscpbota

13.9

11.7

N. s. morleyi

6.3

30.5

D. a. modest us

3.9

14.5

L. c. or n at um

33.4

22.9

C. c. constrictor

8.0

8.1

C. c. flaviventris

10.7

8.7

C. c. mormon

3.2

8.8

M. /. flagellum

-10.4

3.8

M. /. testaceus

7.6

6.0

M. /. pi ecus (Cape)

-9.1

-1.0

M. t. taeniatus

-19.3

0.8

P. d. perkinsi (Arizona)

31.4

50.0

E. rufodorsata

25.7

24.3

A. e. elegans

7.0

4.1

P. s. affinis (C. Arizona)

10.9

12.0

P. s. affinis (Yuma)

6.0

11.8

L. in ulticincta

-9.0

4.3

R. 1. lecontei (Arizona)

14.2

8.8

R. /. tessellatus

10.2

6.7

S. e pisco pa

17.7

19.5

S. m. linearis (not Laguna)

8.9

18.1

N. t. tigrina

9.1

12.8

C. /. punctigularis

18.1

7.8

T. gracilis

8.1

16.0

7 . albolabris

17.5

43.7

There is evident a tendency of sexual dimorphism to increase with age;
that is, the regression lines of the sexes tend to converge in the juvenile
range. But this tendency is by no means universal; out of a total of 88
series, 2 5, or about 2 8 per cent, have a contrary trend. This divergence
from what might be considered a normal scheme of variation, is particu-
larly evident in certain genera, Arizona, Pituopbis, and Rhinocheilus being
conspicuous examples. It might be thought that this juvenile convergence
of the lines is due to inaccurate sexing of the juveniles. This may be true
to a very minor degree in some series, but there are others, Phyllorhynchus
for example, in which mistakes in sexing are virtually impossible, yet the
same tendency is in evidence.

50

Bulletin 18: Zoological Society of San Diego

TABLE 16

Sexual Dimorphism in Tail Length of T bamnophis
(Expressed as a Percentage in Terms of the
Coefficient of Divergence)

T.

Species

o . vagrans

Locality

Nevada, Utah

Juvenile

5.2

Adult

11.6

T.

o. biscut at us

Klamath region

10.9

11.6

T.

o. elegans

Lassen, Modoc, Shasta, Plumas Cos.

8.1

12.6

T.

o. hydro phila

S\V. Oregon

7.5

12.2

T.

o. hydro phila

\'\V. California

7.1

10.7

T.

o. couchii

Cromberg, Plumas County

19.5

19.2

T.

hammondii

San Diego County

1.2

10.7

T.

hammondii

N¥. Lower California

-4.7

9.8

T. digueti

Lower California

6.2

9.7

T.

o. at rat us

NW. California

8.5

13.1

T.

o. at rat us

San Francisco Bay region

2.9

12.4

T.

o. ordinoides

Canada

16.3

20.5

T.

o. ordinoides

Washington

5.3

14.8

T.

o. ordinoides

Portland

6.7

17.1

T.

o. ordinoides

SW. Oregon

11.3

15.9

T.

mar cianus

Arizona

3.2

11.0

Certain generic tendencies in the adult sexual dimorphism are evident
in these tables. Phyllorhynchus has the highest dimorphism of the forms
investigated. Others outstandingly high are Micrurus, Toluca, Geophis, and
Boaedon. The racers are conspicuously low. These genera with high dimor-
phism are all short-tailed, while the racers are long-tailed. This at once
suggests that there may be a correlation between tail proportions and sexual
dimorphism. To investigate this possibility each subspecies was represented
by an ordinate determined by the mean between the male and female tail-
to-body ratio, while the abscissa was taken as the coefficient of adult
sexual divergence.

It is found that there is a rough correlation — long-tailed snakes do
tend to have low sexual dimorphism and vice versa. However, the correla-
tion is not particularly close and many species are exceptions, among the
most conspicuous being Lichanura, Arizona, Pituophis, Rhinocheilus,
Elapsoidea, and Agkistrodon. All of these are short-tailed snakes, yet they
have a low sexual dimorphism. Ignoring these, the regression equation is
found to be approximately S= 56— 1.33T, where S is the coefficient of
sexual dimorphism in percent, while T is the mean adult tail proportion-
ality, that is, the mean between the sexes at standard adult body length.
A somewhat better fit is given by the parabola S = 78.76 — 3.1 5 T + 0.03 1 T2.
But it should be emphasized that the relationship is only approximate; while
the correlation is not to be questioned, many species do not fall close to
either of these lines.

Klauber: Tail-length Dieferences in Snakes

51

This relationship is not entirely unexpected. In the short-tailed snakes
the male sex organs would require a certain minimum space which would
tend to increase dimorphism. On the other hand, the long-tailed snakes,
in which the tails serve various useful purposes, such as for balance, pro-
pulsion, holding, or climbing, the necessity for the space for primary
sex organs would obviously have a more incidental effect on tail length.

Rattlesnake Tail Proportionalities

The rattlesnakes, with their relatively short tails, do not comprise a
particularly fruitful field for the study of tail proportionalities. However,
sexual dimorphism is considerable, and there is some interspecific varia-
tion. I shall therefore make a survey of the rattlesnake data. This section
of the tail-length study is to be considered the appropriate tail-length
chapter of the series of papers I have hitherto published under the general
title of "A Statistical Study of the Rattlesnakes”.31

In determining the tail length of a rattler, the measurement is made
from the center of the anal plate to the anterior edge of the proximal
rattle. Specimens with incomplete tails are few. On the whole, the rattle-
snake material, in many species, is fairly adequate. As all correlations are
high and tail lengths are usually less than 10 per cent of body length, I
shall neglect the matter of spurious correlation and use L rather than B
as a basis for all calculations. Therefore, the regression equations, if they
prove to be linear, will be expressed in the form T = a + bL.

As an initial approach it may be of interest to neglect ontogenetic
variation and determine the mean and dispersion of the tad proportion-
ality, using the tail ratios of the individual specimens as the variates, a
method not employed in the study of the other snakes. The results of
such calculations, of obviously limited utility owing to the probable effects
of the ontogenetic distribution of the samples, are presented in Table 17.
For this reason the coefficients of variation V are to be considered of more
value than the mean ratios. Not all of the series are territorially homo-
geneous.

A substantial sexual dimorphism will be noted. Variability tends to be
higher among females than males. Specific differences in proportionality
are evident, but not so great as those found within genera of colubrids,
where the tails have many functions quite different from that of a
rattle vibrator.

Having made this preliminary survey, I now proceed to an investiga-
tion of the rattlers by the methods previously employed with the colubrids,
first testing a few series analytically, then making graphic determinations
on other less adequate series.

31 Occ. Papers S. D. Soc. Nat. Hist., Nos. 1, 3, 4, 5, 6.

52

Bulletin 18: Zoological Society of San Diego

TABLE 17

Dispersions of the Tail Length to Length Over-all Ratio in Crotalus
(Neglecting Ontogenetic Variation)

Males

Females

Species

Territory

N

M

V

N

M

V

C. cinereous

Texas- Arizona

247

0.0768

6.16

186

0.0593

9.67

C. ruber

San Diego Co.

79

0.0687

5.66

81

0.0539

11.80

C. s. scutulatus.

All

160

0.0730

7.10

93

0.0543

11.74

C. v. viridis

Pierre

143

0.0745

5.50

151

0.0547

8.66

C. v. viridis

Platteville

453

0.0742

6.23

387

0.0536

9.70

C. v. lutosus

All

146

0.0707

6.88

103

0.05 58

9.26

C. v . oreganus. ..

San Diego Co.

137

0.0740

7.19

135

0.0599

9.55

C. cerastes

All

101

0.08 53

7.09

74

0.0620

10.46

N is the

number of specimens, M

the mean of the ratios calculated

separately,

and V the coefficient of

variation

of the ratios

in per

cent.

From both the analytic and graphic studies, linearity of regression is
again evident among the rattlers, as it was in the other genera. Six of the
largest series were tested for linearity by an F- test:'2 A probable departure
from a straight line was found in only one ( ruber females). In the interest
of simplicity in the treatment of difference problems we are fully justified
in the assumption of linearity throughout.

Table 18 presents an analytical study of seven large, territorially homo-
geneous series of rattlers. Comparing the results with those found in the
taper-tailed snakes previously discussed, we find that the correlation co-
efficient, r, tends to be somewhat lower in the rattlers. Of course, as the
rattlers have short tails, the values of b in the regression equation
T = a + bL are lower than in most other snakes. But the greatest differ-
ence lies in the values of a; these are low in the rattlers, thus indicating
a reduced ontogenetic variation. In fact, a is found to be significantly dif-
ferent from zero in only three out of 14 cases, all three being females.
Hence, to ignore the change in tail proportionality with age would not
be as serious in rattlesnake investigations as in most other genera. The
variability of tail proportionality in the rattlers is high as compared to
most colubrids, as shown by the mean coefficient of variation (compare
with Table 5). Sexual dimorphism, as indicated by the coefficient of
divergence between adult males and females, is high, as is to be expected
in short-tailed snakes, although not reaching the differences attained by
some colubrids (compare with Table 14).

6- The modern test for linearity will be found in most recent statistical texts, as for
example: Rider, 1939, p. 130; Yule and Kendall, 1937, p. 45 5 ; Goulden, 1939, p. 211.
I find it desirable to omit Sheppard’s correction in the r computations, if a linearity test
is to be made.

1

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V/-s

Uh

04

04

r\

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NO

o

o

NO

>o

»o

o

NO

^t"

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Ky-\

»

O

rO

ON

O

ON

o

04

04

a\

o

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■3

04

04

cK

1—^

bC

w

o

Q

c

o

Is.

OO

04

rt

04

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OO

rO

o

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t\

ls~s

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OO

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54

Bulletin 18: Zoological Society of San Diego

As before, it is found that the proportionate dispersion about the re-
gression line remains substantially constant throughout life. The follow-
ing results, for example, were derived from a study of restricted size
groups of the Platteville series, C. v. viridis :

Coefficient of Variation, Per Cent

Males Females

Juveniles 6.19 7.38

Adults 6.16 8.30

It will be seen that there is no evidence of a decrease in the coefficient
of variation with increased size, such as would be the case if the dispersion
about the regression line remained constant in dimension rather than pro-
portion. In general, we find the same conditions to exist in Crotalus as the
other genera: Ontogenetic change in tail proportionality (of reduced im-
portance, however) ; linearity of regression; and a constant coefficient of
variation through life.

Table 19 contains the results of graphic studies on the other series of
rattlers available to me. The data on the analytic series have been repeated
to render comparisons easier. Where a territorially restricted series has
been at hand I have used it to represent its species, thus avoiding the possi-
bility of intrasubspecific variability. But often I have had to go farther
afield and have used all available specimens to secure enough points for a
trend line. In many of the rarer forms the material is too inadequate to
be trustworthy. Some series are poorly distributed ontogenetically, lack-
ing either juveniles or adults. Altogether, I believe the values of b are
moderately reliable, those of a not very dependable. In general, while
we may draw the conclusion that the ontogenetic change in the rattlers is
less important than in most colubrids, it still is worthy of some attention.
If we average all values of a we find the mean of the males to be —0.5,
and of the females + 1.1. I think we may conclude that male tail ratios in-
crease slightly with age, while female tail proportionalities decrease some-
what more, and that a values of -0.5 in the case of the males and +1.1
in the females probably indicate a situation approximated in most species.
To give an idea of the extent of ontogenetic change thereby involved,
these average values of a are applied to a hypothetical species 300 mm.
long at birth and 1000 mm. at maturity, with b values (close to cinereous
or viridis) of 0.075 for the males and 0.05 5 for the females. The results

follow:

Males

Females

Sexual
divergence
per cent

Tail ratio at birth

.... 0.0733

0.0587

22.2

Adult tail ratio

0.0745

0.0561

28.2

Ontogenetic divergence, per cent

1.6

-4.5

TABLE 19

Relationship of Tail Length to Length Over-all in Rattlesnakes

(General Series) Standard Ratio of Tail Adult

Species or Subspecies

Area or Series

Number of
Specimens

M F

C. durissus durissus

All

29

22

C. durissus terrificus

All

31

34

C. uni col or

All

8

5

C. basiliscus

All

68

46

C. enyo

All

34

21

C. molossus molossus

All

90

56

C. molossus nigrescens

All

62

56

C. adamanteus

All

31

43

C. cinereous

Texas and Oklahoma

149

124

C. cinereous

Arizona

87

95

C. cinereous

California

42

29

C. tortugensis

All

30

17

C. lucasensis

Cape Region, B. C.

161

140

C. ruber

San Diego Co

156

104

C. exsul

All

17

7

C. scutulatus scutulatus

Arizona

156

97

C. scutulatus scutulatus

California

84

52

C. viridis viridis

Montana

80

93

C. viridis viridis

Pierre, S. D

364

342

C. viridis viridis

Platteville, Colo

452

392

C. viridis viridis

New Mexico

84

53

C. viridis nuntius

Winslow

69

37

C. viridis abyss us

All

13

9

C. viridis hit os us

Utah

84

58

C. viridis hit os us

Nevada

52

33

C. viridis decolor

All

19

19

C. viridis oreganus

Pateros, Wash.

314

266

C. viridis oreganus

San Diego Co.

354

354

C. viridis oreganus

Arizona

51

22

C. mitchellii mitchellii

Cape Region, B. C —

50

30

C. mitchellii pyrrhus

San Diego Co

68

34

C. mitchellii pyrrhus

Arizona

28

13

C. mitchellii stephensi

All

67

42

C. tigris

All

28

13

C. cerastes

Arizona

52

28

C. cerastes

Nevada and Utah

10

7

C. cerastes

Colorado Desert

134

114

C. cerastes

Mohave Desert

101

55

C. poly st ictus

All

9

8

C. horrid us horridus

All

43

69

C. horridus atricaudatus

All

13

28

C. lepidus lepidus

All

16

5

C. lepidus klauberi

Arizona

53

50

C. triseriatus triseriatus

All

38

39

C. triseriatus pricei

All

39

20

C. ivillardi

All

16

14

S. ravus

All

11

7

S. miliarius miliar ius

All

36

20

S. miliarius barbouri

All

103

80

S. miliarius streckeri

All

55

49

S. eaten at us catenatus

All

177

170

S. catenatus tergeminus

All

43

65

Adult

Length to Length

Coefficient

Regression

Reg

ression

Length

Over-all at Standard

of Sexual

Coefficient, b

Constant, a

Over-all

Adult Length

Divergence

M

F

M

F

mm.

M

F

Per Cent

.1030

.0684

-4.0

1.0

1450

.1002

.0691

36.7

.1006

.0669

- 1.9

0.7

1150

.0989

.0675

37.8

.1042

.0677

- 1.5

2.7

800

.1023

.0711

36.0

.0927

.0644

- 2.1

1.7

1450

.0913

.0652

33.4

.0944

.0629

0.6

1.2

750

.0952

.0645

38.4

.0725

.0532

- 1.0

3.1

1000

.0715

.0563

23.8

.0757

.0560

0.3

2.5

1000

.0760

.0585

26.0

.0830

.0619

0.7

1.1

1700

.0834

.0625

28.7

.0808

.0578

- 1.5

1.6

1450

.0798

.0588

30.3

.0769

.0560

- 1.3

1.6

1200

.0758

.0573

27.8

.0810

.0585

- 1.8

-0.3

1200

.0795

.0582

30.9

.0790

.0588

- 3.4

- 0.8

850

.0750

.0579

25.8

.0720

.0522

1.0

1.3

1050

.0730

.0534

31.0

.0692

.0514

-0.6

0.7

1100

.0686

.0521

27.4

.0707

.0520

- 1.0

1.5

800

.0694

.0539

25.1

.0746

.0503

- 0.6

2.2

950

.0740

.0526

33.8

.0742

.0536

- 0.5

0.6

950

.0737

.0542

30.5

.0730

.0485

1.8

3.5

1000

.0748

.0520

36.0

.0740

.0496

0.4

2.6

950

.0745

.0523

35.0

.0725

.0476

0.8

3.0

850

.073 5

.0510

36.0

.0772

.0560

0.9

1.7

800

.0783

.0581

29.6

.0751

.0504

1.1

2.7

550

.0771

.05 53

32.9

.0722

.0590

1.0

- 0.2

850

.0734

.0588

22.1

.0697

.0522

1.0

2.1

950

.0708

.0544

26.2

.0737

.05 56

- 0.9

0.9

950

.0728

.0565

25.2

.0803

.0591

- 1.4

0.0

650

.0781

.0591

27.7

.0736

.0562

- 0.1

0.3

850

.0735

.0566

26.0

.0730

.0524

0.7

3.0

950

.0737

.05 55

28.2

.0679

.0520

1.1

1.8

850

.0692

.0541

24.5

.0849

.0676

- 2.4

- 3.0

800

.0819

.0638

24.9

.0743

.0578

- 1.8

0.2

950

.0724

.0580

22.1

.0698

.0606

2.2

0.2

900

.0722

.0608

17.1

.0823

.0592

- 1.2

0.6

750

.0807

.0600

29.4

.0861

.0621

- 2.2

2.0

700

.0830

.0650

24.3

.0886

.0624

- 2.1

- 0.8

600

.0851

.0611

32.8

.0998

.0698

- 3.1

- 1.7

600

.0946

.0670

34.1

.0932

.0655

- 3.9

- 1.3

600

.0867

.0633

31.2

.1017

.0660

- 3.8

- 0.4

600

.0954

.0653

37.4

.0763

.0578

2.5

0.5

800

.0794

.0584

30.5

.0766

.0570

0.6

1.8

1050

.0772

.05 87

27.3

.0763

.0608

1.5

1.0

1250

.0775

.0616

22.9

.0850

.0713

0.2

0.2

550

.08 54

.0717

17.4

.0807

.0655

1.8

2.4

560

.0839

.0698

18.4

.0963

.0705

0.2

1.5

530

.0967

.0733

27.5

.0893

.0727

-0.8

0.2

530

.0878

.073 1

18.3

.1023

.0820

0.6

3.8

475

.1036

.0900

14.1

.0967

.0818

- 0.4

- 0.4

520

.0959

.0810

16.9

.1180

.0958

0.7

1.0

450

.1196

.0980

19.9

.1202

.1034

1.7

1.2

535

.1234

.1056

15.5

.1276

.1060

- 0.4

2.1

520

.1268

.1100

15.6

.1065

.0780

- 0.7

1.8

700

.1055

.0806

26.8

.1109

.0837

- 0.7

0.4

680

.1099

.0843

26.4

Klauber: Tail-length Dielerences in Snakes

55

An ontogenetic difference of 4.5 per cent from birth to maturity is
present in the females. It will be observed that sexual dimorphism in-
creases with age to a considerable degree; this results from the divergence
of the two regression lines inherent in the negative value of a in the male
line, whereas it is positive in the female.

In assigning standard adult lengths over-all, I have taken as a guide
large, but not record-breaking, sizes attained by the smaller sex — the
females in all species except cerastes.

Some racial trends of interest will be noted in Table 19. It will be seen
that the males of many species have a tail ratio of between 0.07 and 0.08,
while most females fall between 0.0 5 and 0.06. These figures may be con-
sidered to represent the rattlesnake mode; they are characteristic, for ex-
ample, of the cinereous and viridis groups. Durissus and its allies (includ-
ing enyo) are definitely long-tailed, as is adamanteus to a lesser degree.
Stephens i, figris, and cerastes are moderately long-tailed; this is also the
case with the smaller rattlers, lepidus, triseriatus, and willardi. Small species
are generally long-tailed. All members of the genus Sistrurus are long-
tailed, especially miliarius, which also has a low sexual divergence. The
shortest-tailed species is probably ruber.

It is to be regretted that sufficient specimens of the interesting form
stejnegeri, the longest-tailed member of the genus Crotalus, were not
available to permit ascertaining its regression lines. Three males have an
average tail proportionality of 0.1209, thus approximating the tail ratio
of miliarius. Other forms, of which only a few specimens were available,
are the following:

3 female totonacus average 0.681
1 male vegrandis 0.1010; 1 female 0.070 5
1 male omiltemanus 0.0868; 2 females 0.0682

Tail length is sometimes useful as a diagnostic character. The most
important case among the rattlers is the differentiation of basiliscus from
molossus. These two species have a considerable superficial similarity.
There is some difference in tail proportionality, especially in the males,
for basiliscus belongs to the long-tailed durissus group, while molossus is
more normal. The easiest way to diagnose questionable specimens is to
make a graphic determination by drawing the regression lines of the two
species (the sexes must be handled separately), using the coefficients
a and b in Table 19; then plot the positions of the doubtful specimens,
allocating each to whichever species it more nearly approaches. This
should give a correct determination in most cases. Of course, other char-
acters, especially those of pattern, should be used to confirm the diagnosis.

This brings up a problem of some interest which may be solved ana-
lytically: What is the overlap in the molossus and basiliscus tail-length
ranges? As the regression lines are not parallel, there is a slightly different

5 6

Bulletin 18: Zoological Society of San Diego

answer for every snake-length over-all. I shall use, as an example, male
snakes of 1000 mm.; normal dispersion about the regression line is
assumed, as well as an 8 per cent coefficient of variation, which is a fair
average among the rattlers. From the regression line we calculate the
mean tail-length of basiliscus at 1000 mm. to be 90.6 mm., with a stand-
ard deviation (8 per cent) of 7.248 mm.; the corresponding figures for
molossus are 71.5 mm. and 5.720 mm. The difference between the means,
d, is 19.1 mm. The ratio between the standard deviations, placing the
larger in the numerator, is assigned the symbol k; in this case it equals
1.267. The equation for the point of intersection between two normal
curves (which also defines the minimum overlapping areas) is found to be

— d + \k2d2 + 2(T22 {kd - 1 ) log£k \
x =

k--l

where OL is the larger of the two standard deviations. Solving this we find
that 6.92 per cent of the molossus would fall in the basiliscus area and
7.13 per cent of the latter in the area of the former; that is, these figures
represent the tails of the curves beyond their point of intersection. This
gives an idea of the number of specimens which would be incorrectly
identified by the use of this criterion.

I should call attention to the fact that the same method of analysis
may be used to determine the percentage of specimens of any species
which would be incorrectly sexed by the use of tail length alone. Of
course, I do not suggest this as a practical method. Rather, it may be
recommended that the available specimens be plotted graphically. Aber-
rants and doubtful specimens will be clearly apparent, and should be
rechecked. In fact, one can, if he wishes, draw a straight line which seems
best to separate the sexes; the equation of this line is then determined and
may thereafter be employed as a guide in rechecking the sexing. For ex-
ample, I find that the line T = 0.786L — 2.0 separates the sexes in the
Colorado Desert specimens of C. cerastes quite consistently, few specimens
of either sex straying to the wrong side of the line. Thus, I know that a
specimen 5 00 mm. long with a tail less than 37.3 mm. long is quite
unlikely to be a male and, if so recorded, should be rechecked. This may
appear complicated when the equation is cited, but it is rapid and simple,
once the specimens have been plotted on cross-section paper.

Acknowledgments

I wish to express my sincere appreciation to those who sent me many
excellent series of measurements, of both American and foreign species
of snakes which would not otherwise have been available to me, includ-
ing especially, Dr. Howard K. Gloyd, Mr. Arthur Loveridge, Mr. Karl P.
Schmidt, Dr. Hobart M. Smith, and Dr. Edward H. Taylor. Dr. Henry
S. Fitch allowed me to use the measurements of the garter snakes studied

Klauber: Tail-length Differences in Snakes

57

in preparation of his 1940 monograph. Mr. Joseph R. Slevin worked with
me for a number of days measuring specimens in the fine collection of
the California Academy of Sciences. To Mr. Charles E. Shaw I am much
indebted for the measurements of most of the series of snakes contained
in my own collection. As usual, I have had the advantage of the pertinent
criticisms of Mr. C. B. Perkins of the Zoological Society of San Diego. The
diagram was prepared by Mr. L. C. Kobler, who likewise prepared the
figures for the succeeding paper.

Summary and Conclusions

1. Various statistical equations defining the relationship of parts and
wholes are given.

2. The coefficient of divergence is defined and its standard error is
presented.

3. The quantitative effects of spurious correlation are discussed.

4. Examples of the use of the coefficient of divergence, both as a meas-
ure of sexual dimorphism in scalation, and of racial difference are presented.

5. Methods and precautions necessary in accumulating data on the
tail-lengths of snakes are cited. Graphic and analytic methods of de-
termining trends and dispersions are discussed. The importance of the
graphical method in eliminating errors and aberrants is pointed out.

6. The tail-length proportionalities of 48 series of snakes are analyzed,
together with 40 others which are treated graphically. The statistics are
set forth. The relationship between tail length and body length, or length
over-all, is linear, or substantially so, in every species studied. An ontogenetic
variation in tail proportionality of considerable extent is usually present,
so if the tail length is to be used in taxonomic problems, provision must
be made for this variation or the results may be seriously in error. The
dispersion of individual specimens about the regression line remains sub-
stantially constant in proportion (rather than in absolute measure) during
life.

7. Ontogenetic trends differ in the various genera and species. Often
there is a tendency toward an increase in tail proportionality with age,
especially in the males, but some forms show contrary effects. Phyllorhyn-
chus is a genus of high ontogenetic change.

8. Parabolic tail-length equations are deduced for a few species. How-
ever, as they seem to have no advantage in accuracy, the simpler straight-
line equations are to be preferred in taxonomic studies.

9. There is a negative correlation between tail proportionality and
body thickness — slim snakes tend to have long tails; however, the corre-
lation is not particularly close.

58

Bulletin 18: Zoological Society of San Diego

10. Methods of determining the significance of differences, taking
into account both ontogenetic change in proportionality, and dispersion
about the regression lines, are evolved. Example problems are presented.

11. The sexual dimorphism in tail proportions of a number of species
and subspecies are stated in terms of the coefficient of divergence. There
is a tendency of sexual dimorphism to increase with age, but this is not
universal. There is a moderate correlation between sexual dimorphism and
tail ratio, short-tailed snakes having a higher sex difference, but there
are exceptions to the rule.

12. Similar methods were employed in a study of rattlesnake tail
lengths. The same trends are disclosed as in the snakes with tapered tails,
although the ontogenetic change is reduced. Some boundary problems
are analyzed.

The question naturally arises as to how far the methods herein out-
lined are applicable to ordinary taxonomic problems.

The purpose of this paper has been dual: To determine the nature of
the tail-length proportion in such series of snakes as are available, to see
if there are any standardized modes of variation; and, to suggest statistical
methods for the use of the tail-length proportion in taxonomy.

Probably the most important conclusion, insofar as ordinary studies
are concerned, is that the tail-length ratio is usually not ontogenetically
constant. Such being the case, if interspecific comparisons are to be made,
it is essential that either statistical methods, along the lines suggested
herein, be employed, or else the material used in the differential studies
be limited to specimens within a narrow range of age, adults preferred.
The most serious errors will occur if two series are compared, in one of
which juveniles predominate, while the other is made up largely of adults.
With these precautions, and having in mind the limited material avail-
able in most taxonomic problems, it will be sufficiently accurate to em-
ploy as the variates, the calculated ratios of the several specimens, using
the formulas for the significance of the difference between two means.
But in monographic work, with plenty of material available, the more
thorough methods are to be recommended; at least they should be employed
with one or two of the largest homogeneous series, to determine the nature
of the variation of tail length within the genus. The analytic method will
be found particularly useful in the study of subspecies and races, but even
then it is only valuable when relatively large series are at hand. Graphic
studies, because of the clarity with which they disclose aberrants, mis-
identifications, and errors of sexing or measurement, are to be recommended
in all cases where tail length is to be used as an identifying character, and
particularly when it is to be employed in differential diagnosis.

Klauber: Tail-length Differences in Snakes

59

Bibliography

Burt, C. E.

193 6. The Nomenclature of Western Coral King Snakes, Lam propeltis
zona fa Versus L. multicincta. Copeia, No. 2 of 193 6, p. 94-98.

Cochran, Doris M.

1941. The Herpetology of Hispaniola. Bull. U. S. Nat. Mus., No. 177,
pp. VII + 398.

Dahlberg, Gunnar

1940. Statistical Methods for Medical and Biological Students. London.
Do Amaral, Afranio

1921. Contribution towards the Knowledge of Snakes in Brazil. Part I.
Four New Species of Brazilian Snakes. Mem. Inst. Butantan,
Vol. I, pp. 51-88.

1934. Sobre a especie Bothrops alternata D. & B., 18 54 (Crotalidae) .
Mem. Inst. Butantan, Vol. 8, pp. 161-182.

Fitch, H. S.

1940. A Biogeographical Study of the Ordinoides Artenkreis of Garter
Snakes (Genus Tbamnophis) . Univ. Calif. Pubs, in Zook, Vol. 44,
No. I, pp. 1-150.

Goulden, C. H.

1939. Methods of Statistical Analysis. New York.

Huxley, J. S.

1932. Problems of Relative Growth. London.

Kenney, J. F.

1939. Mathematics of Statistics. 2 Vols., New York.

Klauber, L. M.

1936-40. A Statistical Study of the Rattlesnakes. Occ. Papers San
Diego Soc. Nat. Hist., Nos. 1, 3, 4, 5, 6.

Linsdale, Jean

1936. Variations in the Dotted Burrowing Snake Chilomeniscus stram-
ineus. Copeia, No. 4 of 193 6, pp. 2 32-2 34.

Pearl, Raymond

1940. Introduction to Medical Biometry and Statistics. Third Edition.
Philadelphia.

Peters, J. L.

193 8. The Name of the Western Coral King Snake. Copeia, No. 2 of
1938, p. 93.

Rider, P. R.

1939. An Introduction to Modern Statistical Methods. New York.

60

Bulletin 18: Zoological Society of San Diego

Simpson, G. G. and Roe, Anne

193 9. Quantitative Zoology. New York.

Slevin, J. R.

1939. Notes on a Collection of Reptiles and Amphibians from Guate-
mala. Proc. Cal. Acad. Sci., 4th Ser., Vol. 23, No. 26, pp.
393-414.

1942. Notes on a Collection of Reptiles from Boquete, Panama, with
the Description of a New Species of Hydromor pirns . Proc. Cal.
Acad. Sci., 4th Ser., Vol. 23, No. 32, pp. 463-480.

Smith, H. M.

1941. Notes on the Mexican Snakes of the Genus Geophis. Smithsonian
Misc. Coll., Vol. 99, No. 19, pp. 1-6.

1942. A Review of the Snake Genus Adelphicos. Proc. Rochester Acad.
Sci., Vol. 8, pp. 175-195.

Thompson, D. W.

1942. On Growth and Form. New Edition. Cambridge (England).
Yule, G. U. and Kendall, M. G.

1937. An Introduction to the Theory of Statistics. Eleventh Edition.
London.

Klauber: Graphic Method of Showing Relationships

61

A GRAPHIC METHOD OF SHOWING RELATIONSHIPS

The problem of determining the relative degrees of the relationship of
a single specimen (or group of specimens) with two other populations,
subspecies, or species, is of frequent occurrence in taxonomy and phylogeny.
A simple example may be stated as follows: Suppose there are two large
islands inhabited by the species K. obesus and K. tenuis. A specimen, which
we shall call x, is found on a small island nearby. Which of the two
species does this new specimen more nearly resemble? This paper suggests
and tests a graphic method of appraising the evidence by the use of co-
ordinates.

Some of the complications involved in this problem lie in evaluating
characters, and in assessing degrees of difference. The new specimen may
be nearer K. obesus in Character A, and nearer K. tenuis in Character B,
but A may be more variable than B. Obesus and tenuis may be much alike
in Character C, yet the new specimen may differ from both in this
character.

Some characters are measurable, others are countable. Units of measure-
ment differ not only in nature, but in value; for example, if a certain
species of snake averages 100 ventrals and 20 subcaudals, it is obvious
that a difference of one ventral is of less importance than a difference of
one subcaudal. The degree of variability of each character within a homo-
geneous population is of importance in determining the weight or sig-
nificance to be assigned to any difference in that character between in-
dividuals or groups.

Assuming that the values of character differences have been placed on
a comparable basis, some method of combining them into a single value,
or a single estimate of relative importance, must be utilized. Thus, assume
that we find the differences between our new island form x and K. obesus
in seven different characters, and from K. tenuis in the same seven char-
acters. How shall these two groups be combined into a single pair of
differences which shall give the final answer to our problem? How may
we prevent dilution by characters in which all three ( obesus , tenuis, and x)
are much the same?

Several methods of combining differences numerically have been pro-
posed. At one time Pearson’s Coefficient of Racial Likeness (Pearson, 1926),
enjoyed a considerable vogue; more recently it has been considered inade-
quate (Seltzer, 1937). The additive feature of chi-square has been utilized
by Fisher (1932), and in modified form by Tippett (1937). This seems
to be the best method of combining character values now available. It
has been discussed at length by Wallis (1942).

The co-ordinate method herein suggested is not a substitute for these
schemes for totalizing differences. Rather, it is a means of presenting the
relationships graphically in such a way as to facilitate a final judgment.

62

Bulletin 18: Zoological Society of San Diego

It gives a single comprehensive picture of the relationship between our
new island specimen a, and K. obesus and K. tenuis, with respect to all
the characters which we can count, measure, or otherwise convert to
numerical values.

The system proposed does not purport to make any automatic numeri-
cal determination as to whether a certain difference shall be considered
specific, subspecific, or below the level of nomenclatorial recognition, as
does the criterion of Ginsburg ( 1938). That is, it does not draw con-
clusions as to whether the new specimen a shall be considered conspecific
with obesus or with tenuis, or shall be described as a new species or sub-
species. It merely serves to show the character relationhips graphically and
comprehensively; the interpretation rests with the judgment of the
taxonomist.

In the discussion which follows, I shall continue to refer to the two
already-known species as obesus and tenuis, and to the specimen which it is
desired to diagnose as a. By the use of these abbreviated terms the discus-
sion will be clarified. But it is to be understood that obesus and tenuis
represent any two populations, whether species, subspecies, or mere
localized aggregations of animals, and whether island or mainland; and A
represents any specimen, or group of specimens, whose relative similarity
or likeness to obesus and tenuis is to be appraised.

I shall first discuss the problem when only a single specimen of A is
available. The co-ordinate method of graphically illustrating differences
involves plotting each character of a as a point in a rectangular chart.
Differences from obesus are plotted as abscissas; differences from tenuis as
ordinates. The relative positions of the points, each representing a separate
character, present a diagrammatic summary of the comparative relation-
ships. If a third possible relative, albus, is involved, other charts comparing
a with dibits and obesus, or with albus and tenuis may be prepared.

The co-ordinate method is not restricted to a single scale of measure-
ment for evaluating differences; any one of at least four may be used,
although one, the plotting of relative probabilities, will be found prefer-
able in problems wherein A comprises a single specimen, or only two or
three. The important criterion is that the evaluating system shall make
the co-ordinates of the points independent of the units by which each
character is measured and thus place them on a comparable basis.

The first of the four scales suggested is to measure the departure of
each character in a from the mean of that character in each of the major
populations, obesus and tenuis, in terms of the standard deviation of the
character in each population. That is, locate the point for Character A
by the co-ordinates

MAo - XA MAt - XA

tQ and A =

Ta.o oAl

Klauber: Graphic Method of Showing Relationships

63

where Ma0 and Oao are the mean and standard deviation of Character A
in obesus , and Mai and 0At are the same parameters of tenuis. Xa is the
value of this character in a. A similar method is used in deriving the co-
ordinates of the points for Characters B, C, etc. If the samples of obesus
and tenuis are small, it is well to make the usual correction in deriving
the optimum value of the standard deviation of the population, from the
equation 0 = 5 [N / (N — 1 ) ] 1 2 where 5 is the standard deviation of the
sample.

The use of this scale of t in plotting character points is subject to the
criticism that values of t are independent of the number of specimens in
the major populations, except for the minor correction of O.1 A second
suggested scale is to use probabilities as co-ordinates, that is, the values
of Q0 and Ot as found in a /-table, the arguments being t0 and 1 t as de-
rived above, at N0 — 1 and N t— 1 degrees of freedom (0=1— P).
P should be taken as the ratio of both tails of the curve to the entire
area. For most small-sample problems this scale of co-ordinates is to be
preferred to the others. The co-ordinates have definite upper limits at
unity, and comparisons of probabilities are readily apparent. The char-
acters by which a differs from obesus or tenuis, either significantly
(O )> .95), or highly significantly (O i> .99) are clearly indicated.

A third scheme is to go one step farther and utilize Fisher’s method of
converting the probability values P into chi-square values at two degrees
of freedom, these chi-square results being used as the co-ordinates. The
additive quality of chi-square suggests the utility of this scheme; and
it may be useful because chi-square tends to spread the higher values of
Q (lower values of P). But this very effect is a disadvantage in some
problems, since some of the co-ordinates will be so large that the others
must be unduly compressed.

In cases wherein large series of specimens are available, not only of
obesus and tenuis, but of a as well, even small differences in characters
will be found to be statistically significant, that is, both Oo and Qt will
closely approach unity in almost every character. The result, if prob-
abilities be the co-ordinates, is a chart in which all points cluster about
the upper right hand corner and conclusions are drawn with difficulty.

There is a rather important difference between problems of small samples
and those of large. In the former, means, and therefore mean differences,
are known only approximately — that is, the samples may or may not
accurately define or represent the populations, and it is the populations
and not the samples with which we are really concerned." The confidence

1 A further modification may be made by finding the value of P which corresponds to
each value of t in a t- table, and retranslating this to the value of t on the normal
curve, thus correcting for small samples.

2 The samples comprise the specimens available for study; the populations are the
creatures remaining in the wild, from which the samples were collected.

64

Bulletin 18: Zoological Society of San Diego

to be assigned to a mean may be deduced from its standard error. Inevitably
the sampling variation of the difference between two means is of first
importance in judging the reality of the difference. This reality or sig-
nificance is dependent on the extent of the difference, and also on the in-
ternal dispersions of the character under consideration, in each of the two
populations between which the relationship is being judged. Thus Q be-
comes a measure of both extent and validity, and is a necessary compro-
mise when only small samples are available. But where the samples are
relatively large, say about 100 specimens or more, and internal population
dispersions are of the order of magnitude usually found in these biological
problems, the means are known within a narrow confidence range, and
we are then more interested in the direct measure of differences than in
a statement of their significances. In this case a method of reducing diverse
characters to some comparable standard, independent of the unit of meas-
urement, is still necessary. Such a criterion is furnished by the coefficient
of divergence, that is, the difference between two means, divided by half
their sum. This comprises the fourth scale of co-ordinates which may
be suggested. But it should be remembered that the coefficient of diverg-
ence, in itself, contains no indication of the significance of a difference,
that is, whether it may not be the result of the chance composition of the
available samples; it serves merely as a yard-stick to place a variety of
characters on a comparable basis of measurement. It may be thought that
there is something fundamentally inaccurate in using for co-ordinates a
statistic which is based entirely on means, thus ignoring the intrapopu-
lation dispersions of the character under consideration. But investigation
indicates that there is only a slight correlation between intrapopulation
dispersion and interpopulation differences.

The use of the coefficients of divergence as co-ordinates occasionally tends
to crowd the points of some characters because a few others take high
values — a situation which also occurs with / or chi-square. In such situa-
tions the use of log-log paper may be recommended.

In calculating probabilities (if x be a single specimen) one may determine
the position of x directly from the place where it falls in the arrays of
each of the major populations, using twice the percentage values of the
tails beyond that point as the co-ordinates. In case there is more than one
specimen in the class in which x falls, the tail should be considered as be-
ginning at the midpoint of the class. For example, assume this distribution
in the subcaudals of obesus:

Number of

subcaudals 20 21 22 23 24 2 5 26 Total

Number of

specimens 1 13 27 47 41 8 3 140

Klauber: Graphic Method of Showing Relationships

65

Suppose that x has 25 subcaudals. Then, splitting the 2 5 class in obesus,
we take the tail of the distribution to be

4 + 3= 7, and find P0 = 2 X 7/140 = .10, orQo = .90

While this modification of the probability method is simple and avoids
the necessity of computing the means and standard deviations of each
character of obesus and tenuis, the result depends rather too much on the
accidental distributions of the samples of obesus and tenuis available to us.
Such a method, however, may be justified if the distributions of several of
the characters are badly skewed.

The use of co-ordinates based on probabilities permits the inclusion of
any character the x- value of which may be expressed as a probability (as,
for example, by a four-fold contingency table) ; it is not restricted to the
difference between two means.

If we have more than one specimen of x available, two courses are open.
We can plot a O-value separately for each character of each specimen,
thus finding as many points for each character as there are specimens. In
plotting, suitable colors or symbols may be used to identify either speci-
mens, characters, or both. Or, we may calculate the means and standard
deviations of each character of the specimens comprising x; and then, by
one of the formulas for the significance of the difference between two
means, find the values of t, Q, or chi-square corresponding to the dif-
ference between the mean of x and the mean of obesus, and between the
means of * and tenuis. Thus, as was the case when v was only a single
specimen, we shall have a pair of co-ordinates defining a single point for
each character. This latter is the only practical alternative if x contains
more than three or four specimens. Of course, the fourth scale suggested
uses means in any case, but is not to be considered adequate if v is a
small sample.

As a first illustrative problem I shall take a single rattlesnake specimen
from Shoup, Lemhi County, Idaho. This is on the upper Salmon River.
It is known that rattlesnakes are found along this river down to its con-
fluence with the Snake River, and that the lower reaches of the Salmon and
beyond are inhabited by Crotalus viridis oreganus. In the opposite direction
Crotalus viridis viridis has been collected across the Continental Divide
(near the Bitter Root Range) in the Valley of the Beaverhead River,
Montana. The Divide can be crossed at a number of points at about 8000
feet elevation; and, as rattlers have been observed in this vicinity at least
up to 7000 feet, there is little doubt that they range across the Divide.
The problem is whether the specimen from Shoup more nearly resembles
C. v. oreganus down the river to the west, or C. v. viridis across the
mountains to the east.

We shall take, as representative populations of these two subspecies,
the two nearest available series; these are a collection of oreganus from

66

Bulletin 18: Zoological Society of San Diego

Pateros, in eastern Washington, and a series of viridis from Montana. The
essential data on these series are set forth in Table 1, ten characters being
given. Only males are included in the statistics of those characters wherein
sexual dimorphism is present, because our specimen from Shoup is a male.
Table 2 gives the derived differences, and the values of Q calculated there-
from. Chi-square results are also included for comparative purposes.

TABLE 1
Character Data

The Relationship of a Specimen from Shoup, Idaho, to
C. v. viridis, and C. v. oreganus

Montana viridis Pateros oreganus Specimen

— - — — - from

Character

N

M

(7

N

M

(7

Shoup

Scale rows

174

25.443

0.913

615

25.511

0.873

27

Ventrals

70

174.943

4.363

326

173.313

3.125

175

Subcaudals

72

24.93 1

1.417

324

22.985

1.446

26

Supralabials

340

14.406

1.185

1229

15.218

0.832

14/2

Infralabials

345

15.149

1.093

1230

16.178

0.892

14/2

Scales on crown

151

18.728

4.516

267

24.985

4.839

18

Intersupraoculars

151

2.775

0.799

278

5.813

0.986

2

Body blotches

171

45.819

3.557

616

33.195

2.293

48

Tail rings

70

9.786

1.512

326

5.497

0.774

10

Width of button, mm.

25

5.248

0.2 56

252

5.611

0.365

4.8

* Mean of the two sides.

TABLE 2
Difference Data

The Relationship of a Specimen from Shoup, Idaho, to
C. v. viridis and C. v. oreganus

Differences from viridis Differences from oreganus

Character D t Q Chi-square D t Q Chi-square

Scale rows 1.557 1.705 .912 4.860 1.489 1.706 .912 4.861

Ventrals 0.057 0.013 .010 0.020 1.687 0.540 .41 1 1.059

Subcaudals 1.069 0.754 .546 1.579 3.01 5 2.085 .963 6.594

Supralabials 0.094 0.079 .063 0.130 0.718 0.863 .612 1.893

Infralabials 0.649 0.594 .447 1.185 1.678 1.881 .940 5.627

Scales on crown 0.728 0.161 .128 0.274 6.985 1.443 .851 3.808

Intersupraoculars 0.775 0.970 .668 2.205 3.813 3.867 .999+ 18.226

Body blotches 2.181 0.613 .460 1.232 14.805 6.457 .999+ 45.916

Tail rings 0.214 0.142 .112 0.238 4.503 5.818 .999+ 37.858

Width of button, mm. 0.448 1.750 .907 4.750 0.811 2.222 .974 7.299

Klauber: Graphic Method of Showing Relationships

67

1.0

.8

to

D

Z

<

O

Id

cr

o

to

O

tr

UJ

h-

<

CL

2

O

o:

z

UJ

a:

UJ

u

— .4
O

u

O

>■

f-

CD

<

CD

O

T2

•

TAIL RIN

SS

•

BOD

•

INFR

—

Y BLOTCHE

•

SUBC

^LABIALS

• —

S IN

AUDALS

FERSUPRAO

RATT

CULARS

LE BUTTOr'

SCALE ROW

c

S

•

•

SCALES

ON CROWf'

SUP

•

RALABIALS

VENTRAL

9

S

FIG. 1

RELATIONSHIP OF A SPECIMEN FROM SHOUP, IDAHO

TO

C.v.viridis AND C.v. oreganus
(basis of comparison, q)

.2 .3 .4 .5 .6 .7 .8

PROBABILITY OF DIFFERENCE FROM MONTANA VIRIDIS

Using the paired values of O as co-ordinates, the graphic results are
shown in Fig. 1. There can be no question that the specimen from Shoup
more closely follows viridh than oreganus. With the single exception of the
scale rows, all characters fall on the viridh side of a diagonal from 0-0
to 1-1. Our specimen v differs from oreganus significantly in no less than
five characters; in three of these the differences are highly significant. It
is of interest to note that this conclusion is substantiated by other non-
measurable differences, such as the width of the postocular light stripe,
the nature of the blotch borders, etc. Other specimens of rattlers from
the Lemhi area are available; they have been omitted from considera-
tion in this study in order to reduce our problem to an example in which
x comprises a single specimen. Parenthetically, it may be stated that the
additional specimens verify the conclusion that the Lemhi snakes are
nearer viridh than oreganus, although with some suggestions of intergrada-
tion with both. There seems little doubt that a full series collected down
the Salmon would complete the link.

68

Bulletin 18: Zoological Society of San Diego

As a second illustrative problem I shall consider the relationships of
Crotalus viridis abyssus of the Grand Canyon in Arizona, with its neigh-
bors to the north and south. This will serve as an example wherein a few
specimens of each of three forms are available, as is typical of many
taxonomic problems.

To the south of the Grand Canyon, between El Tovar and Williams, lies
the Coconino Plateau. This is inhabited by a form of Crotalus viridis
nuntius, a somewhat larger and differently colored snake from the typical
nuntius of the vicinity of Winslow. Intergradation of nuntius and abyssus
seems evident along the south rim of the Canyon. Most of the available
specimens of this Coconino Plateau form of nuntius were collected some
miles south of the rim, at Anita and Valle.

North of the Canyon occurs Crotalus viridis lutosus, the Great Basin
Rattler. While considerable areas of the north rim may be too high for
rattlers, there is a ready means of access between the territory of lutosus
and the Canyon via the gorge cut by Kanab Creek. Fortunately, we have
available a small series of specimens from the upper Kanab in western Kane

TABLE 3
Character Data

The Relationship of C. v. abyssus with C. v. nuntius and C. v. lutosus

Coconino nuntius Grand Canyon abyssus Kanab lutosus

Character

N

M

jj.

cr

N

M

o'

N

M

a*

Scale rows

23

24.217

0.930

16

25.375

0.780

18

25.667

1.154

Vent rals males

13

167.33 8

1.678

11

177.818

2.249

11

180.3 5 5

2.307

females

10

172.900

1.921

4

180.750

1.480

7

184.571

4.370

Subcaudals males

13

24.308

1.726

11

25.545

1.827

11

24.091

1.564

females

10

18.700

1.418

5

20.200

0.748

7

20.143

1.884

Supralabials

46

14.935

0.919

32

15.781

0.892

35

15.486

0.873

Infralabials

46

15.478

1.246

32

16.281

1.123

35

15.629

0.928

Scales on crown

23

3.739

0.529

15

6.067

0.998

18

5.778

1.272

Intersupraoculars

23

23.435

4.726

15

36.067

5.423

18

31.278

9.056

Body blotches

23

41.000

2.904

13

42.385

2.923

18

39.722

3.228

Tail rings males

13

8.923

0.917

11

8.635

1.494

12

7.167

0.898

females

10

7.000

0.894

4

6.750

0.435

6

5.833

1.067

Width of button, mm...

4

4.600

0.123

5

5.280

0.232

5

5.680

0.117

* Standard deviation of sample, not estimated standard deviation of the population.
Correction was made elsewhere in the computations.

Klauber: Graphic Method of Showing Relationships

69

TABLE 4

Difference Data

*

The Relationship of C. v. abyssus with C. v. nun tius and C. v. lutosus

Character

Differences from nuntius

Differences from lutosus

D

t

Q

CD%

D

t

Q

CD%

Scale rows

1.158

3.97

.999

+

4.7

0.292

0.82

.58

1.1

Ventrals males

10.280

9.65

.999

+

6.0

2.537

2.49

.98

1.4

females

. 7.850

6.80

.999

+

4.4

3.821

1.54

.84

2.1

Subcaudals males

1.237

1.63

.88

5.0

1.454

1.92

.93

5.9

females

1.500

2.06

.94

7.7

0.057

0.06

.05

0.3

Supra labials

0.846

3.99

.999

+

5.5

0.295

1.35

.82

1.9

Infralabials

0.803

2.88

.995

5.1

0.652

2.56

.99

4.1

Scales on crown

2.328

9.05

.999

+

47.4

0.289

0.70

.51

4.9

Intrasupraoculars

12.632

6.77

.999

+

42.4

4.789

1.63

.89

14.2

Body blotches

1.385

0.92

.63

3.3

2.663

2.28

.97

6.5

Tail rings males

0.288

0.55

.42

3.3

1.468

2.75

.99

18.5

females

0.250

0.49

.37

3.6

0.917

1.46

.82

14.6

Width of button, mm..

0.680

4.69

.998

13.8

0.400

3.08

.98

7.3

County, Utah. Our problem then, is to determine the degree of relation-
ship between abyssus and Coconino nuntius, on the one hand, and abyssus
and the Kanab lutosus on the other. I have omitted from the abyssus
sample, specimens from along the south rim that might be intergrades;
thus, the sample comprises only true abyssus from within the Canyon.
To make the problem more typical, not all available nuntius from the
Coconino Plateau have been included, since such large series are rarely at
hand in taxonomic studies of this kind.

Tables 3 and 4 present the statistics of the problem, while Fig. 2 gives
the graphic result. The significances of the differences between means have
been computed by the null method. Because of dimorphism the sexes have
been segregated in ventrals, subcaudals, and tail rings. It is not to be
deemed inconsistent that the sexes do not agree in the significance of re-
sults; it should be remembered that Q depends on N as well as on means
and standard deviations.

The results in pictorial form show conclusively that the Canyon speci-
mens are more closely related to the population to the north than that
to the south. In several characters (male ventrals, infralabials, and rattle
buttons) abyssus differs significantly from both adjacent populations.
(Note how the points representing these characters crowd into the upper
right-hand corner.) The relationship with nuntius (as measured by prob-

70

Bulletin 18: Zoological Society of San Diego

•

SUBC

AUDALS 9

SCALES

—

ON CROWr

•

Si sc

1 r*~* — *

ALE ROWS | |/ i t

SUPRALABIALS' /
VENTRALS 9
INTERSUPRAOCULARS

1 _ L

••

y '•

RA

VENTRAL.
TTLE BUT '
INFRALAf
SUBCA

o'

roN' x
3IALS "
UDALS

•

4

BODY BL

DTCHES

/

TAIL

UNGS J _

©

•

TAIL RIN

;s 9

FIG. 2

RELATIONSHIP OF C.v. abyssus
TO

C.v. nuntius AND C.v. lutosus
(basis of comparison, q.)

.8

D

Z

O

cr

^ .6

Id

O

Z

UJ

o:

w .5

u_

Ll.

Q

U-

O

CD

<

CD .3

O

cr

CL

.3 .4 .5 .6 .7

PROBABILITY OF DIFFERENCE FROM LUTOSIJS

.8

.9

1.0

ability) is closer than with lutosus in four characters (three of which
are independent3) : Male subcaudals, body blotches, and the tail rings of
both sexes. It will be observed that the link with nuntius is closer in all
pattern characteristics; to this extent pattern gives a different answer
than scale counts. I should consider the latter the more important.

As a final example I take one in which large series of all three forms
to be compared are available, so that even relatively small differences are
likely to be statistically significant, yet may be unimportant in extent. In
this example the populations compared are territorially more scattered
than those previously discussed, as is generally the case in such taxonomic
problems.

3 It is obvious that two independent characters should be given greater weight, in
reaching a decision, than two which are correlated. This matter of character correlation
will be the subject of a paper now in course of preparation.

Klauber: Graphic Method of Showing Relationships

71

The diamondback rattlesnakes, C rot aim cinereous , C. lucasensis and
C. ruber are obviously related. Superficially, lucasensis appears to be inter-
mediate, as might be expected from the evidence of other elements of the
Cape fauna, which were derived from mainland Mexico rather than from
California. I shall endeavor to ascertain the comparative relationship of
lucasensis with ruber and with cinereous. For the lucasensis material I
shall use all the available non-island specimens from southern Lower Cali-
fornia. The ruber material is from southern California, and northern and
central Lower California. The cinereous series is from southern and western
Arizona, southeastern California, and northeastern Lower California.

Since all series are large, the significances of differences have been cal-
culated by dividing the differences by their standard errors and taking the
values of Q directly from a table of the normal curve. Several additional
characters have been introduced, including head length, and tail propor-
tionality. I have discussed elsewhere the method of determining the sig-
nificances of differences in these ontogenetically variable characters at
standard body lengths.

The statistics of this problem are set forth in Tables 5 and 6; the
graphic presentation in Fig. 3.

We should first verify the validity of placing lucasensis between cinereous
and ruber. We find, from a study of Table 5, that lucasensis falls between
cinereous and ruber in 7 independent characters; it falls outside of
cinereous in 2%, and outside of ruber in 5Yg. Most of the characters
in which lucasensis is more highly differentiated from cinereous than is
ruber are head scales — labials, scales on the crown, scales around the rostral,
and loreals. I deem these less important than scale rows, ventrals, and body
proportions, and therefore conclude that lucasensis is intermediate. That
both were derived from cinereous seems clear; but whether separately, or
lucasensis first, and then ruber as an offshoot of the latter, is somewhat
uncertain. The fact that island forms are rather close to one or the other,
suggests a separate development, followed by a gradual invasion of inter-
mediate territories. Whether there is intergradation at present is not
definitely known, but seems probable, from the few scattered specimens
available between Concepcion Bay and La Paz. There is no indication of
intergradation of either lucasensis or ruber with cinereous.

Fig. 3 indicates without question that lucasensis is more closely allied to
ruber than to cinereous, thus solving our problem. In only 8 out of 21
characters (not all independent) are the points closer to cinereous than
ruber and all of these fall near the diagonal which represents the line of
equal separation. On the other hand, 7 of the 13 characters which favor
ruber as the closer relative, are much nearer to the ruber base than that
of cinereous.

This cinereous-lucasensis-rubcr example illustrates the inadequacy of
using co-ordinates based on Q where large numbers of specimens are
available, for many O-values closely approach unity. Values of t are better

Character Data

The Relationship of C. lucasensis with C. cinereous and C. ruber

72

Bulletin 18: Zoological Society of San Diego

b

04

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Total counts per specimen.

At standard body length of 13 50 mm.

Coefficient of variation at standard body length taken at 4 per cent.

TABLE 6
Difference Data

The Relationship of C. lucasensis with C. cinereous and C. ruber

Klauber: Graphic Method of Showing Relationships

73

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74

Bulletin 18: Zoological Society of San Diego

criteria, but I think the extent of the differences, when reduced to a
comparable basis, as afforded by the use of the coefficient of divergence,1
gives the better illustration. However, if one desires to incorporate the
effect of internal dispersion, then t should be employed.

In many genera of snakes there will be fewer countable characters than
in the rattlers. Scale rows and labials, for example, are often invariant. On
the other hand, use can be made of characters which I have not employed
in these rattler examples, such as internasals, oculars, temporals, genials,
the ventral count at locations where scale rows are dropped, or any other
counts in which the forms being compared are not constant within a
population. If there is enough variation in any character to produce an
K x 2 table, a value of Q may be deduced. There are also the variables of
head and tail ratios, although, in using them, care must be taken to avoid
the fictitious effects of ontogenetic variation.

4 The percentages by which the character means of lucasensis are above or below th
corresponding means of cinereous and ruber may also be used as co-ordinates.

100

7

/

<L

/

/

/

/•

LOR

:al

S

/

/

/

/

/

/

/

1

/

f AT T LE W
rHIRD RAT
/

BODY BLO

IDTH,

TLE

TCHES

SECOND RATTLE®

INTERSL

JPRA

OC

ULAR

/

/

SCALE RO

® TAIL

ws"*

LENGT

©RAT

H d*

TLE WIDT

®S

H, BUTTON

SALES

ON

CR

OWN

su

3CAUC

ALS

TAIL RIN

>

/

gs

VENTRA

4 VENT

-s d

RALS

•

s

> •

CAL

AIL

_ES AROUN

LENGTH

D ROSTRA

?

Dl

L IN

VIDE

FRA

D F
LA

IRST ~

BIALS

/

/

/

/

/

/

/

0

SI

BCAU

daC'i

/

d

/

/

/

FIG. 3

RELATIONSHIP OF

TO

C. cinereous AND
(basis of comparison,

C. lucasensis

/

/

/

/

/

/

T

AIL

Rifs

GS 9 ®

INFRAL

ABIALS A

C. ruber

COEFFICIENT

s

UPRALABI

OF DIVERGENCE IN PER CENT)

ALS#

80

60

40

20

10

z

Ld

u

CC

Ld

CL

(T

bJ

CO

D

CC

o

tr

I.J

z

Ld

o

cr

Ld

L_

o

z

Id

U

Li_

!l_

LU

O

o

.2

.4 .6 .8 I. 2 4 6 8 10 20 40 60 80 100 200

COEFFICIENT OF DIVERGENCE FROM CINEREOUS PER CENT

HEAD LENGTH

Klauber: Graphic Method of Showing Relationships

75

It will be observed that in only one of these examples have I taken
account of the direction of differences. Even when our x lies territorially
between K. obesus and K. tenuis, its characters may not always fall be-
tween then means. Quantitatively this is provided for in the values of Q
or the coefficient of divergence. If it be desired to distinguish between
the characters in x which fall between obesus and tenuis, and those which
fall outside of the one or the other, this can be done by the use of colors,
symbols, 01 a diagram designed to show the directions of differences, as
well as their amounts.

Such a diagram can be prepared by employing all of the four quadrants of
analytic geometry, instead of only the first quadrant as in the previous prob-
lems, wherein the directions of differences were ignored. As before, we plot
differences from obesus as abscissas and differences from tenuis as ordinates,
but in addition we follow the standard practices of analytic geometry, plot-
tmg positive values to the right and above the origin, and negative values
to the left and below. Employing the usual designation of the quadrants, the
first being that to the upper right of the origin, the second upper left, the
third lower left, and the fourth lower right, we note that the first quadrant
would contain points representing characters in which a is higher than
either obesus or tenuis, the second quadrant, points in which a is higher
than tenuis but lower than obesus, the third, points in which a is below
either, and the fourth, characters in which a is below tenuis but higher
than obesus. It is evident that when a lies between tenuis and obesus in
any character, the co-ordinate representing that character will fall in the
second or fourth quadrant, depending on whether tenuis or obesus is the
higher. If a point falls in the first or third quadrant, .a lies outside the range
between obesus and tenuis.

While this type of diagram has the advantage of indicating directional
trends, its evaluation, in judging the cumulative effect of all characters
to determine the relative closeness of a to obesus and to tenuis, is not so
easy as with the simpler single-quadrant method. In this more elaborate
directional analysis we must draw two limiting diagonals. Characters
falling within the range of 45 to 135 and 225° to 315° indicate a
closer relationship of a to obesus, while points lying between 13 5° to 225°,
and from 315 through zero to 45" show a to be closer to tenuis. This
may be stated more simply, but less definitely, by saying that, if the
character points of a tend to adhere closely to the vertical axis, the re-
lationship of a is nearer to obesus; whereas, if the majority are nearer the
horizontal axis, a is more like tenuis. The principal advantage of this type
of analysis is that it will indicate whether, in the majority of characters,
a is intermediate between obesus and tenuis or falls outside the means of
both. Such a type of directional presentation is thought particularly suit-
able for a study of hybrids. It also suggests the matter of diagrams to
illustrate dines, a complicated problem beyond the scope of the present dis-
cussion.

76

Bulletin 18: Zoological Society of San Diego

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