Download Trends in ternary petrologic yariation diagrams-fact or fantasy?

American Mineralogist, Volume 64, pages I I l5-I121, 1979
Trends in ternary petrologic yariation diagrams-fact or fantasy?
Josu C. Burlpn
Department of Geology, University of Houston
Houston, Texas 77004
Abstract
For more than 75 yearspetrologists(among others)have made use of triangular variation
diagramsto illustrate comfositional variability among membersof a predefinedigneousrock
suiG and to draw geneticinformation from the spatial positioning of the points on the diagram. when ternary percentagesare formed, the summaryunivariate and bivariate statistics
of the ternary percentageswill usually differ from their counterpartsin the parent data array.
Such changesare due entirely to the eflectsofpercentageformation. For example,reversals
in the rank order of variancesoften occur, with the result that the variable with the smallest
variancein the initial array may have the largestvariancein the ternary array. Such changes
may be understoodby examining the row sum variable (the sum of the valuesof the initial
variablesselectedto createthe ternary array) and the correlationbetweenthe row sum variable and eachof the initial variables(the part-whole correlation).The correlationcoefficients
betweenthe three ternary variablesare likely to differ considerablyfrom their initial values.
The net effect is that any observedternary trends must be consideredto result from a complicated set of interactionsincluding: (l) the sampling strategyused to assemblethe initial
data set;(2) the summarystatisticsin the initial data setwhich themselvesare subjectto modi-ficationby percentageformation; (3) the part-whole correlationsin the initial data seq (4)
restrictionson the variancesofa closedternary anay; (5) modificationsofall univariate and
bivariate statisticsaccompanyingternary percentageformation; (6) petrogeneticcontrols'
Unlessa rational meansof separatingtheseeffectsfrom eachother can be found it would appear to be unprofitableto attempt to basegeneticstatementson ternary variations.
which are readily observed when diagrams are
used to representnumerical figures.Moreover, visual memory is sufficientlydevelopedin most persons to enable them to carry in mind simple geometrical forms, where it does not permit them to
recollect manifold assemblagesof oft-repeated
numbers. Mental impressionsof simple diagrams,
therefore,are more definite and lasting and enable
the student to store up a much greateramount of
quantitative data than he could otherwise acquire."
"It is frequently supposedthat curves drawn in
ternary diagrams reveal fundamental relations
between the variables.. ." (Chayes, 1961, p.
160)
Introduction
For more than 100yearspetrologistshave debated
how to most effectivelyportray the chemistry of igneousrocks for classificationpurposesand for revealing underlying petrogeneticprocesses.Particularly
revealing are lddings' comments (1903, p. 9) concerning the necessityfor reliance on graphical de- Most of Iddings' comments were directed towards
vices:
classificationproblems and strategies,but it seems
numerobvious that his basic philosophy has been adopted
of
the
amounts
in
variations
"The intricate
comby many whose primary coneern lies in elucidating
of
chemical
or
components
ous mineral
petrogeneticrelations.
their
to
comprehend
most
attempts
ponents,bafle
It is hard to disagreewith Iddings' desire to unby
or
contemplation
simple
by
interrelationships
a complicated situation, but it is inexcomplicate
may
be
which
they
in
numbers
study of the
the user to ascertainwhether the maupon
cumbent
overlooked
are
relations
and
facts
pressed.Many
9/09to-l t I 5$02.00
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ll15
BUTLER: TERNARY PETROLOGIC VARIATION DIAGRAMS
nipulations required to prepare the graphical device
have in any way modified the relationsamongthe selected variables.For example,Chayes(1971,p. 7l)
noted that many of the data manipulation schemes
commonly resorted to by petrologistsresult in distortion of such magnitude that many of the trends
are more a measureof numerical consequencethan
petrogeneticreality.
Of particular interest (to the author and most petrologistsit would seem)are the ternary variation diagrams such as CaO-NarO-KrO (CNK) and total
iron-MgO-(Na,O + K,O) (FMA). Iddings (1903)
credits Becke(1897)with being the first to represent
the compositionsof igneous rocks by recalculating
Ca, Na, and K to lOOTo
and plotting this information
on an equilateraltriangle in which the zeropoint was
at the center of the diagram. Reid (1902)appearsto
have been the first petrologistto use the modern triangular diagram in which eachside of the triangle is
divided into 100parts.
Since 1902 many petrologic presentationshave
made use'of either the CNK or the FMA (or both)
typesof variation diagrams,and it appearsthat many
of the users have inferred that the principal factor
governingthe configurationofthe observedtrends is
relatedto someprocessof crystalfractionation.
Wright (1974,p. 236-237)argued against the use
of FMA diagrams for deducing petrogeneticinformation for the following reasons:
Barker (1978) agreed in principle with Wright's
objectionsto the continueduseof the FMA diagrams
but arguedthat "the significanceof trends on alkaliiron-magnesium diagramscan be better understood
by examining the relations betweencompositionsof
rocks and their constituentminerals when projected
on the AFM plane."
An expectedchangein all of the univariate statistics of the selectedvariablesand in any of the pairwise measuresof associationis the most insidious
consequenceof forming ternary percentages.Not
only do the observedtrendslead to non-quantitative
expressionsof mineralogical control (Wright, 1974),
but it is conceivablethat an attempt to extract information of a qualitative nature (Barker, 1978)will fail
to isolatemineralogical/fractionationeffectsfrom the
numerical consequencesof ternary percentageformation.
The purposeof this note is to review the effectsof
percentageformation on the properties of ternary
variablesand to assessthe probability of determining
tlae underlying causesof the trends commonly observedin FMA and CNK variation diagrams.
Preparation of the ternary diagram and the partwholecorrelation
The general topic of the effectsof percentageformation on the properties of the included variables
has been reviewed admirably by Chayes (1971,
p.
36a! and will be referredto only with respectto
(l) For most igneousrock seriesthe amountsof the
the
ternary case.When we decide which variables
oxides combined to form the FMA diagrams
(single
or in some combination) will be used to decompriseconsiderablylessthan 50Vo(by weight)
fine
the
ternary subset,the original values (which
of the samples.
constitute
the initial data set) are recalculated to
(2) Trends which appear to be significant can be
l00%o
(in
order
to keep the notation as simple as poscreatedby mixing togethersampleswhich obvisible,
unprimed
letterswill be usedto refer to the iniously have no geneticrelationships.
tial
variables
primed letters to the ternary perand
(3) Normalization to l00Voof sums that are not the
centages):
samedistorts the plots with the result that trend
orientationsdo not quantitatively expressspecific
Ai: 100(AtlTJ
(l)
mineral controls.
T, is the sum of the valuesof A, B, and C in the ith
Similar statementscan be madeconcerningthe CNK
row of the data matrix:
or any other conceivableternary variation diagram.
Ti:Ai+Bi+Ci
(la)
If one wishesto define total variability as the sum of
the variances of the measured oxide components It is useful to focus attention on the summary statis(Chayes, 1964), the FMA and CNK components tics of the three initial variablesand the row sum (T)
generally account for considerablyless than 25Voof in order to understand and appreciatethe effect of
the total variability of the rock suite being investi- percentageformation in controlling ternary trends.
gated (Butler, 1975),and one gets the feeling that a The importance of the row sum statistic cannot be
considerablequantity of potentially useful informa- overemphasized,although it is unlikely that the intion is being ignored.
vestigatorwould bother to computeits summary sta-
BUTLER: TERNARY PETROLOGIC VARIATION DIAGRAMS
tisticsor to treat it as anyhing other than a necessary
stepin the transformationprocedure.
If the initial variables (A, B, and C) are uncorrelated (in which case the Pearsonproduct moment
correlation coefficient-r---equals zero),there will be
a positive correlation betweenany one of the initial
variables and T (this type of correlation will be referred to as the part-whole correlation as, for example, A is part of T---+quation la):
ro-. : so/st
(So * sgras + s.ro.)/st
MEAN
srD
(2a)
From equation (2a) it is clear that the part-whole
correlation may be positive or negativeif the initial
variablesare correlated,and that the sign ofthe partwhole correlation is controlled by the signsand magnitudes of the correlationsbetween the initial variables.Any data set in which the sum of the valuesof
the variables is a constant for all of the samplesis
consideredto be a closeddata set. The standarddeviation ofthe row sum variable (s') in a set of closed
data is equal to zero, and the part-whole correlation
(equations 2 and 2a) is undefined. Wright's (1974)
third objection to the use of the FMA variation diagramsstemsfrom the fact that the initial valuesof F,
M. and A do not constitutea closeddata setby themselves.
I have examinedthe propertiesof many ternary arrays from a variety of published setsof chemical information (Butler, 1975),and the set of I 14 chemical
analysesof igneous rocks from the Big Bend National Park area of west Texas(Maxwell et al.' 1967)
has been selectedto illustrate the effectsof ternary
percentageformation on the properties of the individual variables and to analyzethe trends that are
frequently observedin the CNK and FMA types of
variation diagrams.Summary statisticsfor the CNK
and FMA subsetsare given in Tables I and 2 respectively.
Effect of percentageformation on the mean of ternary variables
A seriesof approximationscan be derived which
allow estimating the properties of ternary variables
DEV.
cao
4.42
Na2o'
17:18
24.49
3.78
9'67
0 95
0.77
0.86
0-26
o'21
9'63%
O.O9% 26'26"1
61.767. l-47%
% VARIANCI
Na2O xzol
4'55 31.18
cao'
31.35
coE!. vAR.
0.58
o-37
SKEWNESS
-0.32
\5'97
0-51
191tl-lw
12'64
L'63
2'2r
0'44
0'18
0 277"
-0'2E
-0 20
0'05
KZ9
3'68
CORREI,ATIONCOEI'I'ICIENI MATRIX
K2o RolI StM
*20'
Na2o' Na2o
-0'905 0 782
1.OoO 0.910 -o'925 -o'563 -0'973
1.000 -0.925 -0'564 -0'958 -0'888 0'817
cdo
cao'
cao'
(2)
where s is the standard deviation of the variable
identified by the subscript.Similar equationscan be
written for variables B and C by appropriately
switching the subscripts.If the initial variables are
correlated (r # 0.0) the part-whole correlation betweenA and T is given by:
fnr:
Table l. Summary statisticsfor ll4 west Texas analyses:CaONa"O-K"O
cao
-N- ,2 -o '
1.OOO O 7OI
O 813
O 752 -O 724
Na2O
1-000
0'438
0'629 -0'042
1 .000
0.932 -0 761
"2'
Kzo
l
ooo -0'509
1 ooo
Row sw
AND TERMRY MEANS
@MPARISON OF INITIAI.
I'ACTOR* INltlAL
MEAN
VARIABTES
TTRMRY
MEAN
PRXDICT'D
EO (34)
CaO'CaO
0.908
4.42
31.63
34 97
Na^O'-Na^O
1.012
4.55
37.18
35.97
3I.r8
29.07
l.o7l
? J.68
(1 - rArcAcT + ci)
KrOi-K2O
*-FAcrOR -
COM?ARISONOT INITIAL
I'ACTOR*
VARIABLES
AND TERMRY VARIANCESAND S(EIINESS
INITIAL
VARIANCE
TERNARY
VARIANCE
I*
cao'-cao
0.516
74.29
599.62
-0.52
Na20'-Ne20
K2O',-K2O
0.078
0.90
93.51
0.18
0.306
2.65
254.99
0.40
= R< ' o *
* FACTO
*
"f,
- 2.Ar"oar)
I- = (CT-rATCA)
from the properties of the initial variables (Chayes,
1971; Butler, 1978). Chayes' first-order approximation for the mean of a percentage is given by:
F:
100(F/D
(3)
That is, the mean of a ternary variable is approximated by 100times the ratio of the mean of the initial variable to the mean of the row sum variable (T).
The ratio of the ternary mean to the initial mean
(R.) is givenby:
R.: F/e:
loo(-AlD/T: roo/T
(3a)
Obviously, the ratio R. will be the same for all three
variables. From Tables I and 2, however, it is clear
that the ratio of the means is not constant for all
three variables. A second-order approximation for
the mean of ternary variables (Butler, 1978) is given
by:
F : 100(tD(l - ro,C^C,+ Ci)
(3b)
where C is the coefficient of variation (the ratio of the
ll18
BUTLER: TERNARY PETROLOGIC VARIATI1N DIAGRAMS
Table 2. Summary statistics for I 14 west Texas analyses:
total
iron_MgO_(Na2o + K2O)
Meo'
MgO ALruIII
HEAN
11.00
2.39
51.81
ATKAI,I IRoN'
lRON
8.23
37.L9
6.93
17.54
ROW S1IM
STD. DEV.
12.34
3.t0
21.94
2.35
12,46
3.43
4.35
COEI. VAR.
1.I2
1.23
0.42
0.28
0.34
0.50
o.25
7. VARIANCE
14.247"
r.r5%
57.67%
1.7r
-0. 17
SKEWNESS
1.32
0.667" ra.60%
-0.32
_0.30
1.4r7.
0.11
MCO'
1.000
Mgo
MgO ALKALI, ALKALI IRON
IRON
0.992 _0.894 _0.829 0.566 o.760
1 . O O O- O 8 4 5 - 0 . 7 9 9 0 . 5 0 5 0 . 7 1 8
ArKAtl'
0.860
0.849
-0.913
1.o00 -o.645_0.723
_O 602
r.ooo 0.940
IRON
1.000
ROWSW
FAcoR*
INITIAL
mN
AND TERNARY MEANS
TERNARY
MEAN
PR.EDTCTED
EQ (3a)
Msor-Mso
0.788
2.39
1t.OO
13.61
ALKALI'-ALKALI
1.104
a.23
51.81
46.88
rRoN,-rRON
0.950
6.93
37.19
39.50
* IAcToR= t r- .o.o", * .f)
COMPARISONSOF INITIAL
FACTOR*
uso'-Meo
INITIAL
VARIANCE
AND TERNARYVARIANCESAND SreWNESS
TERNARY
VA.R.IANCE
]-**
1.201
9.6r
152.27
_0.85
Ar,KALr'-ALKALT 0.227
5.49
48t.44
0.42
11.78
r55 25
_0.20
rRoNf-rRON
0.75:
o.9ll
1.00tJ
CO},IPARISONOF INITIAI
0.083
* F A C T o R= < " 1 , * I -
2ro,coc,)
(4)
ROII SW
1.OOO 0.832 _0.886-0.962
rRoN,
sl. : 10"(-AlD1c1+ c|-
The ratio of the ternary variancesof variablesA and
B is given by:
CORRSI"A.TION
COEFI,ICIEIfI MATRIX
ugo'
mation and that such changes are not easily explained. The variance of a ternary variable can be
approximatedfrom the propertiesof the initial variablesby (Chayes,l97l; Butler, 1978):
2.Ar"A.r)
*1=(CT-TATCA)
standard deviation to the mean) of the variable iden_
tified by the subscript and ro, is the part-whole correlation (equations 2 and 2a). From equation 3b it is
clear that if ro, is negative, the ternary mean will exceed that predicted by equation 3. Sirnilarly, if ro, is
positive and greater than the rutio C,/Co, the ternary
mean will be less than that predicted by equation 3.
For example, the mean of total alkalis (Table 2) is
considerably increased (51.81) compared with the
value predicted by equation 3 (46.89) and those of
MgO and total iron are correspondingly decreased.
From Table 2 the mean of CaO in the parent set
@.42) is considerably greater than that of K,O (3.69),
yet the ternary mean of CaO (31.63) is only slighrly
larger than that of K,O (31.18). Although not illustrated in Tables I or 2, a reversal in the rank order of
the means can occur as the result of percentage formatron.
Effect of percentage formation of the variances of
ternary variables
Chayes (1962) noted that the variances of ternary
variables often change markedly following transfor-
s2^,/s?8,:
IG; XC; )(C^,+ C,, - 2ro"CoC,)l
/lG; XC; XC",+ C,, - 2r",C"C,)l
(4a,1
which was written in a form which emphasizesthe
relationshipbetweenthe initial variancesand the ternary variances.The behavior of MgO and total alkalis (Table 2) illustratesa situation of frequent occurrence.The variable with the smallestvariance in
the initial data set (total alkalis) has the largestvariance in the ternary subset.Although there is a large
number of variablesin equation 4a, severalgeneralizations can be made concerning reversalsin variancerank order:
(l) Reversalsin rank order of variance are favored
when the variable with the largestinitial variance
has the largest initial coefficient of variation
(other quantitiesbeing equal).
(2) Reversalsin rank order of variance are favored
when the variable with the largestinitial variance
has a positive part-whole correlation and the
variable with the smallestinitial variance has a
negativepart-whole correlation (other quantities
beingequal).
In all the FMA diagrams examined by the author
(identified in Butler, 1975),the behaviorexhibited by
the initial and ternary variablesis very similar. F, M,
and A accountfor a very small percentageof the total initial variance (usually lessthan l}Vo), and total
alkalis have the smallestvariance in the initial set
and the largestin the ternary subset.
Even when reversalsin rank order of the variances
do not occur, changesin the ratio ofthe variancesalmost always occur when ternary percentagesare
formed (Tables I and 2).It seemsintuitively reasonable to expectthat the trend and scatterappearingon
any form ofternary variation diagram in part reflect
the variances of the variables. Given the fact that
major reversalsof variance can occur simply as the
result of ternary percentageformation, it should be
just as reasonableto expectthat at least part of any
trend may be artificial.
BUTLER: TERNARY PETROLOGIC VARIATION DIAGMMS
ll19
Effect of ternary percentage formation on the skewness of ternary variables
The skewness of a variable is a measure of the
asymmetry of its distribution. Pearson (1926) noted
that a ratio formed from two symmetrically distributed variables (zero skewness)will have an asymmetrical distribution (non-zero skewness). The sign of
the skewness of such a ratio (for example, A/T) will
be given by the sign of L (Merrill, 1928):
Lo, : C, - rorCo
(5)
As the initial variables in a set of chemical analyses
commonly have non-zero values of skewness, L", (5)
does not give the sign of the skewness but does indicate the direction of change of skewness following
ternary percentage formation. For example, MgO
(Table 2) has a skewnessof l.7l in the initial data
set, and the computed L."o. of -0.85 indicates that
percentage formation should and does decrease the
skewness to 1.32. In general, it does not make sense
to worry about a genetic interpretation of skewness
of a percentage, as its sign and magnitude may be
numerical rather than petrologic in origin (Butler,
1978).
Effects of percentage formation on the total variance
of a closed array and on the correlation between a
pair of variables
Chayes (1961, 1962) showed that no closed standard deviation can be as large as the sum ofthe standard deviations of the remainder of the variables.
The situation where one standard deviation equals
the sum of the other two (in the case of ternary percentages) can occur only when that variable is perfectly negatively correlated with the other variables
and the other variables are perfectly positively correlated with each other:
Se.: ss.+ s., if, and onlY if,
rls:
r|c:
-1.0 andr!":
t r2 u
Na20
Fig. l. CaO-Na2O-K2O
data set.
variation diagram for the west Texas
near the CaO apex and extends to the NarO-KrO
limiting binary.
Qualitatively it appears that the linear trend expressed in Figure I evolves into a simple curvilinear
trend as the quantity (s". * s..) departs from so and
rns, rnc, and r". approach 0.0. The FMA variation
diagram for the west Texas data set, given in Figure
2, appears to be typical of curvilinear ternary plots.
The trend is initiated near the apex where the variable with the largest standard deviation is plotted (total alkalis for Fig. 2), arrd the inflection point is nearest the apex where the remaining variable with the
largest mean is plotted (total iron).
One normally considers the correlation coefrcient
(6)
1.9
A plot of the ternary variables which obey these relations will define a straight line in an equilateral triangle. The line heads away from the apex of the triangle at which the variable with the largest standard
deviation is plotted. As the quantity (s", * s.,) begins
to exceed so,, the straight line is replaced by a linear
trend and the three correlations depart from an absolute value of 1.0. The CNK diagram for the west
Texas data set, given in Figure l, is typical of other
CNK plots examined by the author. The trend begins
AtK ATIS
Fig. 2. Total iron-MgO-Na2o
west Texas data set.
T()TAL IR()N
+ K2O variation diagram for the
I 120
BUTLER: TERNARY PETROLOGIC VARIATION DIAGRAMS
to be a measure of the strength of the linear pair-wise
association between a pair of variables. However,
Chayes (197l) showed that the correlation between a
pair ofternary variables can be expressed as a function ofthe variances ofthe three ternary variables:
r",c, : [s1,,- (sr",+ sr.,)]/2sis!
(7)
expected as a function of the variances of the variables which are a function of: (l) the sampling strategy used to construct the data array and (2) the partwhole correlation. Miyashiro (1975), in fact, argued
that a complete gradation exists between the tholeiitic and calc-alkaline suites. Barker (1978) artributed the increased scatter of the typical calc-alkaline
trends in an FMA variation diagram to a more pronounced fractionation of plagioclase. However, as
evidenced by the previous discussion, ternary curvilinear relations are characterized by two correlations
with relatively large negative values and one correlation that is usually positive but may be close to zero.
Thus, one should expect greater scatter with a curvilinear trend than with a linsil trend, regardless of
whether plagioclase is precipitating from or dissolving into a magma; unless, of course, one could relate plagioclase behavior to the part-whole correlation in the initial data set.
where similar expressions can be given for r^,", and
ro.", by appropriately switching subscripts. From
equation 7 it is clear that ternary variables pose a difficult problem in interpretation. The correlation between a pair of ternary variables (for example, B, and
C') is positive if, and only if, the sum of the variances
of B' and C' is less than the variance of ternary variable A'. If all three initial variables are positively correlated, only one pair of the ternary variables will be
positively correlated and, commonly, all three pairs
will be negatively correlated. In the CNK and FMA
diagrams examined there are no reversals in the sign
of the correlation coefficients, but there certainly is a
Extension of the preceeding arguments to the initial
change in their magnitudes (see Tables I and 2). For
data set
example, the correlation coefficient between initial
values of MgO and total iron (Table 2) is 0.760 and
Chayes (1971, p. 4445) has argued that a chemthat between the two ternary variables is 0.566. If the
ical analysis is closed by observation. Errors of omisother two correlations (between MgO and total alsion and/or commission account for the fact that a
kalis, and between total iron and total alkalis) were good analysis will comnonly total between 98.5 and
to remain constant and if the correlation between
101.5.The point is, one must recognize that the uniMgO and total iron were to approach 1.0, the trend
variate and bivariate statistical properties ofthe parwould approach linear behavior, whereas if the value
ent data set contain some contribution from the clowere to approach 0.0 the trend would become curvisure effect.
linear with an even more pronounced curvature
Wright's (1974) objection concerning normal(compared to Fig. 2).
ization to l00%o of sums that are not constant has
In summarizing the difficulties inherent in working
been discussed previously. In the same discussion
with ternary variables, Chayes (1961, p. 16l) stated Wright suggests that petrologic relations can be best
"many of the alleged ternary trends are probably
discerned by using a bivariate plot of MgO vs. all
best regarded as complicated and camouflaged state- other major oxides, rather than by using SiO, (which
ments of variance relations largely controlled by the
leads to the familiar Harker diagram) for the mafic
criteria set up in the sampling plan." Recalling that
and ultramafic suites. All binary variation diagrams
ternary percentage formation can result in major
(including Harker diagrams and Wright's MgO varichanges in variance and in the rank order of the
ation diagrams), while reasonable on petrologic
three variances, Chayes' reference to ,.camouflaged grounds, fail to consider that an unknown portion of
statements of variance relations" must be taken seri- the observed correlation and scatter is the result of
ously. Perhaps, however, it would be more appropripercentage formation. At this time it does not appear
ate to say that the ternary trends are best thought of
possible to satisfactorily separate the effects of peras reflections of the part-whole correlations present centage formation from fractionation, assimilation,
in the parent data set.
or whatever petrologic model the investigator may
Irvine and Baragar (lgil)
argued that tholeiitic
have in mind.
(simple linear) and calc-alkaline (curvilinear) suites
Wright and Hamilton (1978) suggestedusing MgO
could be easily distinguished by their trends on FMA
variation diagrams to define chemical types of igvariation diagrams. As noted above, however, a graneous rocks. Samples from an igneous unit that is
dation between these two end-member trends is to be stratigraphically restricted are plotted on such a vari-
BUTLER: TERNARY PETROLOGIC VARIATION DIAGRAMS
ation diagram and used to define a polygonal field
which may aid in characterizingthe chemicaltype of
unit and aid in identifying unknown samples.The
spread of data points on such a binary diagram reflects the variancesof the plotted variables and the
correlation betweenthem. As both the variance and
correlation are likely to have been modified by percentageformation, the shapeof the defining polygon
may have a componentdue to the closureeffect.
tt2l
unprofitable to attempt to basegeneticstatementson
ternary variationsby themselves.
Acknowledgments
Earlier drafts were reviewed by Max F. Carman and Dan
Barker. Felix Chayes gave a detailed critical review ofthe manuscript which is greatly appreciated. Judy Butler assisted with the
drafting. The Geology Foundation of the University of Houston
supported this work.
References
Summaryand conclusions
The preceding discussion shows that when one
forms percentagesthe summaryunivariate and bivariate statisticschange.This effectmay be so drastic as
to changethe rank order ofthe means,variances,and
correlation coefficientsso that, for example,the variable with the smallestvariancein the initial data set
has the largestvariance in the ternary data set. This
combination of changes in means, variances, and
correlationsmanifestsitself in the nature of the distribution of the sample points in an equilateral triangle.Simple linear and curvilinear trendsare gradational as revealedby the following relations:
l. straight line:
fec :
fen :
-1.0 and r". :
+1.0
2. linear trend:
3. curvilinear:
ro. and rAB are large negative;
r". is large positive
rr.. and rAB are intermediate
to large negative;
r". is intermediate to small
positive
Therefore, the observedternary trend must be considered to result from a complicated set of interactions including: (l) the sampling strategyused to
assemblethe initial data set; (2) the summary statistics in the initial data set. which themselvesare subject to modification by percentageformation effects;
(3) the part-whole correlationsin the initial data set;
(4) restrictionson the variancesofa closedsubset;(5)
modifications of all univariate and bivariate summary statisticsaccompanyingternary percentageformation; (6) petrogeneticcontrols.
Unless a rational meansof separatingtheseeffects
from each other can be found, it would appearto be
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