a) Introduction
Bivariant statistics describe the relationships between
pairs of variables. A variable in this case is either the amount
of a major or minor element in a sample. In most cases the most
useful device for the presentation of the data is a scatter
diagram. Usually, the data can be quantified in terms of
regression and correlation coefficients.
The coefficient of correlation between two variables that
is conventionally used in scientific studies is the product moment
correlation coefficient. There are however a number of objections
to using this coefficient. These are discussed in Appendix C. In
this work the Kendall rank order correlation coefficient is also
used because it is less sensitive to the constraints that apply to
the former.
b) The correlation matrices
The correlation coefficients for both methods of
calculation are presented in Table 3-10. This is a matrix for all
combinations of variables. The matrix is based on the maximum
possible number of comparisons within the data set eg. the
coefficient for Si with Al is based on 107 points and that for Si
with C on 60 points. It should be noted that the values of the
coefficients may be affected by the phenomenon of closure which
arises when data are referred to a constant total such as 100 in
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coefficient calculated from such data may be quite anomalous, with
the effect being most marked between the most abundant variables
(Chayes 1971). Caution must accordingly be exercised in the
interpretation of such matrices.
It is pertinent to observe here that when the two
coefficients of correlation are compared, the sign is the same for
each but in most cases the degree of estimated correlation of the
rank order correlation coefficient is lower than that of the
product moment coefficient. There are exceptions however, eg. the
correlation coefficient calculated between K and Al is 0.48 when
calculated by the product moment method and 0.60 when calculated
by the Kendall rank order method.
Sulphur has not been included in the correlation matrix
because only 27 analyses are available. It however correlates
well with Fe. Correlation analysis provides a product moment
correlation coefficient of 0.81 and a Kendall rank order
correlation coefficient of 0.50.
c) Element associations
Most of the comparisons yield positive correlations except Si,
which has negative correlation coefficients with all other elements
except C. The strongest correlations generally appear to be
associated with the major elements; this may be partly due to the
closure phenomenum. If an arbitrary figure of 0.50 is chosen for
the Kendall rank order correlation coefficient, it is found that
the linear regression for all element pairs exceeding this value
and regression lines for those element pairs which have a
correlation coefficient equal to or greater than 0.50 are presented
in Figs. 77-85.
There are 31 strongly correlated pairs of elements out of
a possible 136 pairs. Although Sr is not well correlated with Mn,
Mg and Ca when the total sample suite is considered, it is found
that when only those samples from Dobb’s Linn are considered,
these elements correlate well. Correlation analysis for the
Dobb’s Linn sub-set of samples yields Kendall rank order correlation
coefficients of 0.55, 0.53 and 0.59 for the variation of Sr with
Mn, Mg and Ca respectively. When these correlations and that of
Fe with S are included, there is a grand total of 35 correlatable
pairs of elements within the data set (Fig. 86a).
It is possible to assign most of the better correlated
element pairs to one of four groups (Fig. 86b), The first group
consists of all ten possible combinations of the elements Al, K\
Ti, Rb and Zr. Attention has already been drawn above (Section 6b)
to the similarity of the vertical profiles of the elements in this
group. The second group consists of all six possible combinations
of the elements Ca, Mg, Mn and Sr. This group of elements is
most significant at Dobb’s Linn. The third group of elements may
be divided into two parts. The first part consists of all six
possible combinations of the elements Fe, Al, Mg and Zn whereas
the second part consists of two of the three possible combinations
between the elements Fe, S and Cu. The fourth group is composed
of negative correlations between Si and the elements Ti, Al, Fe,
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to any of the four groups are Zn with Mn and Zr with Mg.
The first three groups, with their high positive internal
correlations, are clearly very significant and are considered
below in greater detail. The fourth group with its negative
correlations based on Si, may be attributed to a greater or lesser
extent, to closure. Where elements are presented in a given
mineral species in approximately constant proportions, high
positive correlations may reflect the variation in abundance of
that mineral in a range of rocks.
The regressions for all 10 combinations of the Al - K -
Ti - Rb - Sr group pass close to the origin and suggests that the
majority, but not all, of these elements are found in one mineral
species. X-ray diffraction studies have confirmed the presence
of a mica conforming to the 2M type of the illite group. If the
element associations are indicating this mineral species, then the
mica must contain trace quantities of Rb and Sr.
The approximate proportions of each element within the
mineral species have been calculated from the slope of the
regression lines. A composition of Al 75.6%, K 19.1%, Ti 5.0%,
Rb 0.1% and Zr 0.2% resulted. These proportions are not
dissimilar to those of the type Fithian illite which contains
71.5% Al, 26.2% K and 3.7% Ti (Weaver and Pollard 1973). This
lends support to the diagnosis of illite.
In the sedimentary environment the elements Ca, Mg, Sr
and Mn are commonly associated as components of carbonate minerals
X-ray diffraction studies have shown that only a few samples have
that Mn correlates to a limited extent with Fe (Fig. 83). The
regression slopes for the correlation of these five elements
indicate the presence within the Moffat Shales of a carbonate with
63.1% Ca, 7.2% Mn, 25.7% Mg, 1.4% Fe and 0.1% Sr.
Element proportions calculated from the regression slopes
for the third group of elements are not as meaningful as those
computed for the first two groups. This is partly due to the link
through Al with the group one elements, partly to the common link
through Fe with the group two elements and partly to the common
link with Fe within the group (Fig, 86a).
The regression slopes of the Fe - S - Cu sub-group indicate
the presence of a pyritic mineral with 42.3% Fe, 57.2% S and
0.4% Cu. Similarly an iron-rich chlorite, related to clinochlore,
of composition Fe 36.3 : Mg 14.7 : Al 49.0 : Zn 0.05, is indicated
by the regression slopes of the Fe - Mg - Al - Zn sub-group.
d) Association of elements with minerals
The variation of element concentration has been correlated
with mineral content as calculated from X-ray modal analysis.
Product moment and Kendall rank order correlation coefficients are
given in Table 3-10. The correlation matrix indicates 13 mineral-
element pairs with a Kendall rank order coefficient greater than
0.50. Not included in the matrix is the variation of sulphur with
pyrite. This combination gives a product moment correlation
coefficient of 0.81 and a Kendall rank order correlation coefficient
of 0.50.
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illustrated in Fig. 86c and scatter plots with superimposed
regression lines are given in Figs. 87-89. It is observed that
Si exhibits negative correlations with albite and sericite but is
positively correlated with the amount of quartz in the sediments.
Al, Fe and Mg yield negative correlations with quartzjand Ti, Al
and Mg are positively correlated with the amount of sericite.
Albite, although not recognised as a distinct mineral species when
considering element associations, is found to correlate with both
Al and Mg. Pyrite, iron and sulphur intercorrelate lending
confidence to the observed vertical mineral distribution
(cf Figs. 22,49,57). Correlations within the mineral group itself
indicate that as the amounts of sericite and albite increase
within the sediments there is a decrease in the amount of quartz
present.
In certain horizons within the sequence, notably within
the M.cyphus Zone, pyrite is visible in hand specimen. This
observation plus the good statistical correlations between Fe, S
and modal pyrite indicate that Fe is principally bound up in the
sulphide phase. Cu, Pb and Zn can be present in pyrite in the
form of admixtures of other minerals such as chalcopyrite, galena
and sphalerite (Fleischer 1955) or as small amounts in solid
solution with pyrite (Deer et al,, 1962).