Steve M., Darby D.M., Geostatistics Explained - An Introductory Guide for Earth Scientists
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These graphs show that tooth decay increases with age, but the pattern
differs between cities – in Hale the increase is fairly steady, but in Yarvard it
remains low in children up to age seven but then suddenly increases. This
led to several hypotheses including that there might have been a child dental
care program, or water fluoridation, in place in Yarvard for the past eight
years compared to no action on decay in Hale.
Of course, there is always the possibility that the samples are different due
to chance, so perhaps the first step in any further investigation would be to
repeat the sampling using much larger numbers of children from each city.
Subsequent investigation found that the Yarvard municipal drinking
water had been fluoridated for the past eight years, but this treatment had
Location of tornado (US state)
(a)
100
50
0
AL AR FL GA LA MS NC OK SC TN TX
Number of tornados
Location of tornado (US state)
(b)
TX AR MS OK AL GA SC TN LA NC FL
100
50
0
Number of tornados
Figure 3.4 (a) Preliminary data on tornado occurrence in southeastern US
states (listed alphabetically) from 1998–2007. (b) The same data but with the
number of cases ranked in order from most to least.
3.5 Bivariate data 23
not been introduced in Hale. The fluoride program works because your
teeth are made of the mineral hydroxyapatite (the same mineral that binds
to heavy metals). In this case the apatite in your teeth binds fluorine ions
which substitute for hydroxyls in the apatite structure, making the enamel
of your teeth less soluble and therefore less prone to decay. This seems a very
Table 3.3 The number of dental caries and age of 20 children
chosen at random from each of the two cities of Hale and Yarvard.
Hale Yarvard
Caries Age Caries Age
1 3 10 9
12 1 5
4 4 12 9
43 1 2
56 1 2
6 5 11 9
23 2 3
9 9 14 9
45 2 6
21 8 9
78 1 1
34 4 7
98 1 1
11 9 1 5
12 7 8
14 1 7
37 1 6
11 1 4
11 2 6
65 1 2
Table 3.4 The average number of dental caries and
age of 20 children chosen at random from each of the
two cities of Hale and Yarvard.
Hale Yarvard
Caries Age Caries Age
4.05 4.5 years 4.10 5.5 years
24 Collecting and displaying data
plausible reason, but bear in mind that these data are only correlative and
there may be other reason(s) for the difference between the two cities.
3.6 Data expressed as proportions of a total
Data for the relative frequencies in two or more categories that sum to a
total of 1.0, or 100%, can be displayed as a pie diagram – a circle in which
each of the categories is displayed as a “slice,” the size of which is propor-
tional to its value. For example, a sample containing four different min-
erals that are equally abundant would be shown as a circle subdivided into
four equal 90
o
slices. Pie diagrams are easily interpreted when there are 10
or fewer categories and each contains at least 10% of the data (Figure 3.6).
When there are more than 10 categories the display will appear cluttered,
especially when slices are distinguished by their color, but it will be even
harder to differentiate among a lot of categories shown only as black, white
and shades of grey. Categories representing a relatively small number or
proportion of total cases will appear very narrow and may be overlooked.
The procedure for drawing a pie diagram showing either the relative
proportion of cases in several categories, or the values of two or more
variables (e.g. the concentrations of six different ions) is straightforward.
First, the data for each category are listed, summed to give a total, and then
expressed as proportions of this total. Each proportion is then multiplied by
360 to give the width of the slice in degrees, which is used to draw the
appropriate divisions on the pie diagram.
(a)
Age (years)
Number of caries
0
12
8
4
0
510
(b)
Age (years)
Number of caries
0
8
12
4
0
510
Figure 3.5 The number of dental caries plotted against the age of 20 children
chosen at random from each of the two cities of (a) Hale and (b) Yarvard.
3.6 Data expressed as proportions of a total 25
3.7 Display of geographic direction or orientation
Rose diagrams are used to show a summary of the direction or orientation
of a sample of objects such as crystals or fractures in rock, or the geographic
orientation of paleocurrent directions in ancient river systems. For example,
a unimodal paleocurrent implies a river with steep slopes, but a bimodal one
suggests a meandering river with a low slope. Rose diagrams are also
commonly used by meteorologists to report the direction and magnitude
of winds. The procedures for drawing rose diagrams and analyzing data for
direction and orientation are described in Chapter 22.
3.8 Multivariate data
Often earth scientists have data for three or more variables measured on the
same sampling unit. For example, a geologist might have data for mineralogy,
chemical composition, geological age and metamorphic grade for 20 outcrops
across a zone of contact metamorphism, or a paleontologist might have data
for the numbers of several species of brachiopods from a specific formation.
Results for three variables could be shown as three-dimensional graphs,
but direct display is difficult for more than this number of variables. Some
relatively new statistical techniques have made it possible to condense and
summarize multivariate data in a two-dimensional display, and these are
introduced in Chapter 20.
Hornblende
Biotite
Plagioclase
K-feldspar
Quartz
(a)
Hornblende
Biotite
Plagioclase
K-feldspar
Quartz
(b)
Figure 3.6 Pie diagrams comparing the mineralogy of two different granites.
From this type of comparison it is clear that the rock in (a) has far less K-
feldspar and much more hornblende compared to the rock in (b).
26 Collecting and displaying data
3.9 Conclusion
Graphs may reveal patterns in data sets that are not obvious from looking at
lists or calculating descriptive statistics. Graphs can also provide an easily
understood visual summary of a set of results. In later chapters there will be
discussion of data displays such as boxplots and probability plots, which can
be used to decide whether the data set is suitable for a particular analysis.
Most modern statistical software packages have easy to use graphics options
that produce high-quality graphs and figures. These packages are very useful
for writing assignments, reports or scientific publications.
3.9 Conclusion 27
4 Introductory concepts
of experimental design
4.1 Introduction
To generate hypotheses, you often sample different groups or locations
(which is sometimes called a mensurative experiment because you usually
measure something, such as air density, chemical composition or temper-
ature, on each sampling unit) and explore these data for patterns or
associations. To test hypotheses you may do mensurative experiments, or
manipulative ones where you change a condition and observe the effect of
that change upon each experimental unit (like the experiment with apatite
and lead described in Chapter 2). Often you may do several experiments of
both types to test a particular hypothesis. The quality of your sampling and
the design of your experiment can have an effect on the outcome and
determine whether or not your hypothesis is rejected, so it is important to
have an appropriate and properly designed experiment.
First, you should attempt to make your measurements as accurate and
precise as possible so they are the best estimates of actual values.
Accuracy is the closeness of a measured value to the true value.
Precision is the “spread” or variability of repeated measures of the same
value.
For example, a thermometer that consistently gives a reading corre-
sponding to a true temperature (e.g. 20 °C) is both accurate and precise.
Another that gives a reading consistently higher (e.g. +10 °C) than a true
temperature is not accurate, but it is very precise. In contrast, a ther-
mometer that gives a fluctuating readi ng within a wide range of values
around a true temperature is not precise and will usually be inaccurate
except when the reading occasionally happens to correspond to t he true
temperature.
28
The distinction between accuracy and precision is particularly important
(and sometimes quite vexing) in geochemical studies. Most types of analytical
instruments are calibrated relative to standards (usually glasses or minerals),
but even many “true” or “standard” values for the compositions of naturally
occurring glasses or minerals have themselves been obtained using analytical
techniques that are subject to error. So an important (though rather prosaic)
part of geochemistry is the characterization of standards. This is usually done
by analyzing a standard in several different ways, repeating these in different
laboratories and using the combined results to determine an accepted “true”
value for it (that is usually an average). Because geoscientists deal with
naturally occurring materials such approximations are often unavoidable
and the errors they contribute to analytical work are an underlying source
of inaccuracy in most types of geochemical data.
Inaccurate and imprecise measurements or a poor or unrealistic sampling
design can result in the generation of inappropriate hypotheses. Measurement
errors or a poor experimental design can give a false or misleading outcome
that may result in the incorrect retention or rejection of an hypothesis.
The following is a discussion of some important essentials of sampling
and experimental design.
4.2 Sampling: mensurative experiments
Mensurative experiments are often a good way of generating or testing
predictions from hypotheses. (An example of the latter is “I think apatite
binds heavy metals. So if I sample groundwater at 500 sites with high
concentrations of naturally occurring apatite and 500 where apatite concen-
trations are low or zero, the groundwater in the first group should, on average,
contain less dissolved heavy metals.”) You have to be careful when interpret-
ing the results of mensurative experiments because you are sampling an
existing condition, rather than manipulating conditions experimentally.
There may be some other difference between your groups (e.g. the “high
apatite” sites may have negligible amounts of heavy metals present anyway).
4.2.1 Confusing a correlation with causality
A correlation between two variables means they vary together. A positive
correlation means that high values of one variable are associated with high
4.2 Sampling: mensurative experiments 29
values of the other, while a negative correlation means that high values of
one variable are associated with low values of the other. For example, the
graph in Figure 4.1 shows a positive correlation between the increasing tilt
of the summit of a volcano and the number of earthquakes measured there.
A volcano with a flat summit has zero tilt.
Unfortunately a correlation is often mistakenly interpreted as indicating
causality. In this example it seems very plausible that the tilt of the summit
could be caused by settling that occurs as a result of the earthquakes, so
volcanoes with more frequent earthquakes are more likely to have tilted
summits. However, even if there is a very obvious correlation between any
two variables, it does not necessarily show that one is responsible for the
other. The correlation may have occurred by chance, or a third unmeasured
factor might determine the values of the two variables studied. In this case,
tilting actually occurs because of increased pressure from fresh magma
moving into the chamber beneath the summit, which inflates the ground
beneath the summit, both increasing the tilt and causing rock fracturing that
is manifested as earthquakes. There is no causal relationship between earth-
quakes and tilt: they are both caused by the volume of magma injected into
the chamber (Figure 4.2).
Another complication in the interpretation of correlation arises often in
geological analyses involving closed data sets, where the variables sum to a
fixed total, percentage or proportion such as 100% or 1.0. This type of
error is particularly common in reports of chemical analyses of rocks and
minerals. For example, summary data for mineral formulae are routinely
Tilt of summit
surface (µ radians)
Number of earth
q
uakes
Figure 4.1 Example of a positive correlation between the tilt of the summit of
a volcano and the number of earthquakes observed there.
30 Introductory concepts of experimental design
summed to the appropriate number of cations. The two olivine minerals
fayalite (Fe
2
SiO
4
) and forsterite (Mg
2
SiO
4
) rarely occur in these two “pure”
forms. Instead, most specimens contain a mixture of iron and magnesium,
each of which can fit in the two available octahedral sites in the structure,
which is why you might see the formula for olivine written as
Fe
0.6
Mg
1.4
SiO
4
. The cations Fe and Mg must sum to two, so they will always
be inversely correlated with each other. An increase in Fe will inevitably
be accompanied by a decrease in the percentage of Mg, but this does not
mean one has an effect on the other. Rather, the formula probably reflects
the amounts of Mg and Fe in the original melt when the mineral crystallized.
Care must be taken to avoid being misled by such spurious negative correla-
tions in closed systems.
4.2.2 The inadvertent inclusion of a third variable: sampling
confounded in time
Occasionally researchers have no choice but to sample different sites at
different times. These results should be interpreted with great caution,
because changes occurring over time may contribute to differences (or the
lack of them) among samples or sampling units. The sampling is said to be
confounded in that more than one variable may be having an effect on the
results. Here is an example.
An oceanographer measured the sodium content of carbonate shelf sedi-
ments that are being formed in the Holyoke estuary, by taking 100 cores
running in a line down the length of the estuary. Coring is time-consuming
and the oceanographer only had access to a boat for one day per week and
Size of ma
g
ma chamber
Tilt of summit Number of earthquakes
Figure 4.2 The involvement of a third variable “size of magma chamber” that
determines the tilt of the volcano’s summit and the number of earthquakes
that occur. Even though there is no causal relationship between tilt and the
number of earthquakes, these two variables are positively correlated.
4.2 Sampling: mensurative experiments 31
could only take five cores per day. Therefore the most upstream part of the
estuary was sampled in May, the middle in June and the most downstream
sites in August. The sampling showed a positive correlation between the
sodium content of newly laid down sediment and increasing distance
downstream. Unfortunately, however, the change in sodium content was
actually caused by seasonal variation in salinity that was low throughout the
entire estuary in spring due to groundwater run-off and increased as the
year progressed (and the scientist did not know this). Subsequent work
where the entire estuary was sampled on the same day found no difference
in the sodium content of the sediments, so the correlation between distance
downstream and sodium content was an artifact of the sampling of different
places being confounded in time. This is an example of a common problem,
and you are likely to find similar cases in many published scientific papers
and reports.
4.2.3 The need for independent samples in mensurative
experiments
Frequently researchers have to accurately describe relatively large areas or
objects as part of a mensurative experiment. For example, annual sedimen-
tation rates in glacier-fed lakes are used to reconstruct up-valley glacier
activity and thereby model environmental change. Sediment traps are
deployed at various locations in these lakes, and the mean particle size,
quantity and sediment flux are calculated. There is an obvious need to
replicate the sampling – that is, to independently measure sedimentation
rate in more than one place.
If you only sampled sediment in one trap at one place (Figure 4.3(a)) the
results would not be a good indication of the sediment accreting across the
whole lake. The sampling needs to be replicated, but there is little value in
repeatedly sampling one small area (e.g. by taking several samples under
“*****” in Figure 4.3(b)) because this still will not give an accurate indication
of variation in sediment rate and composition across the whole lake
(although it may give a very accurate indication of conditions in that
particular part of the lake sampled). This sort of sampling is one aspect of
what Hurlbert (1984) called pseudoreplication, which is still a very com-
mon flaw in a lot of scientific research. The replicates are “pseudo”–sham
or unreal – because they are unlikely to truly describe what is occurring
32 Introductory concepts of experimental design