Page 21 - Statistics and Data Analysis in Geology

Statistics and Data Analysis in Geology - Chapter 1

fossil as a brachiopod and another as a crinoid implies nothing about the relative
importance or magnitude of the two.
The number of observations occurring in each state of a nominal system can
be counted, and certain nonparametric tests can be performed on nominal data. A
classic example we will consider at length is the occurrence of heads or tails in a
coin-flipping experiment. Heads and tails constitute two categories of a nominal
scale, and our data will consist of the number of observations that fall into them.
A geologic equivalent of this problem consists of the appearance of feldspar and
quartz grains along a traverse across a thin section. Quartz and feldspar form
mutually exclusive categories that cannot be meaningfully ranked in any way.
Sometimes observations can be ranked in a hierarchy of states. Mohs' hardness
scale is a classic example of a ranked or ordinal scale. Although the minerals on
the scale, which extends from one to ten, increase in hardness with higher rank,
the steps between successive states are not equal. The difference in absolute hard-
ness between diamond (rank ten) and corundum (rank nine) is greater than the
entire range of hardness from one to nine. Similarly, metamorphic rocks may be
ranked along a scale of metamorphic grade, which reflects the intensity of alter-
ation. However, the steps between grades do not represent a uniform progression
of temperature and pressure.
As with the nominal scale, a quantitative analysis of ordinal measurements is
restricted primarily to counting observations in the various states. However, we can
also consider the manner in which different ordinal classes succeed one another.
This is done, for example, by determining if states tend to be followed an unusual
number of times by greater or lesser states on the ordinal scale.
The interval scale is so named because the length of successive intervals is a
constant. The most commonly cited example of an interval scale is that of tempera-
ture. The increase in temperature between 10" and 20" C is exactly the same as the
increase between 110" and 120" C. However, an interval scale has no natural zero,
or point where the magnitude is nonexistent. Thus, we can have negative temper-
atures that are less than zero. The starting point for the Celsius (centigrade) scale
was arbitrarily set at a point coinciding with the freezing point of water, whereas
the starting point on the Fahrenheit scale was chosen as the lowest temperature
reached by an equal mixture of snow and salt. To convert from one interval scale
to another, we must perform two operations: a multiplication to change the scale,
and an addition or subtraction to shift the arbitrary origin.
Ratio scales have not only equal increments between steps, but also a true zero
point. Measurements of length are of this type. A 2-in. long shell is twice the length
of a 1-in. shell. A shell with zero length does not exist, because it has no length
at all. It is generally agreed that "negative lengths" are not possible. To convert
from one ratio scale to another, such as from inches to centimeters, we must only
perform the single operation of multiplication.
Ratio scales are the highest form of measurement. All types of mathematical
and statistical operations may be performed with them. Although interval scales
in theory convey less information than ratio scales, for most purposes the two can
be used in the same manner. Almost all geological data consist of continuously
distributed measurements made on ratio or interval scales, because these include
the basic physical properties of length, volume, mass, and the like. In subsequent
chapters, we will not distinguish between the two measurement scales, and they

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