Page 20 - Statistics and Data Analysis in Geology

Introduction

is in large part concerned with the effect sampling has on the outcome of statisti-
cal tests. Although Griffiths’ examples are drawn from sedimentary petrology, the
methods are equally applicable to other problems in the Earth sciences. The book
represents a rigorous, formal approach to the interpretation of geologic phenom-
ena using statistical methods. Griffiths’ book, unfortunately now out of print, is
especially commended to those who wish to perform experiments in geology and
can exercise strict control over their sampling procedures. In this text we will con-
cern ourselves with those less tractable situations where the sample design (either
by chance or misfortune) is beyond our control. However, be warned that anuncon-
trolled experiment (ie., one in which the investigator has no influence over where or
how observations are taken) usually takes us outside the realm of classical statistics.
This is the area of “quasi-statistics” or “proto-statistics,” where the assumptions of
formal statistics cannot safely be made. Here, the well-developed formal tests of
hypotheses do not exist, and the best we can hope from our procedures is guidance
in what ultimately must be a human judgment.

Measurement Systems

A quantitative approach to geology requires something more profound than a head-
long rush into the field armed with a personal computer. Because the conclusions
reached in a quantitative study will be based at least in part on inferences drawn
from measurements, the geologist must be aware of the nature of the number sys-
tems in which the measurements are made. Not only must the Earth scientist un-
derstand the geological significance of the recorded variables, the mathematical
significance of the measurement scales used must also be understood. This topic
is more complex than it might seem at first glance. Detailed discussions and refer-
ences can be found in Stevens (1946), the book edited by Churchman and Ratoosh
(1959) and, from a geologist’s point of view, in Griffiths (1960).
A measurement is a numerical value assigned to an observation which reflects
the magnitude or amount of some characteristic. The manner in which numerical
values are assigned determines the scale of measurement, and this in turn deter-
mines the type of analyses that can be made of the data. There are four measure-
ment scales, each more rigorously defined than its predecessor, and each containing
greater information. The first two are the nominal scale and the ordinal scale, in
which observations are simply classified into mutually exclusive categories. The
final two scales, the interval and ratio, are those we ordinarily think of as “mea-
surements” because they involve determination of the magnitudes of an attribute.

The nominal scale of measurement consists of a classification of observations
into mutually exclusive categories of equal rank. These categories may be identified
by names, such as “red,” “green,” and “blue,” by labels such as “A,” “B,” and “C,” by
symbols such as N, 0, and 0, or by numbers. However, numbers are used only
as identifiers. There can be no connotation that 2 is “twice as much” as 1, or that
5 is “greater than” 4. Binary-state variables are a special type of nominal data in
which symbolic tags such as 1 and 0, “yes” and “no,” or “on” and “off” indicate
the presence or absence of a condition, feature, or organism. The classification
of fossils as to type is an example of nominal measurement. Identification of one

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