Geoscience Reference
In-Depth Information
2
Statistical Tools and Concepts
Abstract
Mineral resource estimation requires extensive use of statistics. In our context,
statistics
are
mathematical methods for collecting, organizing, and interpreting data, as well as drawing
conclusions and making reasonable decisions based on such analysis. This chapter presents
essential concepts and tools required throughout the topic.
2.1
Basic Concepts
is that data must be grouped together before any statistical
calculations are attempted. Ideally, a decision of how to
group the data can be made on the basis of clear geological
controls, as discussed in Chap. 4. Some of the statistical tools
presented in this chapter are useful to help make a choice of
stationarity, but most assume that the decision has already
been made and the data have been assembled into reasonable
groups.
In most cases we consider continuous variables that are
mass or volume fractions, that can take any value between
a minimum (0 %) and maximum (100 %). We sometimes
consider categorical or discrete variables that can take spe-
cific values from a closed set. A typical categorical variable
would be lithology or mineralization type.
Statistical tools are used for several reasons, including
(1) an improved understanding of the data and the mineral
deposit, (2) to ensure data quality, (3) to condense infor-
mation, and (4) to make inferences and predictions. In gen-
eral, we are not interested in the statistics of the samples.
Our goal is to go beyond the limited sample to predict the
underlying population. Additionally, creative visualization
of data is an important component of mineral resource es-
timation, partly because of its usefulness as a tool to un-
derstand data, but also to help validate spatially distributed
models.
There are many good references on basic statistics. One
accessible reference is Lapin (
1983
). This topic uses a few
notation conventions. Lowercase letters (
x
,
y
,
z
,…) denote
actual values such as a measured value or a specified thresh-
old. Uppercase letters (
X
,
Y
,
Z
,…) denote a random variable
(RV) that is unknown. We characterize the uncertainty in a
random variable with a probability distribution. A random
A conventional presentation of statistics includes the no-
tion of a
population
that is the virtually infinite collection
of values that make up the mineral deposit. A
sample
is a
representative subset selected from the population. A good
sample must reflect the essential features of the population
from which it is drawn. A r
andom sample
is a sample where
each member of a population had an equal chance of being
included in the sample. The
sample space
is the set of all
possible outcomes of a chance experiment, for example a
drilling campaign. The
event of a sample space
is a group of
outcomes of the sample space whose members have some
common characteristic.
Statistically independent events
are
such that the occurrence of one event does not depend on the
occurrence of other events. Sampling mineral deposits rarely
fits nicely in the framework of representative samples from a
statistical population; nevertheless, many concepts and tools
from conventional statistics are used routinely.
Inductive statistics
or
statistical inference
is attempted if
the sample is considered representative. In this case, conclu-
sions about the population can often be inferred. Since such
inference cannot be absolutely certain, the language of prob-
ability is used for stating conclusions.
Descriptive statistics
is
a phase of statistics that describes or analyses a given sample
without inference about the population. Although our goal in
mineral resource estimation is almost always inference, we
use many descriptive statistics for viewing, understanding,
and evaluating data.
An essential concept in statistics is
stationarity
, that is, our
choice of data to pool together for common analysis. Chap-
ter 6 describes stationarity more formally, but the concept
