Probability and Statistics - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 41

Chapter 2
Probability and Statistics
Abstract From a historical perspective, the theory of statistics was developed
relatively recently. Calculating the mean from a number of measurements was a
procedure first practiced in 1581 but even this simple method, which is understood
by most people to-day, remained highly controversial until the end of the eighteenth
century. Traditionally, the upper part of the Earth's crust has been viewed as a
complex three-dimensional mosaic of numerous rock units with different compo-
sitions and ages. However, as emphasized in this chapter, many geological features
display random characteristics that can be modeled by adapting methods of math-
ematical statistics. The idea that random samples can be taken from statistical
populations for the estimation of parameters remains paramount. The main para-
meters of geoscience data to be estimated are their mean and variance. Frequency
distribution analysis is applicable to many different types of geological data.
Discrete distributions include the binomial and Poisson, but also the geometric
and negative binomial distributions. The normal and lognormal distributions are
most important in modeling continuous data although the Pareto distribution is
becoming increasingly important because of its close connection to fractal model-
ing. Many methods of statistical inference including Student's t -test, analysis of
variance and the chi-squared test for goodness of fit are based on the normal
distribution. These statistical methods remain important in geology if used in
an exploratory manner because the random variables considered often are not
independent and identically distributed (iid), which is a requirement for statistical
problem-solving as practiced in most other fields of science. Especially, numbers of
degrees of freedom commonly used in statistical tests are strongly affected by
spatial autocorrelation due to the continuous nature of most geological variables.
Keywords Mathematical statistics • Probability calculus • Frequency distributions
• Statistical inference • Significance tests • Q-Q plots • Geostatistics • Sampling
• Grenville Province age determinations • Pulacayo zinc values • Witwatersrand
gold assays