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 compositions and ages. However, as emphasized in this chapter, many geological features display random characteristics that can be modeled by adapting methods of mathematical statistics. The idea that random samples can be taken from statistical populations for the estimation of parameters remains paramount. The main parameters 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 modeling. 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.