Extreme value cumulative probability function

A number of statistical transformations have since then been proposed to quantify the distributions in pitting variables. Gumbel is given the credit for the original development of extreme value statistics (EVS) for the characterization of pit depth distribution [10]. The EVS procedure is to measure maximum pit depths on several replicate specimens that have pitted, then arrange the pit depth values in order of increasing rank. The Gumbel or extreme value cumulative probability function [f(x)] is shown in Eq (6.1), where A and a are the location and scale parameters, respectively. This probability function can be used to characterize data sets and estimate the extreme pit depth that possibly can affect a system. 

The normal probability relationship and its familiar beU-shaped curve represent a totahty of data, all of the scores on a test, average soil resistivities, or all pit depths form the basis for the curve. Application of the cumulative probability function for an exponential extreme value distribution of a standard variate to practical situations requires statistically valid collection of data. A practical and consistent sample size must be selected and enough samples must be taken to attain reliable results. 

When the phenomenon is known only poorly, the collected data may contain some degree of randomness or even reflect beliefs and bias on the part of the collector. To mitigate this problem, analysts may rely on probability, which relates statistical concepts to stochastic variables by mean of large data sets. An example of the use of one statistical and probabilistic method is the use of extreme value distributions applied to cumulative distribution functions to analyze pitting corrosion as performed by Macdonald and colleagues and shown in Figure 3.5. ... 

In days gone by this was achieved using probability paper, specially ruled graph paper which took care of the normal pdf. Nowadays, spreadsheets have functions to perform this calculation in Excel it is NORMSINV(x), where x is the normalized cumulative frequency. If the data are normally distributed this graph should be linear. Obvious outliers are seen as points at the extremes of the x-axis, that is, at values much greater than would be expected. Example 3.1 shows how to determine whether data are normally distributed using a Rankit plot in Excel. 

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