Some basic statistics

Despite the conflicting philosophies between statisticians and chemometricians, it is an unavoidable fact that statistics provides a strong foundation for chemometrics. One could easily argue that, without classical statistics, there could be no chemometrics. 

One underlying principle of classical statistics is that any observation in nature has an uncertainty associated with it. One extension of this principle is that multiple observations of the same object will result in a distribution of values. One common graphical representation of a distribution of values is the histogram, where the frequency of occurrence of a value is plotted versus the value. Many statistical tools are based on a specific type of distribution, namely the Gaussian, or Normal distribution, which has the following mathematical form  

Although there are many useful statistical tools, there are two that have particular relevance to chemometrics the t-test and the f-test [22,23]. The t-test is used to determine whether a single value is statistically different from the rest of the values in a series. Given a series of values, and the number of values in the series, the f-value for a specific value is given by the following equation  

The t value is the number of standard deviations that the single value differs from the mean value. This t value is then compared to the critical t value obtained from a t-table, given a desired statistical confidence (i.e., 90%, 95%, or 99% confidence) and the number of degrees of freedom (typically iV-1), to assess whether the value is statistically different from the other values in the series. In chemometrics, the t test can be useful for evaluating outliers in data sets. 

The f-test is similar to the t-test, but is used to determine whether two different standard deviations are statistically different. In the context of chemometrics, the f-test is often used to compare distributions in regression model errors in order to assess whether one model is significantly different than another. The f-statistic is simply the ratio of the squares of two standard deviations obtained from two different distributions  

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If error is random and follows probabilistic (normally distributed) variance phenomena, we must be able to make additional measurements to reduce the measurement noise or variability. This is certainly true in the real world to some extent. Most of us having some basic statistical training will recall the concept of calculating the number of measurements required to establish a mean value (or analytical result) with a prescribed accuracy. For this calculation one would designate the allowable error (e), and a probability (or risk) that a measured value (m) would be different by an amount (d). 

Process Analytical Technology 12.2.2 Some basic statistics... 

We cannot possibly provide a detailed account of the full range of statistical methods and strategies that constitute the armamentarium of modern science. Yet, there is value in considering some basic statistical principles. 

Some basic statistical concepts are defined in this section. For a more comprehensive treatment, the reader is directed to standard statistical texts. ... 

The real problem in utilizing expressions (58) to (63) is the assignment of the weights, Wy. To proceed further, it is necessary to digress into a discussion of some basic statistical ideas. 

In the following we will thus present some basic statistical methods useful for determining turbulence quantities from experimental data, and show how these measurements of turbulence can be put into the statistical model framework. Usually, this involves separating the turbulent from the non-turhulent parts of the flow, followed by averaging to provide the statistical descriptor. We will survey some of the basic methods of statistics, including the mean, variance, standard deviation, covariance, and correlation (e.g., [66], chap 1 [154], chap 2 [156]). 

To check that the downloaded assemblies are valid and to get some basic statistics about them, we will use the faCount, f aLen, and stats utilities (Mercator distribution). The faCount utility calculates nucleotide frequencies within each input FASTA record (chromosomes or contigs in our case) and... 

The purpose of this chapter is to review some basic statistical concepts. Statistical calculation and algorithms are not covered since, as has already been explained, they do not form part of the subject matter of this book. I shall, however, attempt an illustration of the nature of an important divide in statistics that between Bayesian or subjectivist statistics on the one hand and classical or frequentist on the other. 

It is impossible to evaluate model fitting without some basic statistical concepts. But do not be alarmed. You wiU not have to become a master statistician to benefit from the techniques presented in this book. A few notions derived from the (deservedly) famous normal distribution will suffice. These, presented in Chapter 2, are very important if we wish to understand and to correctly apply the methods scussed in the rest of the book. In an effort to lighten the dullness that often plagues the discussion of such concepts, we base our treatment of the normal distribution on solving a practical problem of some relevance to the culinary world. 

Analyzer training Subjects should include how the analyzer works, includes some basic chemistry, physics, and electronics some basic statistics knowledge, such as of standard deviation, is very helpful. The technicians should be familiar with the analyzer capability, such as the sensitivity, response time, analytical accuracy, and repeatability. The QA program and troubleshooting procedures should also be included. 

Prichard, E. (editor). 2001. Analytical Measurement Terminology, Royal Society of Chemistry, Cambridge. (A simple introduction to analytical measurement and quality principles, including some basic statistics.)... 

Because of this variation in results it is necessary to apply some type of statistical treatment to the data (e.g., the arithmetic averaging of replicate values). When analyzing data, however, it helps to know some basics. Statistical techniques can be misused and give the researcher a false sense of security. One must be aware of limitations as well as advantages. 

First we need to state some basic statistics, which is of crucial importance in the following sections. When determining a continuous variable such as fat or moisture content, etc. with a wet chemistry laboratory reference method, it is important to estimate the standard error of the laboratory reference method (SEL or Sref). Assume that there are A[l, 2,A] samples, each measured in M[l, 2,... y,... A/] replicates and that the average from the replicates is used in the multivariate NIR... 

Some basic statistical requirements were also overlooked when quantitative models based on coefficients were developed. The most important assumption underlying the linear regression model is that the dependent variable contains all the errors in each data pair (Maes 1984 Irvin and Quickenden 1983). The range for 1-octanol/water partition coefficients obtained for each molecule clearly demonstrates that this basic assumption is not met. It has been demonstrated (York 1966) that if this basic assumption is violated, the fitted slopes can deviate by as much as 40% from the correct value. Thus, the validity and applicability of published quantitative models describing the relationship between the Kq coefficients and soil sorption coefficients are highly questionable. 

In addition to the graphical techniques that have been illustrated in previous sections, some basic statistical tools should be brought to bear in the analysis of kinetic data. In fact, in most cases, graphical analyses merely set the stage for the efficient use of statistical analysis. Some of the most useful statistical tools are illustrated in the following example. 

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