Statistical Concepts

The first step in the establishment of reference values is the selection of a group of reference individuals. It is usually not feasible to obtain observations on aU possible reference individuals of a certain category of the general population. We therefore hope that the smaller group examined, the subset (sometimes called the reference sample group), can give us the desired information about the characteristics of the complete set of individuals (the reference population). 

The larger set is often considered hypothetical because its characteristics are not observed directly we know neither the number (the set size) nor the properties of aU its individuals. We therefore want to infer from observations made on the subset to the hypothetical set. An obvious requirement is that the individuals in the subset are typical of those in the complete set. Statistical theory usually assumes that the items in the subset are selected at random from among those in the set otherwise, the subset may be biased. If the items are not randomly selected, we can still use statistical techniques, but only with due caution and remembering the possible bias mtroduced. 

There are two main types of inferences made from values obtained from the subset (sample group) to the set (total reference population). 

The true (unknown, hypothetical) value of a property of the set is often called a parameter and given a lower-case Greek letter symbol. For example, the standard deviation of the population is symbolized by the Greek sigma The corresponding property determined in die 

Tfriother inference is to test hypotheses regarding properties of the set. We might, for example, state the hypothesis that the distribution of values for serum triglyceride 

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Moore, D. S., Statistics Concepts and Controversies, W. H. Freeman, New York, 1985. MuUiolland, H., and C. R. Jones, Fundamentals of Statistics, Plenum Press, New York, 1968. Taylor, J. K., Statistical Techniques for Data Analysis, Lewis, Boca Raton, FL, 1990. 

Our purpose in this introduction is not to trace the history of polymer chemistry beyond the sketchy version above, instead, the objective is to introduce the concept of polymer chains which is the cornerstone of all polymer chemistry. In the next few sections we shall introduce some of the categories of chains, some of the reactions that produce them, and some aspects of isomerism which multiply their possibilities. A common feature of all of the synthetic polymerization reactions is the random nature of the polymerization steps. Likewise, the twists and turns the molecule can undergo along the backbone of the chain produce shapes which are only describable as averages. As a consequence of these considerations, another important part of this chapter is an introduction to some of the statistical concepts which also play a central role in polymer chemistry. 

Statistical Process Control Statistical process control (SPG), also called statistical quahty control (SQC), involves the apphcation of statistical concepts to determine whether a process is operating satisfactorily The ideas involved in statistical quahty control are over fifty years old, but only recently with the growing worldwide focus on increased productivity have applications of SPG become widespread. If a process is operating satisfactorily (or in control ), then the variation of product quahty tails within acceptable bounds, usually the minimum and maximum values of a specified composition or property (product specification). 

Theoretical efforts a step beyond simply fitting standard statistical curves to fragment size distribution data have involved applications of geometric statistical concepts, i.e., the random partitioning of lines, areas, or volumes into the most probable distribution of sizes. The one-dimensional problem is reasonably straightforward and has been discussed by numerous authors... 

Although progress in continuum and computer modeling of dynamic fracture and fragmentation is encouraging, it is apparent that further advancements are needed. Many of the emerging physical and statistical concepts, some of which have been discussed in the present chapter, are not yet included in these... 

It is not sufficient to train staff solely in the techniques they need to use - a wider appreciation of the concepts will facilitate improved application. The staff assigned to quality planning need an even wider appreciation of statistical concepts and it is probably use-... 

Statistical concepts in many quality systems provide the basis for identifying the source of problems and directing improvement ef-... 

Factor S3 is based on statistical concepts and can be varied from 1.0 to account for structures whose probable lives are shorter (or longer) than is reasonable for the application of a 50-year return-period wind. 

Thermodynamics describes the behaviour of systems in terms of quantities and functions of state, but cannot express these quantities in terms of model concepts and assumptions on the structure of the system, inter-molecular forces, etc. This is also true of the activity coefficients thermodynamics defines these quantities and gives their dependence on the temperature, pressure and composition, but cannot interpret them from the point of view of intermolecular interactions. Every theoretical expression of the activity coefficients as a function of the composition of the solution is necessarily based on extrathermodynamic, mainly statistical concepts. This approach makes it possible to elaborate quantitatively the theory of individual activity coefficients. Their values are of paramount importance, for example, for operational definition of the pH and its potentiometric determination (Section 3.3.2), for potentiometric measurement with ion-selective electrodes (Section 6.3), in general for all the systems where liquid junctions appear (Section 2.5.3), etc. 

In order to appreciate how understanding new statistical concepts can help us, let us look at an example of where we can better apply known statistical concepts, to understand phenomena currently afflicting us. To this end, let us pose the seemingly innocuous question When doing quantitative calibration, why is it that we use the formulation of the problem that makes the constituent values the dependent (i.e., the Y) variable, and make the spectroscopic data the X (or independent) variable, called the Inverse Beer s Law formulation (sometimes called the P-matrix formulation) (For that matter, why is the formulation that we most commonly use called Inverse Beer s Law instead of the direct Beer s Law )... 

In preparing the book, a special effort has been made to create self-contained chapters. Within each one, numerical examples and graphics have been provided to aid the reader in understanding the concepts and techniques presented. Notation, references, and material related to that covered in the text are included at the end of each chapter. It is assumed that the reader has a basic knowledge of matrix algebra and statistics however, an appendix covering pertinent statistical concepts is included at the end of the book. 

Here, several general statistical concepts are briefly discussed as a complement to the material covered in this book. The books of Davis and Goldsmith (1972) and Mikhail (1976) are excellent sources for such information. Most of the concepts and definitions presented in this Appendix were extracted and summarized from these references, and for more detailed information, the reader is referred to these publications. 

Model of a supramolecular structure of polymolecular ensembles or clusters, determined by interaction and mutual arrangement of the forming molecules. At this level, the specific mechanisms of supramolecular chemistry, including molecular recognition, self-assembly, etc. [4] can be allocated. In most cases, it is possible to limit this area to objects with the sizes under 1 to 2 nm, since further increase in the sizes admits application of statistical concepts like phase and interphase surface. 

The method detection limit is, in reality, a statistical concept that is applicable only in trace analysis of certain types of substances, such as organic pollutants by gas chromatographic methods. The method detection limit measures the minimum detection limit of the method and involves all analytical steps, including sample extraction, concentration, and determination by an analytical instrument. Unlike the instrument detection limit, the method detection limit is not confined only to the detection limit of the instrument. 

The these d agregation ends with a very brief chapter Time and entropy," which contains the root of Prigogine s future preoccupations. He defines a thermodynamic time" related to the entropy production. It is interesting to point out one of the last conclusions of this chapter Originating from the second principle, the thermodynamic time necessarily appears as a statistical concept. It loses its meaning at the scale of elementary processes. This... 

Moore, D.S. (1979), Statistics Concepts and Controversies, Freeman, San Francisco, CA. 

The Z)-statistic concept can be readily extended to the two-factor case, where (A, B,) are corresponding ranks, i.e. if A and B are the ordinal form of observation sets (the A7B notation is used instead of Lehman s (R.S) notation in order to avoid confusion between similar symbols). If rank positions are tied, they are replaced by their mid-rank, yielding modified distributions A and B. By a straightforward extension of Eq.(6), the modified Z)-statistic is computed as [17]... 

The statistical concept of consistency embodies the idea we can obtain as much accuracy and precision as desired by collecting enough data. Technically, there are different dehnitions of consistency recognized by mathematical statisticians. Consistency is one example of an asymptotic property. ... 

Kaplan EL and Meier P (1958) Non-parametric estimation from incomplete observations Journal of the American Statistical Association, 53, 457-M81 Kaul S and Diamond GA (2006) Good enough a primer on the analysis and interpretation of non-inferiority trials Annals of Internal Medicine, 145, 62-69 Kay R (1995) Some fundamental statistical concepts in clinical trials and their application in herpes zoster Antiviral Chemistry and Chemotherapy, 6, Supplement 1, 28-33 Kay R (2004) An explanation of the hazard ratio Pharmaceutical Statistics, 3, 295-297... 

Taguchi. He has developed both a philosophy and a methodology for the process of quality improvement, which depend heavily on statistical concepts and tools. Many Japanese firms have applied these methods with great success. 

Like the other nonlinear constrained methods, the maximum-entropy method has proved its capacity to restore the frequency content of 6 that has not survived convolution by s and is entirely absent from the data (Frieden, 1972 Frieden and Burke, 1972). Its importance to the development of deconvolution arises from the statistical concept that it introduced. It was the first of the nonlinear methods explicitly to address the problem of selecting a preferred solution from the multiplicity of possible solutions on the basis of sound statistical arguments. 

Credible pharmaceutical product expiration dates are obtained by rigorous, scientifically designed studies using reliable, meaningful, and specific stability-indicating assays, appropriate statistical concepts, and computers to analyze the resulting data [2]. A comprehensive review of all aspects of pharmaceutical product stability has been published by Lintner [3] and more recently by Connors et al. [4]. 

The shelf life of a formulation is currently based on rigorous physical-chemical laws and statistical concepts to obtain reliable estimates. McMinn and Lintner have... 

Use of multiple regression techniques in the study of functional properties of food proteins is not new I76) Most food scientists have some familiarity with basic statistical concepts and some access to competent statistical advice. At least one good basic text on statistical modelling for biological scientists exists (7 ). A number of more advanced texts covering use of regression in modelling are available (, ). ... 

This formulation may look somewhat unfamiliar, but we shall see that both the weighting factor and the quantity being averaged can mean many different things in colloid science. Appendix C presents a more detailed presentation of basic statistical concepts for both discrete and continuous distributions for those who desire more information on these topics. 

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