Statistical Approaches

Systems with one type of branch unit A statistical approach was used by Flory (1941, 1943, 1953) and Stockmayer (1943, 1953) to derive an expression for the prediction of the extent of reaction at the gel point. In this approach, it is assumed that (1) the reactivity of all functional gronps of the same type is the same and independent of molecular size and (2) there are no intramolecular reactions between functional groups of the same molecule. 

Consider polymerization of bifunctional molecules A—A, B—B, and trifunctional molecule (—A)s in a mixture, not necessarily in equimolar quantities. This will lead to trifunetionally branched network polymer (VI)  

A portion of the trifunctionally branched network (VI) of gelled polymer is sehematically represented in Fig. 5.7, showing a series of contour lines or envelopes on whieh the branehes end, either in a branch point or in a terminal bifunctional group (nonbranch unit). For convenience, the chain sections are shown as being equal in length. However, this will not affect our following analysis. 

Consider the chain section in envelope 1 (Fig. 5.7) having a branch point at each end. The four new chains originating from the two branch points on envelope 1 happen to lead further to three new branch points and one nonbranch unit on envelope 2. The resulting six new chain sections that emanate from envelope 2 happen to lead to three branch points and three nonbranch units on 

X Terminal nonbranch unit — Chain section or chain segment 

Along with the isomerism of linear copolymers due to various distributions of different monomeric units in their chains, other kinds of isomerisms are known. They can appear even in homopolymer molecules, provided several fashions exist for a monomer to enter in the polymer chain in the course of the synthesis. So, asymmetric monomeric units can be coupled in macromolecules according to head-to-tail or head-to-head — tail-to-tail type of arrangement. Apart from such a constitutional isomerism, stereoisomerism can be also inherent to some of the polymers. Isomers can sometimes substantially vary in performance properties that should be taken into account when choosing the kinetic model. The principal types of such an account are analogous to those considered in the foregoing. The only distinction consists in more extended definition of possible states of a stochastic process of conventional movement along a polymer chain. 

---

There are two different aspects to these approximations. One consists in the approximate treatment of the underlying many-body quantum dynamics the other, in the statistical approach to observable average quantities. An exlmistive discussion of different approaches would go beyond the scope of this introduction. Some of the most important aspects are discussed in separate chapters (see chapter A3.7. chapter A3.11. chapter A3.12. chapter A3.131. 

These equations apply when an entire population is available for measurement. The most common situation in practical problems is one in which the number of measurements is smaller than the entire population. A group of selected measurements smaller than the population is called a sample. Sample statistics are slightly different from population statistics but, for large samples, the equations of sample statistics approach those of population statistics. 

J. W. Whalen, Molecular Thermodynamic.s A Statistical Approach John Wiley Sons,... 

Each observation in any branch of scientific investigation is inaccurate to some degree. Often the accurate value for the concentration of some particular constituent in the analyte cannot be determined. However, it is reasonable to assume the accurate value exists, and it is important to estimate the limits between which this value lies. It must be understood that the statistical approach is concerned with the appraisal of experimental design and data. Statistical techniques can neither detect nor evaluate constant errors (bias) the detection and elimination of inaccuracy are analytical problems. Nevertheless, statistical techniques can assist considerably in determining whether or not inaccuracies exist and in indicating when procedural modifications have reduced them. 

By proper design of experiments, guided by a statistical approach, the effects of experimental variables may be found more efficiently than by the traditional approach of holding all variables constant but one and systematically investigating each variable in turn. Trends in data may be sought to track down nonrandom sources of error. 

The drawback of the statistical approach is that it depends on a model, and models are bound to oversimplify. Nevertheless, we can learn a great deal from the attempt to evaluate thermodynamic properties from molecular models, even if the effort falls short of quantitative success. 

There are two ways to arrive at the relationship between aj and the concentration expressed as, say, a mole fraction. One is purely thermodynamic and involves experimental observations the other involves a model and is based on a statistical approach. We shall examine both. 

Comparing Eqs. (8.29) and (8.30) also leads to the conclusion expressed by Eq. (8.22) aj = Xj. Again we emphasize that this result applies only to ideal solutions, but the statistical approach gives us additional insights into the molecular properties associated with ideality in solutions ... 

Equation (9.74) is a one-dimensional version of Pick s second law. We shall presently consider a statistical approach to solving this equation. If c is measured as a function of x and t in an experiment which corresponds to the boundary conditions of the mathematical solution to Eq. (9.74), then D can be evaluated for the solute. We shall consider this below also. 

Design of experiments. When conclusions are to be drawn or decisions made on the basis of experimental evidence, statistical techniques are most useful when experimental data are subject to errors. The design of experiments may then often be carried out in such a fashion as to avoid some of the sources of experimental error and make the necessary allowances for that portion which is unavoidable. Second, the results can be presented in terms of probability statements which express the reliabihty of the results. Third, a statistical approach frequently forces a more thorough evaluation of the experimental aims and leads to a more definitive experiment than would otherwise have been performed. 

Statistical Approach Ignoring any discrepancies between the imphcit model used to establish the constraints and the actual unit, the measurements are adjusted to close the constraints. This adjustment effectively superimposes the known process operation embodied in the constraints onto the measurements. Minimum adjustments are made to the measurements. 

In general, tolerance stack models are based on either the wor.st case or statistical approaches, including those given in the references above. The worst case model (see equation 3.1) assumes that each component dimension is at its maximum or minimum limit and that the sum of these equals the assembly tolerance (initially this model was presented in Chapter 2). The tolerance stack equations are given in terms of bilateral tolerances on each component dimension, which is a common format when analysing tolerances in practice. The worst case model is ... 

The inadequacy of the worst case approach to tolerance stack design compared to the statistical approach is evident, although it still appears to be popular with designers. The worst case tolerance stack model is inadequate and wasteful when the capability of each dimensional tolerance is high > 1.33). Some summarizing comments on the two main approaches are given below. 

Here, the statistical approach will be used to predict the range distribution of the fragments. 

Failure rates are computed by dividing the total number of failures for the equipment population under study by the equipment s total exposure hours (for time-related rates) or by the total demands upon the equipment (for demand-related rates). In plant operations, there are a large number of unmeasured and varying influences on both numerator and denominator throughout the study period or during data processing. Accordingly, a statistical approach is necessary to develop failure rates that represent the true values. 

Statistical Approaches in Machinery Problem Solving Table 62.2 Primary machinery component failure... 

These data show how the method was applied in a case where standard and unknown were presumably identical in composition, and they emphasize the necessity for a statistical approach in all w ork of this kind. 

---