Graphical and Numerical Data Reduction

One significant digit suffices in an expected error. The digit 8 after the decimal point in the value of M in the previous example is not quite significant, but since the error is smaller than 1.0 g moP, it provides a little information, and we include it. The accepted value is 86.17gmol , so that our expected error is larger than our actual error, as it should be about 95% of the time. 

In the cryoscopic determination of molar mass, the molar mass in kg moP is given by 

There are a number of functional relationships in physical chemistry that require data reduction that is more involved that substituting values into a formula. For example, thermodynamic relations imply that the equilibrium pressure of a two-phase system containing one substance is a function of the temperature. If we control the temperature and measure the pressure, we write 

This equation represents a function that we assume to be piecewise continuous. 

Chapter 11 The Treatment of Experimental Data Experimental Vapor Pressures of Pure Ethanol at Various 

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We also discussed graphical and numerical data reduction procedures. The most important numerical data reduction procedure is the least squares, or regression, method, which finds the best member of a family of functions to represent a set of data. We discussed the propagation of errors through this procedure and presented a version of the procedure in which different data points are given different weights, or importances, in the procedure. 

Available experimental data for various flow situations have been correlated in terms of the above dimensionless variables and the results fitted by empirical equations or simply presented in graphical form. Despite the reduction in the number of significant variables achieved by the introduction of the dimensionless products, a considerable amount of experimental work has to be carried out in order to arrive at a useable correlation for most geometries. For this reason it is usually worthwhile to attempt to carry out some form of analytical or numerical solution of the problem, even if the solution is a very simplified one, because this solution may indicate the general form of the correlation equation or, at least, indicate where the major emphasis should be placed in the experimental program. 

Assumptions may be made or models adopted (often by implication) about a system being measured that are not consistent with reality. The selection of the method of data reduction may be partly on the basis of the model adopted and partly on the basis of features such as computation time and simplicity. Kelly classified data processing methods as direct, graphical, minmax, least squares, maximum likelihood, and bayesian. Each method has rules by which computations are made, and each produces an estimate (or numerical result) of reality. 

Many of the sulfur oxoacids and their salts are connected by oxidation-reduction equilibria some of the more important standard reduction potentials are summarized in Table 15.19 and displayed in graphic form as a volt-equivalent diagram (p. 435) in Fig. 15.28. By use of the couples in Table 15.19 data for many other oxidation-reduction equilibria can readily be calculated. (Indeed, it is an instructive exercise to check the derivation of the numerical data... 

Some quantities in which we are interested can be measured directly. More often, a quantity must be calculated from other quantities that can be measured. This calculation process is called data reduction. The simplest form of data reduction is the use of a formula into which measured values are substituted. Other forms of data reduction include analysis of a set of data that can be represented by data points on a graph. Construction of such a graph and analysis of features of the graph, such as slopes and intercepts of lines, can provide values of variables. Statistical analysis done numerically can replace graphical analysis, providing better accuracy with less effort. We discuss both of these approaches. 

Most physicochemical properties and biological activities are multidimensional in essence. Their study in terms of structure-property or structure-activity relationships requires the examination of multidimensional spaces which are hardly perceivable by humans. Under these conditions, numerous linear and nonlinear methods are routinely used in environmental QSAR for data reduction and graphical display (Figure 2). In the same way, even if in most cases linear statistical methods allow the derivation of powerful QSAR models, it has been shown that numerous environmental phenomena are better simulated by means of nonlinear statistical tools such as artificial neural networks. ... 

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