Graphical Methods of Data Presentation

The advantage of the following methods of graphical representation is the clear and simple presentation of the essential facts. Simple charts, like bar charts, x-y scatter diagrams or pie diagrams, which are also available in 3D-form are also suited to visual representation of data. They are not described because this section is devoted to treatment of multivariate data. Graphs for control charts, particulary for quality assurance and control, can be found in [FUNK et al., 1992 AQS, 1991]. 

---

To interpret sieve data, graphical and statistical methods of data presentation are used. The distribution curve that is widely used in industrial practice was developed by Rosin, Rammler, Sperling, and Bennett in the 1930s (Rosin, Rammler, Sperling, 1997 Rosin and Rammler, 1933 Bennett, 1936). They found out that the size distribution of coal dust and of other crushed and milled materials like cement follows a probability curve with a similar pattern as well-known physical functions such as the Maxwdl-BoUzmann distribution (Section 3.1.4) of the speed of gas molecules (Schubert and Waechtler, 1969). The so-called Rosin-Rammler-Sperling-Bennett (RRSB) function is given by ... 

The theory of transformation matrices is described in detail in [9]. In [11] universal and efficient graphical methods of operating with Clebsch-Gordan coefficients and transformation matrices are described. Their use allows one easily to find expressions for the most complex matrix elements of the operators, corresponding to physical quantities, in various coupling schemes, and to carry out their processing if necessary. Here we shall present only the minimal data on transformation matrices and methods of their evaluation. 

To produce this kind of grid map by quickly measuring the concentrations at some points for immediate processing and graphical presentation is a simple and often effective way to communicate the results to persons w ho are not trained to analyze primary results of measurements. The method must, however, be used with care since there is a risk that the sampling of data itself may affect the airflow in the studied area. I he equipment needed is relatively expensive, and the method is therefore of interest when the prerequisites are already available for other reasons. 

Graphical analysis of failure data is most commonly plotted using probability. However, in order to understand the hazard plotting method presented here, is not necessary to understand probability plotting. While it is difficult to utilize probability plotting for multiply-censored data, it is... 

The plotting of Dixon plot and its slope re-plot (see 5.9.5.9) is a commonly used graphical method for verification of kinetics mechanisms in a particular enzymatic reaction.9 The proposed kinetic mechanism for the system is valid if the experimental data fit the rate equation given by (5.9.4.4). In this attempt, different sets of experimental data for kinetic resolution of racemic ibuprofen ester by immobilised lipase in EMR were fitted into the rate equation of (5.7.5.6). The Dixon plot is presented in Figure 5.22. 

The data presented in Figure 8 graphically illustrate the tremendous and rapid growth in interest in FOSS chemistry, especially for patented applications. This looks set to continue with current applications in areas as diverse as dendrimers, composite materials, polymers, optical materials, liquid crystal materials, atom scavengers, and cosmetics, and, no doubt, many new areas to come. These many applications derive from the symmetrical nature of the FOSS cores which comprise relatively rigid, near-tetrahedral vertices connected by more flexible siloxane bonds. The compounds are usually thermally and chemically stable and can be modified by conventional synthetic methods and are amenable to the usual characterization techniques. The recent commercial availability of a wide range of simple monomers on a multigram scale will help to advance research in the area more rapidly. 

While methods validation and accuracy testing considerations presented here have been frequently discussed in the literature, they have been included here to emphasize their importance in the design of a total quality control protocol. The Youden two sample quality control scheme has been adapted for continuous analytical performance surveillance. Methods for graphical display of systematic and random error patterns have been presented with simulated performance data. Daily examination of the T, D, and Q quality control plots may be used to assess analytical performance. Once identified, patterns in the quality control plots can be used to assist in the diagnosis of a problem. Patterns of behavior in the systematic error contribution are more frequent and easy to diagnose. However, pattern complications in both error domains are observed and simultaneous events in both T and D plots can help to isolate the problems. Point-by-point comparisons of T and D plots should be made daily (immediately after the data are generated). Early detection of abnormal behavior reduces the possibility that large numbers of samples will require reanalysis. 

An interesting method of fitting was presented with the introduction, some years ago, of the model 310 curve resolver by E. I. du Pont de Nemours and Company. With this equipment, the operator chose between superpositions of Gaussian and Cauchy functions electronically generated and visually superimposed on the data record. The operator had freedom to adjust the component parameters and seek a visual best match to the data. The curve resolver provided an excellent graphic demonstration of the ambiguities that can result when any method is employed to resolve curves, whether the fit is visually based or firmly rooted in rigorous least squares. The operator of the model 310 soon discovered that, when data comprise two closely spaced peaks, acceptable fits can be obtained with more than one choice of parameters. The closer the blended peaks, the wider was the choice of parameters. The part played by noise also became rapidly apparent. The noisy data trace allowed the operator additional freedom of choice, when he considered the error bar that is implicit at each data point. 

Because of the very low scattered intensity, the data at the shortest sampling interval is usually the poorest in quality. Arbitrary renormalization of the data followed by the graphical representation outlined above is most likely to amplify errors in the data analysis, focus attention on the inherent errors in the construction of the composite relaxation function, and give undue importance to the worst data. When the data is as limited in quality as it is for this problem, any method of analysis should be as numerically stable as possible and the maximum allowable smoothing of the data should be employed. This procedure may obscure subtle features, but only very high quality data could reliably demonstrate their presence anyway. At the present time a conservative approach seems more sensible. 

Conventionally storage modulus versus frequency and temperature results are presented by extrapolated isotherms called master curves. These are plotted by shifting frequency data with a temperature dependent shift factor, a. Most published results for are based on manual graphic methods. Computer work first aimed at systemizing and automating determinations of shift factor was initiated by a desire to efficiently process the hundreds of data points from each FTMA run. 

A method of numerical integration (or quadrature, as it is also called) is required to evaluate I in any of these cases. The specific techniques we will present are algebraic, but the general approach to the problem is best visualized graphically. For the moment, we will suppose that all we have relating x and y is a table of data points, which we may graph on a plot of y versus x. 

---