Data normalization, function graphs

Biaxial testers measure shear stresses related to normal stresses. However, the flow-function graph and its interpretation for silo design requires the calculation of principal stresses (fc and aO from the normal and shear stress data by the use of yield loci and Mohr s circles (a mathematical tool). In concept, a perfect imiaxial tester can directly apply and measure principal stresses, making the construction of yield loci and use of Mohr circles unnecessary. This would expedite the completion of flow functions and reduce testing time. 

A set of experiments was performed at variable droplet sizes. The graph in Fig. 4.7 shows the dependence of the normalized (by Kint/a) osmotic resistance as a function of the oil volume fraction. The normalized values fall onto a single curve within reasonable experimental uncertainty. The results were compared to the normalized data obtained by Mason et al. [7] in the presence of surfactants. These latter are represented as a solid line that corresponds to the best fit to the experimental points (Eq. (4.18)). It is worth noting that the normalized pressures in solid-stabilized emulsions are much larger than the ones obtained in the presence of surfactants. 

Figure 6.3 Quantal effects. Typical set of data after administration of increasing doses of drug to a group of subjects and observation of minimum dose at which each subject responds. Data shown are for 100 subjects dose increased in 0.2 mg/kg of body weight increments. Mean (ji) (and median) dose is 3.0 mg/kg standard deviation (v) is 0.8 mg/kg. Results plotted as histogram (bar graph) showing number responding at each dose smooth curve is normal distribution function calculated for ji of 3.0 and v of 0.8.

When [Rh(phi)2(phen)]3+ is titrated into a solution containing [Ru(phen)2(dppz)]2+ and B-form DNA, the photoinduced luminescence of the ruthenium(II) complex is quenched dramatically (53). In these experiments, luminescence is monitored by laser flash as quencher is added. Data are then plotted in Stern-Volmer format, where the ratio of initial intensity/intensity (I0/I) is given as a function of quencher concentration [Q. The degree of lifetime quenching can also be described by plotting the inverse of the lifetime (r0/r) versus [Q. Normally, when chromophore and quencher interact bimolecularly, Stern-Volmer graphs are linear with [Q] and the slope for r0/r is the same as that for I0/I. 

In days gone by this was achieved using probability paper, specially ruled graph paper which took care of the normal pdf. Nowadays, spreadsheets have functions to perform this calculation in Excel it is NORMSINV(x), where x is the normalized cumulative frequency. If the data are normally distributed this graph should be linear. Obvious outliers are seen as points at the extremes of the x-axis, that is, at values much greater than would be expected. Example 3.1 shows how to determine whether data are normally distributed using a Rankit plot in Excel. 

Double photoionization spectra from TPEsCO are simple one-axis graphs, but complete data sets from double photoionization are inherently three-dimensional intensity as a function of two electron energies. Presentation is in the form of two-axis maps, with grayscale or color points to indicate intensity, or by projection into normal spectra of intensity against a single variable. Many choices of axes are possible, and different authors prefer... 

As was done with the log-normal distribution, it is desired to devise a set of coordinates for graph paper on which a set of data that conforms to a Weibull distribution will plot as a straight line. To do so we can work with the complementary distribution function, Fwic). The complementary distribution function of the Weibull distribution is... 

From such measurements, surface areas (normalized cumulative and relative), pore radii (choice of three measuring units), pore volumes (raw, normalized, cumulative and relative) and pore-size distribution functions of samples can calculated. Figure 8 presents the graphs of mercury-penetrated volume versus pressure in pores of Na- and La-montmorillonite samples. Figure 9 shows pore-size distribution functions from porosimetry data. 

If the particle size distribution is normal or log normal, then the data can be linearized by plotting the particle frequency as a function of particle rize on arithmetic or logarithmic probability graph p r respectively. The 50% value of sudi plots yields the geometric median diameter and the geometric standard deviation is the ratio of the 84.1% m the 50% values. 

Control graphs and the concept of rational subgroups can be used successfully in the search and elimination of outliers in harmonics present in electrical systems, provided that the data set follows a normal distribution. When data do not follow a normal distribution. Functional Data Analysis can be utilized effectively in the detection of outliers, also contributing major advantages in the detection of specific variability compared to traditional techniques such as Statistical Control Process. The functional approach greatly enhances the capacity for analysis facilitating massive and systematic analysis of the data. 

Figure 10.20 Normalized order parameter (s is the original order parameter) of aligned PLC networks with different molar contents of crosslinks as shown in the graph as a function of temperature during the first heating and cooling scans. The order parameter was assessed by IR spectroscopy. (After data of Sahlen et al. [129].)...

Plot the data given in (A), (B), and (C) to determine the functional relation between the two variables. That is, obtain an equation to express the second variable as a function of the first variable. Each set of data will yield a straight line when plotted on one of three types of graph paper - normal, semilog, or log-log. (See Appendix D.)... 

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