Distribution curves examples

Fig. 4 Typical differential molar mass distribution curve, example of polystyrene ----------------wide...

In using the table for pore size calculations, it is necessary to read off the values of the uptake from the experimental isotherm for the values of p/p° corresponding to the different r values given in the table. Unfortunately, these values of relative pressure do not correspond to division marks on the scale of abscissae, so that care is needed if inaccuracy is to be avoided. This difficulty can be circumvented by basing the standard table on even intervals of relative pressure rather than of r but this then leads to uneven spacings of r . Table 3.6 illustrates the application of the standard table to a specific example—the desorption branch of the silica isotherm already referred to. The resultant distribution curve appears as Curve C in Fig. 3.18. 

Since significance tests are based on probabilities, their interpretation is naturally subject to error. As we have already seen, significance tests are carried out at a significance level, a, that defines the probability of rejecting a null hypothesis that is true. For example, when a significance test is conducted at a = 0.05, there is a 5% probability that the null hypothesis will be incorrectly rejected. This is known as a type 1 error, and its risk is always equivalent to a. Type 1 errors in two-tailed and one-tailed significance tests are represented by the shaded areas under the probability distribution curves in Figure 4.10. 

Interpreting Control Charts The purpose of a control chart is to determine if a system is in statistical control. This determination is made by examining the location of individual points in relation to the warning limits and the control limits, and the distribution of the points around the central line. If we assume that the data are normally distributed, then the probability of finding a point at any distance from the mean value can be determined from the normal distribution curve. The upper and lower control limits for a property control chart, for example, are set to +3S, which, if S is a good approximation for O, includes 99.74% of the data. The probability that a point will fall outside the UCL or LCL, therefore, is only 0.26%. The... 

For example, the proportion of the area under a normal distribution curve that lies to the right of a deviation of 0.04 is 0.4840, or 48.40%. The area to the left of the deviation is given as 1 - P. Thus, 51.60% of the area under the normal distribution curve lies to the left of a deviation of 0.04. When the deviation is negative, the values in the table give the proportion of the area under the normal distribution curve that lies to the left of z therefore, 48.40% of the area lies to the left, and 51.60% of the area lies to the right of a deviation of -0.04. 

The variability or spread of the data does not always take the form of the true Normal distribution of course. There can be skewness in the shape of the distribution curve, this means the distribution is not symmetrical, leading to the distribution appearing lopsided . However, the approach is adequate for distributions which are fairly symmetrical about the tolerance limits. But what about when the distribution mean is not symmetrical about the tolerance limits A second index, Cp, is used to accommodate this shift or drift in the process. It has been estimated that over a very large number of lots produced, the mean could expect to drift about 1.5cr (standard deviations) from the target value or the centre of the tolerance limits and is caused by some problem in the process, for example tooling settings have been altered or a new supplier for the material being processed. 

As temperature increases, the speed of the molecules increases. The distribution curve for molecular speeds (Figure 5.9) shifts to the right and becomes broader. The chance of a molecule having a very high speed is much greater at 1000°C than at 25°C. Note, for example,... 

An illuminating example is the effect of Ostwald ripening on pore size distribution in a sintered body, resulting from vacancy transfer from the smaller to the larger pores, where the decrease in the number and the increase in average diameter of the pores can be clearly seen. The distribution curve for... 

The pore size distributions of the molded monoliths are quite different from those observed for classical macroporous beads. An example of pore size distribution curves is shown in Fig. 3. An extensive study of the types of pores obtained during polymerization both in suspension and in an unstirred mold has revealed that, in contrast to common wisdom, there are some important differences between the suspension polymerization used for the preparation of beads and the bulk-like polymerization process utilized for the preparation of molded monoliths. In the case of polymerization in an unstirred mold the most important differences are the lack of interfacial tension between the aqueous and organic phases, and the absence of dynamic forces that are typical of stirred dispersions [60]. 

The polymerization temperature, through its effects on the kinetics of polymerization, is a particularly effective means of control, allowing the preparation of macroporous polymers with different pore size distributions from a single composition of the polymerization mixture. The effect of the temperature can be readily explained in terms of the nucleation rates, and the shift in pore size distribution induced by changes in the polymerization temperature can be accounted for by the difference in the number of nuclei that result from these changes [61,62]. For example, while the sharp maximum of the pore size distribution profile for monoliths prepared at a temperature of 70 °C is close to 1000 nm, a very broad pore size distribution curve spanning from 10 to 1000 nm with no distinct maximum is typical for monolith prepared from the same mixture at 130°C [63]. 

The second factor which would tend to make effective ranges larger than those recorded is that these designated ranges for single items often purport to apply to only 95 per cent of the population, 16 and it seems unjustified to disregard, for example, for each item measured, the approximately 8 million people in the United States who would happen to fall near the end of the distribution curve for one item but not for another. 

Example The thermal energy distribution curves for 1,2-diphenylethane, C14H14, 5 = 3 X 28 - 6 = 78, have been calculated at 75 and 200 °C. [34] Their maxima were obtained at about 0.3 and 0.6 eV, respectively, with almost no molecules reaching beyond twice that energy of maximum probability. At 200 °C, the most probable energy roughly corresponds to 0.008 eV per vibrational degree of freedom. 

This theory clearly predicts that the shape of the polymer length distribution curve determines the shape of the time course of depolymerization. For example Kristofferson et al. (1980) were able to show that apparent first-order depolymerization kinetics arise from length distributions which are nearly exponential. It should also be noted that the above theory helps one to gain a better feeling for the time course of cytoskeleton or mitotic apparatus disassembly upon cooling cells to temperatures which destabilize microtubules and effect unidirectional depolymerization. Likewise, the linear depolymerization kinetic model could be applied to the disassembly of bacterial flagella, muscle and nonmuscle F-actin, tobacco mosaic virus, hemoglobin S fibers, and other linear polymers to elucidate important rate parameters and to test the sufficiency of the end-wise depolymerization assumption in such cases. 

Fig. 5, Nucleosome relaxation data on the two DNA minicircle series, (a)-(c) Nucleosomes were reconstituted with control (Control) or acetylated histones (Acetylated) on ALk = —2.4 to —3.3 topoisomers of pBR 351-366 bp eleven DNA minicircles series or 5S 349-363 bp ten DNA minicircles series, and relaxed in Tris (Tris) or phosphate buffer (phosphate), as described in legend to Fig. 4(a). Topoisomer relative amounts in the equilibrium distributions (see examples in Fig. 4(c)) were plotted as functions of their ALk, calculated from Eq. (4) using h = QA94 ( 0.003) and 10.47s ( 0.003) bp/turn for pBR DNA in Tris and phosphate buffers [28], respectively, and 0 = 10.538 ( 0.006) bp/turn for 5S DNA in Tris buffer [29]. Smooth curves were calculated as described in the text. [Drawn from data in Ref [28] (Acetylated/phosphate) and adapted from Fig. 3 in Ref [29] (Control/Tris).]...

Dyrssen and Sill6n [68] pointed ont that distribntion ratios obtained by conventional batchwise techniques are often too scattered to allow the determination of as many parameters as used in Examples 15 and 16. They suggested a simplified graphic treatment of the data, based on the assnmption that there is a constant ratio between successive stability constants, i.e., KJK i = 10 , and that all distribntion cnrves can be normalized so that A log Pn = where N is the number of ligands A in the extracted complex. Thns, the distribution curve log Du vs. log[A ] is described by the two parameters a and b, and the distribution constant of the complex, Tdc- The principle can be nsefnl for estimations when there is insnfficient reliable experimental data. 

The crystal radius thus has local validity in reference to a given crystal structure. This fact gives rise to a certain amount of confusion in current nomenclature, and what it is commonly referred to as crystal radius in the various tabulations is in fact a mean value, independent of the type of structure (see section 1.11.1). The crystal radius in the sense of Tosi (1964) is commonly defined as effective distribution radius (EDR). The example given in figure 1.7B shows radial electron density distribution curves for Mg, Ni, Co, Fe, and Mn on the M1 site in olivine (orthorhombic orthosilicate) and the corresponding EDR radii located by Fujino et al. (1981) on the electron density minima. 

The cumulative distribution curve of the above example shows that approximately 50% of the values are below 48.5 and 50 % are above this value. And for example it can also be seen that only about 10 % of the values are below 46,5 mg/g. Also the above mentioned typical S-shape can be seen. 

Note that the normal distribution curve has a mathematical equation and integrating the equation of this curve, for example between p — 2a and p + 2a, irrespective of the values of p and a, will always give the answer 0.954. So 95.4 per cent of the area under the normal curve is contained between p — 2a and p + 2a and it is this area calculation that also tells us that 95.4 per cent of the individuals within the population will have data values in that range. 

We noted in Section 1.5c that histograms of distributions of quantities such as particle size approach smooth distribution curves as the number of classes is increased to a very large number. Sometimes it is desirable to represent a distribution function by an analytical expression that is a continuous function of the measured variable. We consider only a few examples of such distribution functions here. 

Thus on the distribution curve belonging to a definite photon energy there exists a minimum which appears simultaneously with the rise of the slow electron maximum, but remains stationary. For example, in the case of toluene [Fig. 9(6) ], such a minimum appears when the photon energy is 10.1 e.v. and remains stationary up to that of 11.0 e.v. 

Theories based on the uniformly effective medium have the practical advantage that they can be extended quite easily to polydisperse systems (227). Viscosity master curves can be predicted from the molecular weight distribution, for example. The only new assumption is that the entanglement time at equilibrium for a chain of molecular weight M in a polydisperse system has the form suggested by the Rouse theory (15) ... 

The absorption curves given by coal macerals approached the horizontal (magnetic field strength) axis more slowly than a Gaussian distribution curve. Shape analysis (16) showed that over much of the curve, the form closely approximated a Lorentzian distribution curve, but both positive and negative deviations were found in the wings of the curves (that is, in various examples, the curves approached the axis either somewhat more or somewhat less rapidly... 

Since an a priori definition of the effective region is hardly possible, each atomic region is usually approximated by a spherical region around the atom, where the radius is taken as its ionic, atomic, or covalent bond radius. The radial distribution of electron density around an atom is also useful to estimate the effective radius of an atom, particularly in ionic crystals. In an ionic crystal, the distance from the metal nucleus to the minimum in the radial distribution curve generally corresponds to the ionic radius. As an example, the radial distribution curves around K in o-KvCrO., (85) are shown in Fig. 19a. The radial distributions of valence electrons (2p electrons) exhibit a minimum at 1.60 A for K(l) and 1.52 A for K(2), respectively. These distances correspond to the ionic radii in crystals (1.52-1.65 A)... 

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