Normal plot

Fig. 1.15 A log-normal plot. Note the irregular shape, arising from the smallness of the sample. (Courtesy, DallaValle )...

FIG. 20-16 Log -normal plot of residence-time distrihiition in Phelps Dodge mill. 

FIG. 28 Log-normal plot of relaxation time t2 vs bias for three different chain lengths (given as a parameter) and a series of medium densities [21]. 

Figures 62.8, 62.9, 62.10 show the data for generator fan failure plotted on exponential, normal and log normal hazard paper respectively. The exponential plot is a reasonably straight line which indicates that the failure rate is relatively constant over the range of the data. It should be noted that the reason the probability scale on the exponential hazard plot is crossed out is because that is not the proper way to plot data. (This will be discussed later.) The normal plot is curved concave upward which...

Although a wide choice for the other parameters occurring in Eqs. (3.64) and (3.65) is possible, their temperature dependence is small in the vicinity of T°. In practice either G/p or log (G/P) is normally plotted as some function of temperature which necessarily entails some choice for these parameters. Each case should be examined individually to ascertain the change a different choice would make, and to only rely on the results within these limits. 

In this application of the log-normal plot, note that the "n chanical" separation of "fines" has created anew particle distribution with 62 = 2 p. Even the value of 02 differs from that of the major particle distribution. In the "fines" fraction, it appears that the largest pcirticle does not exceed about 5 p. Needless to say, lamps prepared from this phosphor were inferior in brightness. Armed with this information, one could then recommend that the method of "fines" removal be changed. 

Since Pparaceii is proportional to the molecular sieving function F(r/R), the interrelationships of Ppaiacen between mannitol and sucrose, including mannitol-manitol and sucrose-sucrose, can be put into perspective via a normalized plot of F(r/R) versus r/R for control and perturbed monolayers (Fig. 12). As pointed out before, F(r/R) is a rapid, monotonically decreasing function bounded by 1.0 and zero. One observes that mannitol is less restricted by the pores of the control monolayer than the larger sucrose molecule. However, in the larger pores of the perturbed monolayer, the increase in permeability is less for mannitol than it is... 

Procedures for curve fitting by polynomials are widely available. Bell shaped curves usually are fitted better and with fewer constants by ratios of polynomials. Problem P5.02.02 compares a Gamma fit with those of other equations, of which a log normal plot is the best. In figuring chemical conversion, fit of the data at low values of Ett) need not be highly accurate since those regions do not affect the overall result very much. 

Probability plot Q-Q plot P-P Plot Hanging histogram Rootagram Poissonness plot Average versus standard deviation Component-plus-residual plot Partial-residual plot Residual plots Control chart Cusum chart Half-normal plot Ridge trace Youden plot... 

Figure 45 should be compared with the distorted plots shown previously137, 1411 (Fig. 43) for a strongly cationic polyelectrolyte. Normal plots yielding the correct molecular weight are obtained when salt is included KC1 for aqueous solutions and LiCl for methanolic ones in view of the insufficiently great solubility of KC1 in... 

While most of the Matlab listing in Main EFAl, m is close to self explanatory, a few statements might need clarification. The singular values are stored in the matrix EFA f which has ns rows and ne columns. It is advantageous to plot the logarithms of the singular values their values span several orders of magnitude and cannot be represented in a normal plot. 

Daniel, C. (1959). Use of half-normal plots in interpreting factorial two-level experiments. Technometrics 1, 311-341. 

Zahn, D. A. (1975). Modifications of and revised critical values for the half-normal plot. Technometrics 17, 189-200. 

Another factor that is of great importance for the observed sulfur isotope variations of natural sulfides is whether sulfate reduction takes place in an open or closed system. An open system has an infinite reservoir of sulfate in which continuous removal from the source produces no detectable loss of material. Typical examples are the Black Sea and local oceanic deeps. In such cases, H2S is extremely depleted in " S while consumption and change in " S remain negligible for the sulfate. In a closed system, the preferential loss of the lighter isotope from the reservoir has a feedback on the isotopic composition of the unreacted source material. The changes in the " S-content of residual sulfate and of the H2S are modeled in Fig. 2.21, which shows that 5 S-values of the residual sulfate steadily increase with sulfate consumption (a linear relationship on the log-normal plot). The curve for the derivative H2S is parallel to the sulfate curve at a distance which depends on the magnitude of... 

When there is no replication and the design is sufficiently large either three normal plots can be constructed, one for the design variable contrasts, one for the environmental variable contrasts, and one for the... 

As was discussed with arrangement (I), it is possible to split the two degrees of freedom for Temperature and Humidity into linear and quadratic contrasts and to construct a normal probability plot for the environmental variable contrasts. This would reveal important effects due to the linear components of both Temperature and Humidity. A normal plot for the design contrasts would indicate that there appears to be a real effect due to A. The analysis of the design x environment interactions is obtained by pooling together higher-order interactions to obtain an... 

Figure 13 Normal plot of the effects of the two level full factorial 2 design for the HPLC example...

Normalized plots of 6/6p against x for all available data for this solvent system are shown in Figure 3. It is immediately clear from the curvature of the lines in Figure 3 that Rb+, Cs+, and F are preferentially solvated by peroxide, that Li+ is preferentially solvated by water, and hardly any preferential solvation exists for Na+ and Cl . 

Half-normal plots are a modification in which the absolute value of X is used rather than the actual value itself (20). This technique has the benefit of a plot whose X axis starts at 0. 

Use of Half-Normal Plots with Factorial Data. The application of this method to the factorial data is straightforward. If, for any given compound, the data from the factorial experiment occurred simply as the result of random variation about a fixed mean, and the changes in the levels of the variables had no real effect at all on the percent recovery, then the 15 main effects and interactions, representing 15... 

The real power of the use of half-normal probability plots, however, comes with data that are likely to have embedded outliers. These data profoundly distort the half-normal plots, as illustrated with the data for methyl isobutyl ketone shown in Figure 9. The plot shows neither normal random error nor significant effects cleanly. Thus, this... 

Compound recovery data for duplicate runs differed by 2-15, depending on the compound. Half-normal probability plot analysis of the new data for the anomalous compounds indicated none of the distortion encountered earlier. Results for acetone and tetrachloroethylene now indicated only random variation with no significant outliers. Results for 2,4-dichlorophenol and 2,5-dichlorophenol indicated a significant pH effect. A significant interaction effect (AB) was detected between variables pH and primary column type for the dichlorophenols and also for methyl isobutyl ketone. This interaction effect indicates that at approximately low pH (pH 2), compound recoveries for dichlorophenols will be greater when a C18 phase is used as the primary column. The half-normal plot for 2,5-dichlorophenol is shown in Figure 10. In examining data for all the compounds from the 23 replicate factorials, this interaction consistently appears for phenolic compounds. 

Analysis of data from the factorials indicates that pH has a consistently significant effect on compound recoveries. A summary of the effect of pH level on compounds used in the study is given in Table VI. There is also an interaction between pH and primary column sorbent type for some compounds. This interaction suggests that at low sample pH, a C18 column will produce the best extraction efficiencies for phenolic compounds. The effect of adding methanol to the sample before extraction clearly produced odd results when the recovery data from the 24 factorial was analyzed by using half-normal plots. This effect will be studied in future work. Additionally, different elution solvents will be examined as well as new sorbent phases as they become available. 

Fig. 5. Steady-state cytosolic adenine nucleotide concentrations. Plot of equations (10)-(14) as a function of the degree of coupling in the interval qe(0.9,l). Values of the parameters AGj hos = 8.5 kcal/mole, AGak = 0.15 kcal/mole, Pj = 0.008 M, Xa = 50 kcal/mole, Z = 3, 0 = l. Inserted points experimental values from perfused livers.5 Normalized plots with 2=1.

Mayr initially defined a set of electrophilic parameters for the benzhydryl cations using a reference nucleophile, which was chosen as 2-methyl-1-pentene.268,269 Values of E were then defined as log k/k0, where k0 refers to a reference electrophile (E= 0), which was taken as the 4,4 -dimethoxybenzhydryl cation. Plots of log k against E for other alkenes are thus analogous to the plots of logk against p fR in Fig. 7 except that the correlation is referenced to kinetic rather than equilibrium measurements. However, they differ from plots based on the Swain-Scott or Ritchie relationships in which log k is normally plotted against a nucleophilic parameter, that is, n or N+, rather than E. 

Normalized plots of several power feed profiles are shown in Figure 2. For the special case where x = 1,(Wi=W2,R]=2R2), the feed profile is linear with time. Curvature is introduced by suitable changes of the initial monomer weights in the two tanks when x > 1, the curve is concave to the abscissa when x < 1, the curve is convex. With proper mixing in the near tank, these feed profiles can be verified experimentally. 

The 1 /e widths are found to follow the linear relationship Xl/e = 0.02 Z + 0.12. It is apparent that the normalized plots are fairly similar to one another. The IAM curves may be fitted to an rms deviation of below 5% by a universal free atom curve comprising a Gaussian (accounting for the central region) and a Lorenzian (describing the wings). 

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