Data distribution bimodal

Afterfilter data. As indicated in Table I, the minimum D50 in this study was about 0.5 pm, and particles smaller than this were collected on an afterfilter. Aerosols from combustion of pulverized coal typically are distributed bimodally, with a fine-particle mode at about 0.1 pm and a large-particle mode at supermicrometer sizes the modal diameter of the latter depends strongly on the efficiency characteristics of the control device. The elemental concentrations in the fine-particle mode are of interest in health-impact and source-apportionment studies because of the typically high enrichment of the concentrations of many potentially toxic elements and useful tracer elements in particles in this size range. Large-particle con-taimination of the afterfilter due to particle bounce can, however, limit the value of these data. 

Data with a polymodal distribution If we give a series of patients a standard oral dose of the anti-tuberculosis drug isoniazid, obtain a blood sample from each individual 6 h later and determine the isoniazid concentrations of those samples, the results will probably look like Figure 3.1. The data are bimodal, because the metabolism of isoniazid is genetically controlled and we all fall into one of two groups - fast or slow metabolizers. The fast metabolizers form the cluster at the low end of the concentration scale and the slow metabolizers form a distinct group with higher levels. 

Prior to a formal analysis, a database should be examined for any unusual characteristics of the data distribution. A database may have some number of outliers, an inherent nonnormal or skewed distribution, or a bimodal character due to the presence of two separate underlying distributions. Most tests for normality are intended for fairly large sample sizes of the order of 15 or more. Smaller databases may be reviewed for unusual characteristics by way of the usual statistical algorithms available with spreadsheets. Tests for normality are listed in Part 2. 

Figure A3.9.3. Time-of-flight spectra for Ar scattered from Pt(l 11) at a surface temperature of 100 K [10], Points in the upper plot are actual experimental data. Curve tinough points is a fit to a model in which the bimodal distribution is composed of a sharp, fast moving (lienee short flight time), direct-inelastic (DI) component and a broad, slower moving, trapping-desorption (TD) component. These components are shown...

The mode of distribution is simply the value of the most frequent size present. A distribution exhibiting a single maximum is referred to as a unimodal distribution. When two or more maxima are present, the distribution is caUed bimodal, trimodal, and so on. The mode representing a particle population may have different values depending on whether the measurement is carried out on the basis of particle length, surface area, mass, or volume, or whether the data are represented ia terms of the diameter or log (diameter). 

One further point might be made here. Although the example illustrates the difference between the two types of molecular weight average, the weight average molecular weight in this example cannot be said to be truly representative, an essential requirement of any measure of central tendency. In such circumstances where there is a bimodal, i.e. two-peaked, distribution additional data should be provided such as the modal values (100 and 100000 in this case) of the two peaks. 

Figure 1 and Table 1 show the data obtained using PIB s prepared with the f-BuCl/ Et2AlCl/MeCl system in the -30° to -65 °C range. PIB s prepared in the range from —30° to —45 °C have monomodal MWD while those prepared below -45 °C exhibit bimodal distributions. MWD s broaden with decreasing polymerization temperature. 

One characteristic of shear banded flow is the presence of fluctuations in the flow field. Such fluctuations also occur in some glassy colloidal materials at colloid volume fractions close to the glass transition. One such system is the soft gel formed by crowded monodisperse multiarm (122) star 1,4-polybutadienes in decane. Using NMR velocimetry Holmes et al. [23] found evidence for fluctuations in the flow behavior across the gap of a wide gap concentric cylindrical Couette device, in association with a degree of apparent slip at the inner wall. The timescale of these fluctuations appeared to be rapid (with respect to the measurement time per shear rate in the flow curve), in the order of tens to hundreds of milliseconds. As a result, the velocity distributions, measured at different points across the cell, exhibited bimodal behavior, as apparent in Figure 2.8.13. These workers interpreted their data... 

Our initial studies (23) were performed in toluene, and Table I shows the results from the polymerization of a number of representative monomers. The data reported in Table I are for direct addition of the monomer to the sodium dispersion. Inverse addition often leads to higher molecular weights, although the overall polymer yields are usually lower (15,23). The results in Table I show that, under these reaction conditions, a bimodal molecular molecular weight distribution is normally obtained. Furthermore, it is obvious that the crude polymer yields drop precipitously as the steric hindrance in the monomer increases. 

The bimodal feature in the lower wavelength images is accounted for by the presence of the two processes for which evidence has been presented above. There is also a clear bimodality in the data from 266 nm dissociation. The energy distribution displays peaks at 2.7 and 3.6 eV. This distribution raises further questions since the process creating > 26) at lower... 

In Fischer-Tropsch synthesis the readsorption and incorporation of 1-alkenes, alcohols, and aldehydes and their subsequent chain growth play an important role on product distribution. Therefore, it is very useful to study these reactions in the presence of co-fed 13C- or 14 C-labeled compounds in an effort to obtain data helpful to elucidate the reaction mechanism. It has been shown that co-feeding of CF12N2, which dissociates toward CF12 and N2 on the catalyst surface, has led to the sound interpretation that the bimodal carbon number distribution is caused by superposition of two incompatible mechanisms. The distribution characterized by the lower growth probability is assigned to the CH2 insertion mechanism. 

The L-mode analysis of the simulated taxonic data produced a bimodal distribution, while analyses of the simulated nontaxonic data and the actual data revealed only one mode. What this means is unclear. It is possible that depression is actually taxonic, but the simulated taxonic data does not adequately reflect the structure of the actual data set. For example, higher nuisance correlations and lower indicator validities in the actual data may not have allowed a latent bimodality to emerge, even though this was detected in the simulated data. Another possibility, of course, is that the BDI items do not define a taxon. 

The basis of all performance criteria are prediction errors (residuals), yt - yh obtained from an independent test set, or by CV or bootstrap, or sometimes by less reliable methods. It is crucial to document from which data set and by which strategy the prediction errors have been obtained furthermore, a large number of prediction errors is desirable. Various measures can be derived from the residuals to characterize the prediction performance of a single model or a model type. If enough values are available, visualization of the error distribution gives a comprehensive picture. In many cases, the distribution is similar to a normal distribution and has a mean of approximately zero. Such distribution can well be described by a single parameter that measures the spread. Other distributions of the errors, for instance a bimodal distribution or a skewed distribution, may occur and can for instance be characterized by a tolerance interval. 

Conclusions. Table II compares some characteristics of the semicontinuous and batch techniques. According to 1H NMR and GPC data the semicontinuous technique produces polymers with well defined symmetrical end groups, while the batchy polymerization may yield once-fired and unfired chain ends. The M Mn values of polymers obtained in semicontinuous runs are close to theoretical, whereas those harvested in conventional batch polymerization exhibit broader molecular weight distributions due to changing [I] and [M], presence of once-fired and/or unfired chain ends, and insufficient reactor control. The latter circumstance may even result in bimodal distributions. 

The results of image analysis of macroporous epoxies showing a narrow and bimodal pore size distribution are summarized in Table 3. The volume fraction, ( ), is always calculated from density measurements. The validity of the data obtained with digital image analysis is of utmost importance in order to draw correct conclusions concerning the structure-property relationships. 

Also shown is the capture efficiency of the human upper respiratory tract (nasopharynx). Closed symbols are derived from data on capture efficiency using monodispersed aerosols open symbols are theoretical calculations ( see Ref. 4). The lower section of the diagram is a schematic of the "Pasadena" bimodal distribution. 

Figures 11.1 and 11.2 illustrate mercury porosimetry data of a bimodal size distribution. However, other types of less typical curves are often encountered. For example, samples of controlled porous glass exhibit intrusion-extrusion curves illustrated by Fig. 11.3, in which all the pores are essentially of one radius.

Randrup et al. (55) first postulated a role for dopamine in depressive disorders. More recently, a reanalysis of the data from several groups has found evidence for a bimodal distribution of CSF homovanillic acid (HVA) levels in depressed patients, with one group comparable with normal control subjects and the other with decreased levels (56). Roy and colleagues (57) also reported on the potential predictive value of lower urinary HVA output in depressed patients who attempted suicide versus those who did not. Both reports indicate a decreased turnover in dopamine. 

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