The Histogram

Fortunately, most of the PSD instrumentation in use today employ computer-generated data to display the particle size. Furthermore, the particle-range spread be measured can be adjusted to suit the material at hand. Thus, we ean use the histogram if we so desire, especially if it has certain advantages over the other methods of displaying PSD. 

Measuring particle size and growing single crystals 

Many particle-measuring methods use Stoke s Law to determine particle distributions. By suitable manipulation (see below), we can obtain an equation relating the Stokes diameter, M, with the particle density, rj, and the liquid density,. Stokes Law is given as follows  

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Figure 6 shows the histogram of localized AE events vs axial position for the same time period as in fig.5. The location of the AE source corresponds, within source location errors (< 10-15 cm), to one of the welds under surveillance. The weld was known by ultrasonic examination to be affected by internal discontinuities. However, the position of the source could also correspond to one of the hangers. The steps observed in EA event accumulation have taken place during steady load operation, which normally corresponds to very low background noise conditions. This type of event, however, has not been observed afterwards. 

Localized AE sources appear during load variations, startups or shutdowns, but their positions are uniformly spread over the length of the two bodies of the header this can be seen from the histogram of the localized AE events for the front body (fig.S) and for the rear body (fig.9). 

We therefore use smooth density estimation techniques that are more reliable than the histogram estimates. To improve the reliability for rare amino acid pairs, we use clustering techniques that identify similar pairs that can be modeled by the same density. 

One therefore needs a smooth density estimation techniques that is more reliable than the histogram estimates. The automatic estimation poses additional problems in that the traditional statistical techniques for estimating densities usually require the interactive selection of some smoothing parameter (such as the bin size). Some publicly available density estimators are available, but these tended to oversmooth the densities. So we tried a number of ideas based on numerical differentiation of the empirical cdf to devise a better density estimator. 

The data in Table 4.12 are best displayed as a histogram, in which the frequency of occurrence for equal intervals of data is plotted versus the midpoint of each interval. Table 4.13 and figure 4.8 show a frequency table and histogram for the data in Table 4.12. Note that the histogram was constructed such that the mean value for the data set is centered within its interval. In addition, a normal distribution curve using X and to estimate p, and is superimposed on the histogram. 

Histogram for data in Tabie 4.12. A normai distribution curve for the data, based onX and s, is superimposed on the histogram. 

Vitha, M. F. Carr, P. W. A Laboratory Exercise in Statistical Analysis of Data, /. Chem. Educ. 1997, 74, 998-1000. Students determine the average weight of vitamin E pills using several different methods (one at a time, in sets of ten pills, and in sets of 100 pills). The data collected by the class are pooled together, plotted as histograms, and compared with results predicted by a normal distribution. The histograms and standard deviations for the pooled data also show the effect of sample size on the standard error of the mean. 

The histogram is a graphical device which is both attainable in practice and also an approximation to a theoretical distribution function. 

The mean can be evaluated from the classified data of the histogram it measures the center of the distribution. The mean (whose symbol is an overbar) is defined as... 

The BMS deviation is a measure of the spread of values for c around the mean. A large value of O indicates that wide variations in c occur. The probability that the controlled variable hes between the values of Cl and C9 is given by the area under the distribution between Ci and Cg (histogram). If the histogram follows a normal probabihty distribution, then 99.7 percent of aU observations should lie with 3o of the mean (between the lower and upper control limits). These Emits are used to determine the quality of control. 

While the F-N curve is a cumulative illustration, the risk profile shows the expected frequency of accidents of a particular category or level of consequence. The diagonal line is a line of constant risk defined such that the product of expected frequency and consequence is a constant at each point along the line. " As the consequences of accidents go up, the expected frequency should go down in order for the risk to remain constant. As the example illustrates, if a portion of the histogram sticks its head up above the line (i.e., a particular type of accident contributes more than its fair share of the risk), then that risk is inconsistent with the risk presented by other accident types. (Note There is no requirement that you use a line of constant risk other more appropriate risk criteria for your application can be easily defined and displayed on the graph.)... 

Figure 4 Sample spatial restraint m Modeller. A restraint on a given C -C , distance, d, is expressed as a conditional probability density function that depends on two other equivalent distances (d = 17.0 and d" = 23.5) p(dld, d"). The restraint (continuous line) is obtained by least-squares fitting a sum of two Gaussian functions to the histogram, which in turn is derived from many triple alignments of protein structures. In practice, more complicated restraints are used that depend on additional information such as similarity between the proteins, solvent accessibility, and distance from a gap m the alignment.

We ean now plot the Normal frequeney distribution superimposed over the histogram bars for eomparison. The eurve is generated using equation 15, where the variables of interest, x, are values in steps of 10 on the x-axis from, say, 380 to 540. The Normal frequeney equation is given below, and Figure 6 shows the histogram and the Normal... 

The diserete data ean be analyzed from the residenee time distribution by using either the histogram method or the trapezoidal rule... 

Sequentially adding up each individual segment on the histogram gives the Cumulative mass fraction, m d)... 

The distribution function of the vectors normal to the surfaces,/(x), for the direction of the magnetic field B, in accord with the directions of the crystallographic axis (100) for the P, D, G surfaces, is presented in Fig. 6. The histograms for the P, D, G are practically the same, as they should be the differences between the histograms are of the order of a line width. The accuracy of the numerical results can be judged by comparing the histograms obtained in our calculation with the analytically calculated distribution function for the P, D, G surfaces [29]. The sohd line in Fig. 6(a) represents the result of analytical calculations [35]. 

FIGURE 11.3 One-way ANOVA (analysis of variance). One-way analysis of variance of basal rates of metabolism in melanophores (as measured by spontaneous dispersion of pigment due to G,.-protein activation) for four experiments. Cells were transiently transfected with cDNA for human calcitonin receptor (8 j-ig/ml) on four separate occasions to induce constitutive receptor activity. The means of the four basal readings for the cells for each experiment (see Table 11.4) are shown in the histogram (with standard errors). The one-way analysis of variance is used to determine whether there is a significant effect of test occasion (any one of the four experiments is different with respect to level of constitutive activity). 

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