Sample histogram

The sample histogram in Figure 2.1 provides a visual summary of the distribution of total cholesterol in a group of 100 patients at baseline (artificial data). The x-axis is divided up into intervals of width 0.5 mmol/1 and the y-axis counts the number of individuals with values within those intervals. 

We will sometimes refer to the sample histogram as the sample distribution. 

Fig. 1.4. The potential of mean force (PMF) for the Tyr -isoalloxazine center-to-center distance, (a) Sampling histograms of the Tyr -isoalloxazine distance the solid line is the histogram with a restraint-free simulation. The dashed hues are histograms obtained with umbrella potentials, (b) The PMF generated with the data from (a) is shown as a solid line. Also shown is the experimental PMF open circles) from reference [6]...

A particle size analyzer determines the particle size distribution of powders either dry or dispersed in solvent by laser light scattering based on the Fraunhofer scattering theory. This type of equipment has an optical bench whose combined dynamic range is nominally 0.7-2000 pm. The instrument calculates mean diameters and distribution data. An interfaced computer generates sample histograms. This technique has been applied to the study of particle size and particle size distributions for polymer powders and polymer suspensions in a variety of solvents. 

FIGURE 2 Location map (distance units in feet) of 180 samples, histogram of lead concentration, variogram of the normal scores transform (hence the sill value of 1.0), a map of kriging estimates on a 100-ft grid and an SGS realization over the same domain. 

It is possible to perform volume visualization with different color and opacity tables, which can be designed by the user out of the samples intensity histogram. 

Figure Bl.10.4. Time-of-flight histogram for ions resulting from the ionization of a sample of air with added hydrogen. The ions have all been aeeelerated to the same energy (2 keV) so that their time of flight is direetly proportional to the reeiproeal of the square root of tiieir mass.

The only density estimators discussed in the protein literature are histogram estimates. However, these are nonsmooth and thus not suitable for global optimization techniques that combine local and global search. Moreover, histogram estimates have, even for an optimally chosen bin size, the extremely poor accuracy of only, for a sample of size n. The theo-... 

It will be convenient to deal first with the distribution aspect of the problem. One of the clearest ways in which to represent the distribution of sizes is by means of a histogram. Suppose that the diameters of SOO small spherical particles, forming a random sample of a powder, have been measured and that they range from 2-7 to 5-3 pm. Let the range be divided into thirteen class intervals 2-7 to 2-9 pm, 2-9 to 3-1 pm, etc., and the number of particles within each class noted (Table 1.5). A histogram may then be drawn in which the number of particles with diameters within any given range is plotted as if they all had the diameter of the middle of the... 

Fig. 1.12 Histogram showing the distribution of particle sizes for the sample of powder referred to in Table 1.5. (After Herdan )...

The distribution curves may be regarded as histograms in which the class intervals (see p. 26) are indefinitely narrow and in which the size distribution follows the normal or log-normal law exactly. The distribution curves constructed from experimental data will deviate more or less widely from the ideal form, partly because the number of particles in the sample is necessarily severely limited, and partly because the postulated distribution... 

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. 

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.

The eomponent shown in Figure 4 is a spaeer from a transmission system. The eomponent is manufaetured by turning/boring at the rate of 25 000 per annum and the eomponent eharaeteristie to be eontrolled, X, is an internal diameter. From the statistieal data in the form of a histogram for 40 eomponents manufaetured, shown in Figure 5, we ean ealeulate the proeess eapability indiees, Cp and Cp. It is assumed that a Normal distribution adequately models the sample data. 

Due to its nature, random error cannot be eliminated by calibration. Hence, the only way to deal with it is to assess its probable value and present this measurement inaccuracy with the measurement result. This requires a basic statistical manipulation of the normal distribution, as the random error is normally close to the normal distribution. Figure 12.10 shows a frequency histogram of a repeated measurement and the normal distribution f(x) based on the sample mean and variance. The total area under the curve represents the probability of all possible measured results and thus has the value of unity. 

The experimental sample on which the frequency histogram is based has an experimental mean m and an experimental variance s, which are... 

Fig. 9-8 Histogram of dissolved solids of samples from the Orinoco and Amazon River basins and corresponding denudation rates for morpho-tectonic regions in the humid tropics of South America (Stal-lard, 1985). The approximate denudation scale is calculated as the product of dissolved solids concentrations, mean armual runoff (1 m/yr), and a correction factor to account for large ratios of suspended load in rivers that drain mountain belts and for the greater than average annual precipitation in the lowlands close to the equator. The correction factor was treated as a linear function of dissolved solids and ranged from 2 for the most dilute rivers (dissolved solids less than lOmg/L) to 4 for the most concentrated rivers (dissolved solids more than 1000 mg/L). Bedrock density is assumed to be 2.65 g/cm. (Reproduced with permission from R. F. Stallard (1988). Weathering and erosion in the humid tropics. In A. Lerman and M. Meybeck, Physical and Chemical Weathering in Geochemical Cycles," pp. 225-246, Kluwer Academic Publishers, Dordrecht, The Netherlands.)...

Figure 1.8. Schematic frequency distributions for some independent (reaction input or control) resp. dependent (reaction output) variables to show how non-Gaussian distributions can obtain for a large population of reactions (i.e., all batches of one product in 5 years), while approximate normal distributions are found for repeat measurements on one single batch. For example, the gray areas correspond to the process parameters for a given run, while the histograms give the distribution of repeat determinations on one (several) sample(s) from this run. Because of the huge costs associated with individual production batches, the number of data points measured under closely controlled conditions, i.e., validation runs, is miniscule. Distributions must be estimated from historical data, which typically suffers from ever-changing parameter combinations, such as reagent batches, operators, impurity profiles, etc.

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