Deviations, size

Physical Properties Density average and standard deviation Size average and standard deviation Fluorescence Fluorescence excitation spectrum Scattered light spectrum Absorption spectrum (from microwave to UV) Raman spectrum Electrical conductivity, impedance Acoustic properties... 

TABLE 3-1 Size [(d (average) d (standard deviation)], Size Distribution (Ad/d). Saturation Magnetization (Af,), and Residual Magnetization (Mr) for Magnetite Particles at Various Temperatures... 

The second important atomistic feature in the chemistry of irradiated nuclear fuel is the size of the fission product atom, i. e. the question whether or not a given atom is compatible with the dimensions of the UO2 lattice. Crystal lattices are able to tolerate foreign atoms of deviating size, in particular at trace concentrations ... 

Table 2 compares between the VIGRAL results and mechanical measurements of the simulated FBH defects. The table lists the size of the reflecting surface,, its depth location, the Yd, and the standard deviation of the depth information, o>i( y ). We note an excellent agreement between the VIGRAL and the mechanical measurements both in size and depth of the defects. 

During the calibration the geometric distortions in the images are reduced from a maximum of more than 11 pixels to less than 0.3 pixels (pixel size about 0.28mm 0.28mm). The whole calibration results in a maximum deviation of the projected calibration marks from the measured calibration marks in the image of less than 0.4 pixel (2D-error). 

The combined result of two such determinations yielded a leak size figure of 8.8% of the feed flow (with a relative standard deviation of less than 5%). This figure could sufficiently explain the product quality problems experienced, whose alternative explanation in turn was catalyst poisoning. 

Another way to improve the error in a simulation, at least for properties such as the energy and the heat capacity that depend on the size of the system (the extensive properties), is to increase the number of atoms or molecules in the calculation. The standard deviation of the average of such a property is proportional to l/ /N. Thus, more accurate values can be obtained by running longer simulations on larger systems. In computer simulation it is unfortunately the case that the more effort that is expended the better the results that are obtained. Such is life ... 

In the next several sections, the theoretical distributions and tests of significance will be examined beginning with Student s distribution or t test. If the data contained only random (or chance) errors, the cumulative estimates x and 5- would gradually approach the limits p and cr. The distribution of results would be normally distributed with mean p and standard deviation cr. Were the true mean of the infinite population known, it would also have some symmetrical type of distribution centered around p. However, it would be expected that the dispersion or spread of this dispersion about the mean would depend on the sample size. 

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... 

Confidence intervals also can be reported using the mean for a sample of size n, drawn from a population of known O. The standard deviation for the mean value. Ox, which also is known as the standard error of the mean, is... 

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. 

Because of the diversity of filler particle shapes, it is difficult to clearly express particle size values in terms of a particle dimension such as length or diameter. Therefore, the particle size of fillers is usually expressed as a theoretical dimension, the equivalent spherical diameter (esd), ie, the diameter of a sphere having the same volume as the particle. An estimate of regularity may be made by comparing the surface area of the equivalent sphere to the actual measured surface area of the particle. The greater the deviation, the more irregular the particle. 

The particle sizes of fillers are usually collected and ordered to yield size distributions which are frequendy plotted as cumulative weight percent finer than vs diameter, often given as esd, on a log probabiUty graph. In this manner, most unmodified fillers yield a straight-line relationship or log normal distribution. Inspection of the data presented in this manner can yield valuable information about the filler. The coarseness of a filler is often quantified as the esd at the 99.9% finer-than value. Deviations from linearity at the high and low ends of the plot suggest that either fractionation has occurred to remove coarse or fine particles or the data are suspect in these ranges. 

The quantity of sample required comprises two parts the volume and the statistical sample size. The sample volume is selected to permit completion of all required analytical procedures. The sample size is the necessary number of samples taken from a stream to characterize the lot. Sound statistical practices are not always feasible either physically or economically in industry because of cost or accessibiUty. In most sampling procedures, samples are taken at different levels and locations to form a composite sample. If some prior estimate of the population mean, and population standard deviation. O, are known or may be estimated, then the difference between that mean and the mean, x, in a sample of n items is given by the following ... 

If the standard deviation of the lot caimot be estimated, a sampling program of greater sample size is required to generate an estimate of the standard deviation for future sampling operations. In some cases, sample size can be increased and sampling costs reduced by the use of automatic samplers. These offer a substantial reduction in labor costs but an increase in capital costs. 

Since the t distribution relies on the sample standard deviation. s, the resultant distribution will differ according to the sample size n. To designate this difference, the respec tive distributions are classified according to what are called the degrees of freedom and abbreviated as df. In simple problems, the df are just the sample size minus I. In more complicated applications the df can be different. In general, degrees of freedom are the number of quantities minus the number of constraints. For example, four numbers in a square which must have row and column sums equal to zero have only one df, i.e., four numbers minus three constraints (the fourth constraint is redundant). 

Chi-Square Distribution For some industrial applications, produrt uniformity is of primary importance. The sample standard deviation. s is most often used to characterize uniformity. In dealing with this problem, the chi-square distribution can be used where = (.s /G ) (df). The chi-square distribution is a family of distributions which are defined by the degrees of freedom associated with the sample variance. For most applications, df is equal to the sample size minus 1. 

---