Spread of distribution

The major benefit of 2nd-order Monte Carlo analysis is that it allows analysts to propagate their uncertainty about distribution parameters in a probabilistic analysis. An analyst need not specify a precise estimate for an uncertain parameter value simply because one is needed to conduct the simulation. The relative importance of our inability to precisely specify values for constants or distributions for random variables can be determined by examining the spread of distributions in the output. If the spread is too wide to promote effective decision making, then additional research is required. 

It is often convenient to summarize the features of an aerosol distribution using one or two of its properties (mean particle size, spread of distribution) than by using the full function nN(Dp). Growth of particles corresponds to a shifting of parts of the distribution to larger sizes or simply an increase of the mean particle size. These properties are called the moments of the distribution, and the two most often used are the mean and the variance. 

The residence time distribution curve (RTD) can be inscribed by its statistical moments, of which the centroid of distribution T and spread of distribution a are the most important numerical values. Thus, for a C curve, the zeroth moment is... 

Speed depends on spread of distribution, number of size intervals required to characterize suspension assumes PSD with range from 2 to 100 /xm includes sample preparation and instrument time. 

Much work has been done using those quantities as basic measures, not only for quantifying the level of spreading of distributions but also for many other applications, such as, for instance, maximum-entropy estimation and reconstruction of an unknown distribution from very limited information on it. 

Therefore, the absorjDtion line is massively inlromogeneously broadened at low temperature. An inliomogeneous lineshape can be used to detennine the static or quasistatic frequency spread of oscillators due to a distribution of environments, but it provides no dynamical infonnation whatsoever [94, 95]. As T is increased to 300 K, the absorjDtion linewidth decreases and increases. At 300 K, the lineshape is nearly homogeneously broadened and dominated by vibrational dephasing, because fast dephasing wipes out effects of inliomogeneous environments, a well known phenomenon tenned motional narrowing [951. 

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. 

Developing a meaningful method for reporting an experiment s result requires the ability to predict the true central value and true spread of the population under investigation from a limited sampling of that population. In this section we will take a quantitative look at how individual measurements and results are distributed around a central value. 

In connection with Eq. (1.4), we noted that the standard deviation measures the spread of a distribution now we see that the ratio M /M also measures this polydispersity. The relationship between these two different measures of polydispersity is easily shown. Equation (1.14) may be written as... 

The complete characterization of a particulate material requires development of a functional relationship between crystal size and population or mass. The functional relationship may assume an analytical form (7), but more frequentiy it is necessary to work with data that do not fit such expressions. As such detail may be cumbersome or unavailable for a crystalline product, the material may be more simply (and less completely) described in terms of a single crystal size and a spread of the distribution about that specified dimension. 

The coefficient of variation (cp) of a distribution is a measure of the spread of the distribution about some characteristic size. It is often used in conjunction with dominant size to characterize crystal populations through the equation... 

Figure 15 shows how the population density function changes with the addition of classified-fines removal. It is apparent from the figure that fines removal increases the dominant crystal size, but it also increases the spread of the distribution. 

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. 

In this thesis we suggest a model of the formation of elements, which allows us to make the connection between the spread of elements and their nuclear char ge. We ve got a function of distribution for the state close to the final period of active formation of elements and for the state, corresponding to the period of forming of condensed bodies [2, 3]. 

The variability or spread of the data does not always take the form of the true Normal distribution of course. There can be skewness in the shape of the distribution curve, this means the distribution is not symmetrical, leading to the distribution appearing lopsided . However, the approach is adequate for distributions which are fairly symmetrical about the tolerance limits. But what about when the distribution mean is not symmetrical about the tolerance limits A second index, Cp, is used to accommodate this shift or drift in the process. It has been estimated that over a very large number of lots produced, the mean could expect to drift about 1.5cr (standard deviations) from the target value or the centre of the tolerance limits and is caused by some problem in the process, for example tooling settings have been altered or a new supplier for the material being processed. 

Where is the mean pore size and is a constant reflecting the spread of the distribution. The probability that one tip of a single defect will cause failure under applied stress o may now be defined from equations (48) and (49) as ... 

The magnitude of the variance a represents the square of the distribution spread and has the units of (time). The greater the value of this moment, the greater the spread of the RTD. The variance is particularly useful for matching experimental curves to one of a family of theoretical curves. 

The distribution of tracer molecule residence times in the reactor is the result of molecular diffusion and turbulent mixing if tlie Reynolds number exceeds a critical value. Additionally, a non-uniform velocity profile causes different portions of the tracer to move at different rates, and this results in a spreading of the measured response at the reactor outlet. The dispersion coefficient D (m /sec) represents this result in the tracer cloud. Therefore, a large D indicates a rapid spreading of the tracer curve, a small D indicates slow spreading, and D = 0 means no spreading (hence, plug flow). 

The complete mathematical definition of a particle size distribution is often cumbersome and it is more convenient to use one or two single numbers representing say the mean and spread of the distribution. The mean particle size thus enables a distribution to be represented by a single dimension while its standard deviation indicates its spread about the mean. There are two classes of means ... 

The eoeffieient of variation is a statistieal measure of the spread of the distribution about the mean. In effeet, it is a seeond term to quantify a distribution in terms of its shape sinee many distributions may have a eommon mean size (in some respeet) but differ eonsiderably in the spread about that mean. 

While it is possible to reeover the original distribution from the leading moments, e.g. by matrix inversion, it is often the ease that simply the mean eharaeteristies together with an estimate of the spread of the distribution are suffieient e.g. 

Verbreitung, /. spreading, dissemination, distribution circulation (of journals) range. Verbreittmgsmittel, n. means of dissemination or distribution, verbrennbar, a. combustible. 

In the previous section (4.8) it has been shown that the spread of a series of results obtained from a given set of measurements can be ascertained from the value of the standard deviation. However, this term gives no indication as to the manner in which the results are distributed. 

Calculation of dependence of o on the conducting filler concentration is a very complicated multifactor problem, as the result depends primarily on the shape of the filler particles and their distribution in a polymer matrix. According to the nature of distribution of the constituents, the composites can be divided into matrix, statistical and structurized systems [25], In matrix systems, one of the phases is continuous for any filler concentration. In statistical systems, constituents are spread at random and do not form regular structures. In structurized systems, constituents form chainlike, flat or three-dimensional structures. 

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