Types of distribution

There are four different particle size distributions for a given particulate material, depending on the quantity measured by number fi x), by length fi(x), by surface fs(x), and by mass (or volume)/M(x). From all these, the second mentioned is not used in practice as the length of a particle by itself is not a complete definition of its dimensions. These distributions are related but conversions from one another is possible only in cases when the shape factor is constant, that is, when the particle shape is independent of the particle size. The following relationships show the basis of such conversions  

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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 larger the value of n, the more uniform is the size distribution. Other types of distribution functions can be found in Reference 1. Distribution functions based on two parameters sometimes do not accurately match the actual distributions. In these cases a high order polynomial fit, using multiple parameters, must be considered to obtain a better representation of the raw data. 

When there are more than one sources of supply, it is recommended to distribute the loads also in as many sections as the incomers, and provide a tie-circuit between every two sections, to obtain more flexibility. Now fault on one section or source of supply will not result in the loss of power to the entire system. Figures 13.16 and 13.17 illustrate this type of distribution. 

The value of the eorrelation eoeffieient using the least squares teehnique and the use of goodness-of-fit tests (in the non-linear domain) together probably provide the means to determine whieh distribution is the most appropriate (Keeeeioglu, 1991). However, a more intuitive assessment about the nature of the data must also be made when seleeting the eorreet type of distribution, for example when there is likely to be a zero threshold. 

The various packings have different characteristics for distributing the liquid throughout the bed. Leva [40] show s the results of Baker, et al. [3] which illustrates the effect of various types of distribution on the liquid pattern inside the packing. A general summary is given in Table 9-20. 

The type of distribution to select depends on the sensitivity of the tower performance to the liquid distribution as discussed earlier. Norton s [83] data indicate that the sensitivity of tower performance to liquid distribution quality depends only on the number of theoretical stages in each bed of packing achierable at its System Base HETP [83]. Tower beds of high efficiency packing are more sensitive to liquid distribution quality than shorter beds of medium efficiency packing [83]. It is important to extend the uniformity of the distributor all the way to within one packing particle diameter of the tower wall [85]. 

With this type of distribution about 68 per cent of all values will fall within... 

The catalytic single-step Alfen process has a good space-time yield, and the process engineering is simple. The molecular weight distribution of the olefins of the single-step process is broader (Schulz-Flory type of distribution) than in the two-step Alfen process (Poisson-type distribution) (Fig. 2). As a byproduct 2-alkyl-branched a-olefins also are formed, as shown in Table 6. About... 

The advantage of this technique is that no assumptions need to be made about the type of distribution underlying the data and that many sets of observations can be visually compared without being distracted by the individual numbers. For further details, see Section 1.8. Note that the fraction of all... 

Following this example are distributions that do have limits to the "normal distribution. What this means is that the distributions conform to the limits defined in 5.7.6. Note that in 5.8.1., a straight line is evident. This is the type of distribution usually found as a result of most precipitation processes. But as we shall see, this is not true for the other types of log-normal distributions. 

The molecular weight distribution in Fig. 5.3 a) exhibits a most probable molecular weight distribution , which is characteristic of polymers produced by metallocene catalysts. This distribution contains relatively few molecules with either extremely high or low molecular weights. Products made with this type of distribution are relatively difficult to process in the molten state, exhibit modest orientation, and have good impact resistance. 

Assumption C, that the distribution of r values is known, is not literally correct, but there is enough information available to bound the types of distribution that are plausible, and calculations can be made for various distributions spanning these bounds. They then span the range of plausible results. 

The normal distribution describes the way measurement results are commonly distributed. This type of distribution of data is also known as a Gaussian distribution. Most measurement results, when repeated a number of times, will follow a normal distribution. In a normal distribution, most of the results are clustered around a central value with fewer results at a greater distance from the centre. The distribution has an infinite range, so values may turn up at great distances from the centre of the distribution although the probability of this occurring is very small. 

Two types of distribution coefficients are commonly measured and used in describing the distribution between solid and liquid phases. The first and simplest is the distribution between total solid and liquid phases. This can be represented by Kd, as given in the equation in Figure 5.11. Here, kg is kilogram and L is liter of soil solution. 

A number of types of distributions have been fully studied, because they, or at least close approximations to them, frequently arise in practice. In connection with the theory of measurement errors and least squares adjustments, the normal and chi-square distributions are often used, so they are briefly discussed in the following paragraphs. 

Descriptive statistics are used to summarize the general nature of a data set. As such, the parameters describing any single group of data have two components. One of these describes the location of the data, while the other gives a measure of the dispersion of the data in and about this location. Often overlooked is the fact that the choice of which parameters are used to give these pieces of information implies a particular type of distribution for the data. 

In the last two years a new initial velocity distribution of NSs became standard. It is a bimodal distribution with peaks at 130 kms-1 and 710kms 1 (Arzoumanian et al. 2002). Contributions from the low and high velocity populations is nearly equal. Brisken et al. (2003) confirm this type of distribution, however they give arguments for smaller fraction of low velocity NSs (about 20%). 

The curve is asymmetrical with a longer tail stretching off towards the more positive values. The mean, median and mode are now separated so that X is nearest the tail of the curve the mode is at the peak frequency and the median is in between the two. This type of distribution can sometimes be made normal by logarithmic transformation of the data. 

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