Distributions median

For a given bulk solid, determine particle size distribution, median particle diameter dv50 (e.g., using a Coulter Counter or a Malvern Laser Diffraction Analyser) and ps. 

The pore diameter on the abscissa is calculated by employing a particular pore model, usually to the intrusion branch. As a matter of convenience, a cylindrical pore model is traditionally applied. On the ordinate, steep changes in the cumulative diagram are reflected as peak maxima in the incremental curve. From several possible representations (incremental, differential, log differential), the log differential plot seems to be the most revealing, since the areas under the peaks are proportional to the pore volume [79]. Data that can easily derived from mercury intrusion are the pore size distribution, median or average pore size, pore volume, pore area, bulk and skeletal density, and porosity. 

Figure 3. An example of the log-normal distribution function in normalized linear form for CMD = 1.0 and Og = 2.0, showing the mode, median and mean of the size distribution, the surface area distribution median and mean diameters, the mass distribution median and mean diameters, and the diameter of average mass. Reproduced with permission from Raabe OG (1970). Generation and characterization of aerosols. In Inhalation Carcinogenesis (MG Hanna, P Nettersheim and JR Gilbert, eds), pp. 123-172. Proceedings of a Biology Division, Oak Ridge National Laboratory Conference. Oak Ridge, TN, USA US Atomic Energy Commission...

One can calculate the acceptance criteria for the particle size distribution median and standard deviation by use of a technique described by Hahn and Meeker (11). Their work describes three types of statistical interval confidence, prediction, and tolerance. The authors maintain that the choice of the appropriate statistical interval to use depends on the nature of the parameters to be estimated. The confidence interval is used when hying to find bounds on a population parameter—for example, the population mean or standard deviation. The confidence interval is the most commonly appearing of the three intervals and is the interval... 

The variability of radiocesium concentration in undisturbed environments such as alpine pastures is very high and a high minimum of soil and plant samples would be required to estimate the radiocesium distribution median with a 95% confidence interval and a tolerable error of at least 20%. Low cost sampling with only a few samples would result in a biased estimate of the real radionuclide contamination. There is some evidence that the high variability is caused more by microscale runoff phenomenon directly after the Chernobyl accident rather than by long-term erosion processes. The high spatial variability of the Cs, the random distribution and the concentration in the first centimeter of the soil possibly causes the missing correlation of the radiocesium activity concentrations in soils and plants and therefore does not allow a soil-to-plant transfer factor to be calculated. 

Errors affecting the distribution of measurements around a central value are called indeterminate and are characterized by a random variation in both magnitude and direction. Indeterminate errors need not affect the accuracy of an analysis. Since indeterminate errors are randomly scattered around a central value, positive and negative errors tend to cancel, provided that enough measurements are made. In such situations the mean or median is largely unaffected by the precision of the analysis. 

The term essentially a drag coefficient for the dust cake particles, should be a function of the median particle size and particle size distribution, the particle shape, and the packing density. Experimental data are the only reflable source for predicting cake resistance to flow. Bag filters are often selected for some desired maximum pressure drop (500—1750 Pa = 3.75-13 mm Hg) and the cleaning interval is then set to limit pressure drop to a chosen maximum value. 

Refractories. Calcined alumina is used in the bond matrix to improve the refractoriness, high temperature strength/creep resistance, and abrasion/corrosion resistance of refractories (1,2,4,7). The normal, coarse (2 to 5 )J.m median) crystalline, nominally 100% a-Al202, calcined aluminas ground to 95% —325 mesh mesh are used to extend the particle size distribution of refractory mixes, for alumina enrichment, and for reaction with... 

Size distributions can also be reduced to a single average diameter, such as the mean, median, or mode. 

Distribution Averages. The most commonly used quantities for describing the average diameter of a particle population are the mean, mode, median, and geometric mean. The mean diameter, d, is statistically calculated and in one form or another represents the size of a particle population. It is usefiil for comparing various populations of particles. 

The median particle diameter is the diameter which divides half of the measured quantity (mass, surface area, number), or divides the area under a frequency curve ia half The median for any distribution takes a different value depending on the measured quantity. The median, a useful measure of central tendency, can be easily estimated, especially when the data are presented ia cumulative form. In this case the median is the diameter corresponding to the fiftieth percentile of the distribution. 

Median Diameter. The median droplet diameter is the diameter that divides the spray into two equal portions by number, length, surface area, or volume. Median diameters may be easily determined from cumulative distribution curves. 

The probabihty-density function for the normal distribution cui ve calculated from Eq. (9-95) by using the values of a, b, and c obtained in Example 10 is also compared with precise values in Table 9-10. In such symmetrical cases the best fit is to be expected when the median or 50 percentile Xm is used in conjunction with the lower quartile or 25 percentile Xl or with the upper quartile or 75 percentile X[j. These statistics are frequently quoted, and determination of values of a, b, and c by using Xm with Xl and with Xu is an indication of the symmetry of the cui ve. When the agreement is reasonable, the mean v ues of o so determined should be used to calculate the corresponding value of a. 

In practice most distribution cuiwes are not symmetrical about the median but are inherently skewed. The effect of an advertising campaign is usually to increase the rate of sales in the early years. It may also increase the level of mature demand for the product, but this mature demand must be asymptotic to a finite upper Emit of sales c. Such a cui ve is positively skewed since xm — xd) < x(j —x ). This situation can often be approximated by the Gompertz cui ve defined by Eq. (9-96) ... 

When the estimates are well founded, the skewness may be preserved by using a distribution such as the Gompertz. The median of that curve occurs a.sy = 0.5 c, while the point of inflexion corresponds to the mode at y = c/exp (1) = 0.3679 c. The statistician Karl Pearson suggested as a simple measure of skewness... 

The Gompertz distribution requires the distribution to be positively skewed, which can be achieved by treating —(NPV) as the independent variable and (c — y) as the dependent variable. From Eq. (9-104) the median of the distribution is given approximately as... 

When a distribufion of particle sizes which must be collected is present, the aclual size distribution must be converted to a mass distribution by aerodynamic size. Frequently the distribution can be represented or approximated by a log-normal distribution (a straight line on a log-log plot of cumulative mass percent of particles versus diameter) wmich can be characterized by the mass median particle diameter dp5o and the standard statistical deviation of particles from the median [Pg.1428]

Average Particle Size A powder has many average sizes hence it is essential that they be well specified. The median is the 50 percent size half the distribution is coarser and half finer. The mode is a high-density region if there is more than one peak in the frequency cui ve, the distribution is said to be multimodal. The mean is the center of gravity of the distribution. The center of gravity of a mass (volume) distribution is defined by. Xyw = X XdV/X dV where dV = X dN dV is the volume of dN particles of size X This is defined as the volume-moment mean diameter and differs from the mean for a number or surface distribution. 

The analysis of the frequeney data is shown in Table 4.12. Note the use of the Median Rank equation, eommonly used for both Weibull distributions. Linear reetifieation equations provided in Appendix X for the 2-parameter Weibull model are used to... 

Distributions are characterized by measures of central tendency The median is the value of X (e.g., crap scores) that divides the distribution into equal areas. The value of x at the peak... 

The justification for the use of the lognormal is the modified Central Limit Theorem (Section 2.5.2.5). However, if the lognormal distribution is used for estimating the very low failure frequencies associated with the tails of the distribution, this approach is conservative because the low-frequency tails of the lognormal distribution generally extend farther from the median than the actual structural resistance or response data can extend. 

FIGURE 5.28 Estimated overall airway deposition as a function of initial particle size and particle hygroscopicity for particles with mass median aerodynamic diameters (MMAD) between 0.1 and 10 p.m. ° Geometric dispersion, a measure of particle size distribution, principally affects only smaller MMAD,... 

The size of inhaled particles varies markedly. The size distribution approximates a log-normal distribution that can be described by the median or the geometric mean, and by the geometric standard deviation. For fibers, both... 

The differential and eumulative size distributions are elearly related, as shown in Figure 1.9. Differentiating the eumulative distribution restores the original histogram but in a smoother form. Two important properties ean be defined, the modal and median sizes. 

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