Distribution curve, cumulative

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. 

Table I, euid histogrcims of the distributions observed for two of the mock runs are shown in Figures 1 and 2. The corresponding cumulative distribution curves u e shown in Figure 3.

Approximate value because cumulative distribution curve for the lower diameter distribution was curved concave to the diameter axis. 

Figure 3. Cumulative distribution curves of distributions shown in Figures 1 and 2. Key , humidifier on, PN on O, humidifier off, PN on.

The effects of applying 0.48% of Texspray Compound and 1.09% of Spraycot 8853 are summarized in Table I. In both cases there is a marked drop in the number of particles (an average of 84.8%) and in dust concentration (70.5% for the Texspray and 84.8% for the Spraycot). The dg calculated algebraically decreases from 1.88 un for the control cotton to 1.44 urn for the two lubricated cottons. The cumulative probability curves of the dust emitted from the cottons containing additives become bimodal. These changes in the nature of the cumulative distribution curves suggest that the fraction of particles removed is not constant for all dieuneters. 

Table I also Illustrates the effect of adding 0.60% of Milube N-32 to a low grade, low micronaire and high trash content cotton. The histograms and cumulative distribution curves of the control and treated cottons are shown in Figures 7 and 8. The results are similar to those found for the other cottons containing additives in that there is a drop of 74.7% in particle count and of 90.8% in dust concentration. The presence of the additive lowers d from 1.62 to 1.09 un (based on the cumulative plots). The cumulative distribution of the stock cotton is linear (except for the lowest dicuneter cell) over a ramge including 99.95% of the particles and that of the additive-treated cotton is bimodal. The smaller...

Figure 5. Cumulative distribution curves of dust collected while processing 100% PET. Key O, original bimodal curve d— 1.08 jxm.a = 1.30 n. curve for distribution having the larger diameter as resolved by theTaylormethod(l ) d=2.9fim, 4 /im.

Figure 10. Cumulative distribution curves of dust in card room atmosphere while processing cotton with an electrostatic precipitator (ESP) in filtration line. Key curve for ESP energized O, curve for ESP not energized and A, size distribution removed by ESP.

Summing up all the frequencies gives the cumulative distribution. The cumulative distribution curve is the integral of the distribution curve. This curve can be used to find the proportion of the values that fall above or below a certain value. For distribution curves or histograms, which have a maximum, (typical for all analytical measurements), the cumulative distribution curves show the typical S-shape. 

For distributions (or histograms) with a maximum, the cumulative distribution curve shows the typical s-shape... 

The cumulative distribution curve of the above example shows that approximately 50% of the values are below 48.5 and 50 % are above this value. And for example it can also be seen that only about 10 % of the values are below 46,5 mg/g. Also the above mentioned typical S-shape can be seen. 

FIGURE 53.1 Cumulative distribution curves of final ADHD scores by treatment group in the MTA Study. The SNAP rating is average score of the 18 items corresponding to the DSMTV defining symptoms for attention-deficit-hyperactivity disorder and oppositional defiant disorder (range 0, not present to 3 very much). Beh, Behavior... 

Another way in which these kinds of data are sometimes represented is as a cumulative curve in which the total number (or fraction) of particles nT>, having diameters less (sometimes more) than and including a particular d, are plotted versus dr Figure 1.18b shows the cumulative plot for the same data shown in Figure 1.18a as a histogram. The cumulative curve is equivalent to the integral of the frequency distribution up to the specified class mark. Cumulative distribution curves are used in Chapter 2 in connection with sedimentation. 

FIG. 1.18 Graphical representation of the data in Table 1.5. Data are presented as (a) a histogram and (b) a cumulative distribution curve. 

Lin et al. (1971) used inelastic scattering of plane-polarized light of 632.8-nm wavelength from a He-Ne laser to determine the diffusion coefficient and thereby the hydrodynamic radii of monodisperse caseinate micelle fractions from milk. The cumulative distribution curve of the weight fraction of micelles revealed that about 80% of the casein occurs in micelles with radii of 50 to 100 nm and 95% between 40 and 220 nm, with the most probable radius at about 80 nm. This method has the advantage that the micelles are examined in their natural medium. 

Crushing is applied to large lumps of feed stock and grinding to smaller lumps, often the products of crushing, but the size distinction is not overly sharp. The process of size reduction results in a range of product sizes whose proper description is with the complete cumulative size distribution, but for convenience a characteristic diameter corresponding to 80% pass in the cumulative distribution curve is commonly quoted. 

This was done in the following way. First, two cumulative distribution curves were prepared for the control sample one is based on the calibration No. 3 and the other on No. 6. 

Here, Joi and parameters defining the log-normal distribution. Joi is the median diameter, and median diameter. In Joi and arithmetic mean and the standard deviation of In d, respectively, for the log-normal distribution (Problem 1.3). Note that, for the log-normal distribution, the particle number fraction in a size range of b to b + db is expressed by /N(b) db alternatively, the particle number fraction in a parametric range of Info to Info + d(lnb) is expressed by /N(lnb)d(lnb). 

Both variability and uncertainty in threshold and exposure data can be taken into account by using probability distributions to represent the input variables instead of point estimates. The data are plotted in a cumulative distribution curve. For example, threshold data and intake data can be plotted as probability distributions. By combining the threshold distribution and the intake distribution, the output distribution will describe the probability that a part of the population will be exposed at such levels and under such circumstances that adverse effects may occur. Consequently, an approximation of the percentage of the population likely to experience adverse effects at various exposure levels can be made. 

Unfortunately, most emulsions do not have a single droplet size. There are small, medium and large droplets present, and it is important to be able to characterise the emulsion for this. This is done by counting the number of particles that is smaller than a specific size, for many different sizes. The resulting data can then be plotted on a curve, the cumulative distribution curve. Alternatively, one can count all particles that have a size within an interval of sizes (e.g., 1-2 pm), and do this for all intervals. Plotting all the numbers obtained for all intervals, then results in a frequency distribution. The two distributions are closely related the derivative of the cumulative curve to the particle size, will give a (continuous) curve that is similar to the discrete frequency distribution obtained earlier, and the smaller the intervals are chosen, the closer the derivative will follow the frequency distribution (see Figure 15.4). 

Figure 1.1. Derivation of reference intakes of nutrients from the distribution around the observed mean requirement plotted below as a cumulative distribution curve, permitting estimation of the probability that a given level of intake is adequate to meet an individual s requirement.

Let us consider a product which is sold entirely on the basis of personal recommendation. The rate of sale will depend on the number of people who have already bought the product. Thus initially sales will increase exponentially. Eventually the market will be saturated, and only replacement purchases will be made. If the frequency curve may be assumed to be symmetrical about a single maximum value, the cumulative distribution curve is known as the logistics curve and is defined by Eq. (9-91) ... 

The median divides the distribution into two equal parts, i.e. it is the 50% size on the cumulative distribution curve. 

FIGURE 8.5. (a) The probability P E) of iJ having certain values depends on whether the structure is centrosymmetric (solid line) or noncentrosymmetric (broken line). Thus, if values are known, a plot of the number with different ranges of I E I will indicate the centrosymmetry (or lack of it) in the crystal structure, (b) The cumulative distribution curves for centrosymmetric and noncentrosymmetric crystals N Z) is the fraction of Bragg reflections with intensities (or E p values) less than or equal to Z times the mean intensity. 

Probability paper and linearization of the cumulative distribution curve In general, when calculating an average one uses the diameter d, at the midpoint of an interval times the number or weight of particles in that interval for the equations summarized in Table 3. However, there are situations where this is not possible e,g when one considers sieve data the midpoint of the interval for the largest sieve size is not defined, all that is known is how much material passes through the top sieve. Thus, to analyze this type of data one cannot use the standard methods illustrated in the previous examples. 

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