Root-normal distribution

Wu, Ruff and Faethl249 made an extensive review of previous theories and correlations for droplet size after primary breakup, and performed an experimental study of primary breakup in the nearnozzle region for various relative velocities and various liquid properties. Their experimental measurements revealed that the droplet size distribution after primary breakup and prior to any secondary breakup satisfies Simmons universal root-normal distribution 264]. In this distribution, a straight line can be generated by plotting (Z)/MMD)°5 vs. cumulative volume of droplets on a normal-probability scale, where MMD is the mass median diameter of droplets. The slope of the straight line is specified by the ratio... 

In the breakup regimes, a droplet may undergo secondary breakup when the breakup time is reached. The droplet size distribution after bag or multimode breakup may follow the Simmons root-normal distribution pattern 264 with MMD/SMD equal to 1.1,... 

In many atomization processes of normal liquids, droplet size distributions fairly follow root-normal distribution pattern 264 ... 

In [2,22], holography was used to measure drop-size distributions for DA < 0.1. In the bag and multimode regimes, the root-normal distribution with MMD/D32 1.2 fit... 

Keywords Characteristic drop diameter Cumulative volume fraction Discrete probability function (DPF) Drop size distribution Empirical drop size distribution Log-hyperbolic distribution Log-normal distribution Maximum entropy formalism (MEF) Nukiyama-Tanasawa distribution Number distribution function Probability density function (pdf) Representative diameter Root-normal distribution Rosin-Rammler distribution Upper limit distribution Volume distribution... 

For each breakup regime, the volume distribution of the child-droplet fragments is described by a root-normal distribution... 

In analytical chemistry one of the most common statistical terms employed is the standard deviation of a population of observations. This is also called the root mean square deviation as it is the square root of the mean of the sum of the squares of the differences between the values and the mean of those values (this is expressed mathematically below) and is of particular value in connection with the normal distribution. 

The statistics of the normal distribution can now be applied to give more information about the statistics of random-walk diffusion. It is then found that the mean of the distribution is zero and the variance (the square of the standard deviation) is na2), equal to the mean-square displacement, . The standard deviation of the distribution is then the square root of the mean-square displacement, the root-mean-square displacement, + f . The area under the normal distribution curve represents a probability. In the present case, the probability that any particular atom will be found in the region between the starting point of the diffusion and a distance of J (the root-mean-square displacement) on either side of it, is approximately 68% (Fig. 5.6b). The probability that any particular atom has diffused further than this distance is given by the total area under the curve minus the shaded area, which is approximately 32%. The probability that the atoms have diffused further than 2f is equal to the total area under the curve minus the area under the curve up to 2f. This is found to be equal to about 5%. Some atoms will have gone further than this distance, but the probability that any one particular atom will have done so is very small. 

In this case the summation is the sum of the squares of all the differences between the individual values and the mean. The standard deviation is the square root of this sum divided by n — 1 (although some definitions of standard deviation divide by n, n — 1 is preferred for small sample numbers as it gives a less biased estimate). The standard deviation is a property of the normal distribution, and is an expression of the dispersion (spread) of this distribution. Mathematically, (roughly) 65% of the area beneath the normal distribution curve lies within 1 standard deviation of the mean. An area of 95% is encompassed by 2 standard deviations. This means that there is a 65% probability (or about a two in three chance) that the true value will lie within x Is, and a 95% chance (19 out of 20) that it will lie within x 2s. It follows that the standard deviation of a set of observations is a good measure of the likely error associated with the mean value. A quoted error of 2s around the mean is likely to capture the true value on 19 out of 20 occasions. 

I. A measure, symbolized by square root of the variance. Hence, it is used to describe the distribution about a mean value. For a normal distribution curve centered on some mean value, fjt, multiples of the standard deviation provides information on what percentage of the values lie within na of that mean. Thus, 68.3% of the values lie within one standard deviation of the mean, 95.5% within 2 cr, and 99.7% within 3 cr. 2. The corresponding statistic, 5, used to estimate the true standard deviation cr = (2(Xi - x) )/(n - 1). See Statistics (A Primer)... 

A basic assumption underlying r-tests and ANOVA (which are parametric tests) is that cost data are normally distributed. Given that the distribution of these data often violates this assumption, a number of analysts have begun using nonparametric tests, such as the Wilcoxon rank-sum test (a test of median costs) and the Kolmogorov-Smirnov test (a test for differences in cost distributions), which make no assumptions about the underlying distribution of costs. The principal problem with these nonparametric approaches is that statistical conclusions about the mean need not translate into statistical conclusions about the median (e.g., the means could differ yet the medians could be identical), nor do conclusions about the median necessarily translate into conclusions about the mean. Similar difficulties arise when - to avoid the problems of nonnormal distribution - one analyzes cost data that have been transformed to be more normal in their distribution (e.g., the log transformation of the square root of costs). The sample mean remains the estimator of choice for the analysis of cost data in economic evaluation. If one is concerned about nonnormal distribution, one should use statistical procedures that do not depend on the assumption of normal distribution of costs (e.g., nonparametric tests of means). 

The averages of random samples of a population are normally distributed. Therefore, the standard deviation of the population of sample means is the standard deviation of the population from which the sample is drawn divided by the square root of sample size. If we standardize the data to have a mean of 0.0 and a standard deviation of 1.0, then the standard deviation of the sample mean is 1.0 divided by the square root of the sample size. To be 95 percent confident that the incidence of insomnia in one group is smaller than the incidence in another group, the incidence in the first must be at least 1.64 standard deviations smaller than the incidence in the second. The sample size required to detect any given difference in means is approximately the square of 1.64 divided by the difference—in this case, (1.64/0.05) or 1,075.84. 

The normal distribution is characterized by two parameters. The center of the distribution, the mean, is given the symbol p. The width of the distribution is governed by the variance, The square root of the variance, ct, is the standard deviation. 

Now, everything falls into place We set out to study the laws of random walk by using the simple model of Fig. 18 and found the Bernoulli coefficients. We then saw that for large n (which is equivalent to large times), the Bernoulli coefficients can be approximated by a normal distribution whose standard deviation, a, grows in proportion to the square root of time, tm (Eq. 18-3). And now it turns out that the solution of the Fick s second law for unbounded diffusion is also a normal distribution. In fact, the analogy between Eqs. 18-3b and 18-17 gave the basis for the law by Einstein and Smoluchowski (Eq. 18-17) that we used earlier (Eq. 18-8). The expression (2Dt)U2 will also show up in other solutions of the diffusion equation. 

The dynamic range of OSME and GC-SNIFF data is generally less than a factor of ten, whereas dilution analysis frequently yields data that cover three or four powers of ten. It has been determined, however, that compressive transforms (log, root 0.5, and so on) of dilution analysis data are needed to produce statistics with normally distributed error (Acree and Barnard, 1994). Odor Spectrum Values (OSVs) were designed to transform dilution analysis data, odor units, or any potency data into normalized values that are comparable from study to study and are appropriate for normal statistics. The OSV is determined from the equation ... 

To demonstrate the accuracy, two dust and two soil reference materials were analyzed with the described method. The mean value of the correlation coefficients between the certified and the analyzed amounts of the 16 elements in the samples is r = 0.94. By application of factor analysis (see Section 5.4) the square root of the mean value of the communahties of these elements was computed to be approximately 0.84. As frequently happens in the analytical chemistry of dusts several types of distribution occur [KOM-MISSION FUR UMWELTSCHUTZ, 1985] these can change considerably in proportion to the observed sample size. In the example described the major components are distributed normally and most of the trace components are distributed log-normally. The relative ruggedness of multivariate statistical methods against deviations from the normal distribution is known [WEBER, 1986 AHRENS and LAUTER, 1981] and will be tested using this example by application of factor analysis. 

Thus the mean and variance of a gamma distribution are sufficient to determine its two parameters, a and /3. Note that the coefficient of variation (standard deviation divided by the mean) is equal to the square root of 1/or. The most probable value of k (the mode) occurs at (a - l)//3 if oc > 1, and as a T oo the gamma distribution itself becomes the gaussian or normal distribution for the variable, /8k.13... 

We can try to find a mathematical transformation of the data that shows a better approximation to a normal distribution. With positive skew, either a square-root or a log transform may be useful. With this data, the square-root transform is insufficiently powerful and the data remain distinctly skewed. The results of the more powerful log transform are presented in Table 17.1 and Figure 17.2(b). The latter shows that the distribution for the smokers data is now much more symmetrical. The effect on the non-smokers data is not shown but is also satisfactory. We would then perform a standard two sample f-test, but apply it to the last two columns in Table 17.1. Generic output is shown in Table 17.2. 

The geometric mean size of a number distribution (.xp is the nth root of the product of the sizes of the n particles examined it is of particular value with log-normal distributions ... 

Again, as the sample size increases the SE decreases in proportion to the square root of the sample size and the CI narrows. Also, although the result for sample A is not obviously incorrect, in fact the sample is too small to use the large sample expression. Instead, a distribution known as the t distribution, which describes sampling error in small samples with quantitative data, should be used. The multiplier corresponding to the normal distribution z of 1 -96 depends on the sample size. The values of t are tabulated... 

Standard deviation (population standard deviation), a The square root of the variance, the population standard deviation represents the dispersion of the population. In the normal distribution, 68% of the distribution lies at the mean /u 1 a. (Section 1.8.2)... 

Droplet size distribution plotted on a root-normal scale. (Reprinted from Lee, J., Sallam, K. A., Lin, K.-C., and Carter, C. D., Journal of Propulsion and Power, 25, no. 2, 258-66, 2009b. With permission from American Institute of Aeronautics and Astronautics.)... 

It is preferable to take at least four standards for the calibration curve. From both the standards and samples the homogeneity of the variances may be calculated. If the variances are homogeneous, the triplicate observations are normally distributed. The standard deviation s is the square root of the variance. 

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