Size-frequency distribution standard deviation

Fig. 2 Normal, or Gaussian, size-frequency distribution curve. Percentage of particles lying within 1 and 2 standard deviations about the arithmetic mean diameter are indicated.

The values of ag and 0/ are the same, that is, the geometric standard deviations by count and weight are equal. This is easily proved by substitution of Eq (5-8) and the value of n given by the size-frequency distribution by count in Eq (5-2). Hence,... 

Weber and Moran Method of Calibration—Another method of calibrating sieves consists in measuring random openings in the sieve and obtaining size-frequency distributions as qutlined in Chapter 3. The openings are measured by filar micrometer or by projection methods. Two screens are then identical if the mean size of the openings and the standard deviations are the same. Weber and Moran (1938) use the coefficient of variation... 

Another aspect of matching output to user needs involves presentation of results in a statistical framework—namely, as frequency distributions of concentrations. The output of deterministic models is not directly suited to this task, because it provides a single sample point for each run. Analytic linkages can be made between observed frequency distributions and computed model results. The model output for a particular set of meteorologic conditions can be on the frequency distribution of each station for which observations are available in sufficient sample size. If the model is validated for several different points on the frequency distribution based on today s estimated emission, it can be used to fit a distribution for cases of forecast emission. The fit can be made by relating characteristics of the distribution with a specific set of model predictions. For example, the distribution could be assumed to be log-normal, with a mean and standard deviation each determined by its own function of output concentrations computed for a standardized set of meteorologic conditions. This, in turn, can be linked to some effect on people or property that is defined in terms of the predicted concentration statistics. The diagram below illustrates this process ... 

Figure 1 shows, as a typical plot, the particle size distributions of the size fractions from a Johnie Boy sample. Only the first fraction, containing the largest particles, deviates significantly from lognormality. The standard deviations are almost the same for the first nine fractions as is apparent from the parallelism of the cumulative frequency curves. When the particle size decreases further, the standard deviations of the size distributions in the fractions increase. 

The geometric standard deviation (GSD) is defined as the size ratio at 84.2% on the cumulative frequency curve to the median diameter. This assumes that the distribution of particle sizes is lognormal. A monodisperse, i.e. ideal aerosol, has a GSD of 1, although in practice an aerosol with a GSD of <1.22 is described as monodisperse while those aerosols with a GSD >1.22 are referred to as poly dispersed or heterodispersed. 

It was observed many years ago that particle size data which were skewed and did not fit a normal distribution would very often fit a normal distribution if frequency were plotted against the logarithm of particle size instead of particle size alone. This tended to spread out the smaller size ranges and compress the larger ones. If the new plot then looked like a normal distribution, the particles were said to be lognormally distributed and the distribution was called a lognormal distribution. By analogy with a normal distribution, the mean and standard deviation became... 

In most analytical experiments where replicate measurements are made on the same matrix, it is assumed that the frequency distribution of the random error in the population follows the normal or Gaussian form (these terms are also used interchangeably, though neither is entirely appropriate). In such cases it may be shown readily that if samples of size n are taken from the population, and their means calculated, these means also follow the normal error distribution ( the sampling distribution of the mean ), but with standard deviation sj /n this is referred to as the standard deviation of the mean (sdm), or sometimes standard error of the mean (sem). It is obviously important to ensure that the sdm and the standard deviation s are carefully distinguished when expressing the results of an analysis. 

Before discussing our method for determining particle size, it is necessary to briefly review the definition of size distribution. If all particles of a given system were spherical in shape, the only size parameter would be the diameter. In most real cases of irregular particles, however, the size is usually expressed in terms of a sphere equivalent to the particle with regard to some property. Particles of a dispersed system are never of either perfectly identical size or shape A spread around the mean distribution) is found. Such a spread is often described in terms of standard deviation. However, a frequency function, or its integrated (cumulative) distribution function, more properly defines not only the spread but also the shape of such a spread around the mean value. This is commonly referred to as the particle size distribution (PSD) profile of the dispersed sample. 

The lognormal distribution function can be interpreted physically as the result of a process of breakup of larger particles at rates that are normally distributed with respect to particle size (Aitchison and Brown, 1957). Approximately lognormal distributions also result when the aerosol size distribution is controlled by coagulation (Chapter 7). In this case the value of the standard deviation is determined by the form of the particle collision frequency function. Multimodal aerosols may result when particles from several different types of sources are mixed. Such distributions are often approximated by adding lognormal distributions, each of which corresponds to a mode in the observed distribution and to a particular type of source. 

Size Distributions. Since size is important, so is its distribution. Size distributions can be presented in various ways, e.g., cumulative or as a frequency (which is the derivative with respect to size of the cumulative distribution) as a number or as a volume distribution versus diameter or (molar) mass, etc. Various types of averages can be defined and calculated, and their values can differ by more than an order of magnitude if the size distribution is relatively wide. It depends on the problem involved what type of average should be taken. Distribution width can be defined as standard deviation over average for most food dispersions, it ranges between 0.2 and... 

If the particle size distribution is normal or log normal, then the data can be linearized by plotting the particle frequency as a function of particle rize on arithmetic or logarithmic probability graph p r respectively. The 50% value of sudi plots yields the geometric median diameter and the geometric standard deviation is the ratio of the 84.1% m the 50% values. 

In Figure A2 wiien size is plotted against fiequency a skewed distribution (positive, ri t-hand ew in figure) usually results. Frequency distributions are characterised by the parameters which measure the central tendency in terms of the mean, median or mode and the dispersion about the central tendency which is the standard deviation. 

To pursue this idea, let us return to the nitrate ion determination described in Section 2.2. In practice it would be most unusual to make 50 repeated measurements in such a case a more likely number would be five. We can see how the means of samples of this size are spread about pi by treating the results in Table 2.2 as 10 samples, each containing five results. Taking each column as one sample, the means are 0.506, 0.504, 0.502, 0.496, 0.502, 0.492, 0.506, 0.504, 0.500 and 0.486. We can see that these means are more closely clustered than the original measurements. If we continued to take samples of five measurements and calculated their means, these means would have a frequency distribution of their own. The distribution of all possible sample means (in this case an infinite number) is called the sampling distribution of the mean. Its mean is the same as the mean of the original population. Its standard deviation is called the standard error of the mean (s.e.m.). There is an exact mathematical relationship between the latter and the standard deviation, [Pg.26]

Figure 3.10. Relative and cumulative frequencies as a function of normalized ID (a) Type I, mono-sized particles/voids of 5.35, and a minimum inter-particle/void distance of 1.5 (b) Type II mono-sized particles/voids of 5.35, and a minimum inter-particle/void distance of zero (c) Type III, log-normally sized particles/voids with a mean of 5.57 and a standard deviation of 1.13 (d) Type IV, log-normally sized particles/voids with a mean of 5.91 and a standard deviation of 2.46 and (e) Type V, log-normally sized bi-modal particles/voids with a similar mean particle/void size of 5.74 but different standard deviations of 1.11 and 2.47 and respectively volume fractions of 0.096 and 0.054. The curves represent cumulative Gaussian distribution functions [36]...

The results from the calculations yield a distribution function for the LoC frequency. Ligure 1 shows the results for the base case. The figure should be read as follows. On the x-axis is the frequency of an LoC event of arbitrary size in [year ]. On the y-axis is the percentage of the frequency distribution histogram where the histogram has 100 equidistant size bins between the minimum LoC frequency (5.15 year ) and the maximum LoC frequency (11.2 year ). This means that it is estimated that the LoC frequency is between 5.15 and 11.2. Any leak frequency rate between those Umits may be expected in the observation of the unit. There is a 90% probabiUty interval between 5.91 year and 8.21 year, which narrows the likely occurrence of an observable LoC rate. The mean is 6.68 year the standard deviation is 0.674 year. The median is 6.55 year. ... 

For each of the above particles, (at least) 100 measurements were conducted with random particle orientation within the measuring volume. From the results, the frequency distribution of U (or the respective d ) was determined, together with the mean and accompanying standard deviations. Upon plotting the mean values U as a function of the mean projected area Ap, then in this size range, (when dy ) the values for all particles should comply with the requirements of eq. (3), i.e. all values should sit exactly on a straight line which passes through the zero point. The results in Fig. 4 verify that a universal and unequivocal correlation exists between the mean value U of the relative parameter U and the mean projected area Ap for... 

Most frequently, an aerosol is characterized by its particle size distribution. Usually this distribution is reasonably well approximated by a log-normal frequency function (Fig. 4A). If the distribution is based on the logarithm of the particle size, the skewed log-normal distribution is transferred into the bell-shaped, gaussian error curve (see Fig. 4B). Consequently, two parameters are required to describe the particle size distribution of an aerosol the median particle diameter (MD), and an index of dispersion, the geometric standard deviation (Og). The MD of the log-normal frequency distribution is equivalent to the logarithmic mean and represents the 50% size cut of the distribution. The geometric standard deviation is derived from the cumulative distribution (see Fig. 4C) by... 

---