Probability paper

Fig. 11. WeibuU probability paper A, estimate of population B, estimate of P (line drawn parallel to population line).

Coefficient of Variation One of the problems confronting any user or designer of crystallization equipment is the expected particle-size distribution of the solids leaving the system and how this distribution may be adequately described. Most crystalline-product distributions plotted on arithmetic-probability paper will exhibit a straight line for a considerable portion of the plotted distribution. In this type of plot the particle diameter should be plotted as the ordinate and the cumulative percent on the log-probability scale as the abscissa. 

Frequency analysis is an alternative to moment-ratio analysis in selecting a representative function. Probability paper (see Figure 1-59 for an example) is available for each distribution, and the function is presented as a cumulative probability function. If the data sample has the same distribution function as the function used to scale the paper, the data will plot as a straight line. 

In summary, it becomes obvious that sieving should be continued until region 2 is reached. A good procedure might be to plot the time-weight curve on log probability paper and then... 

To plot a distribution, we use log probability paper. An sample is given in the following diagram ... 

When gel sections are analyzed for solute concentration, a value C is obtained, and when the solution bathing the gel is analyzed, a value C0 is obtained. The ratio C7C0 is plotted versus gel section distance, x, on arithmetic probability paper, and the slope of this plot is used to determine the diffusion coefficient, D, in the gel as... 

Harding, J. P. (1949). The use of probability paper for the graphical analysis of polymodal frequency distributions. Journal of the Marine Biobgical Association, 28, 141-153. 

The upper-limit distribution function assumes a finite minimum and maximum droplet size, corresponding to a y value of -oo and +oo, respectively. The function is therefore more realistic. However, similarly to other distribution functions, it is difficult to integrate and requires the use of log-probability paper. In addition, it usually requires many trials to determine a most suitable value for a maximum droplet size. 

The first is to normalize the data, making them suitable for analysis by our most common parametric techniques such as analysis of variance ANOYA. A simple test of whether a selected transformation will yield a distribution of data which satisfies the underlying assumptions for ANOYA is to plot the cumulative distribution of samples on probability paper (that is a commercially available paper which has the probability function scale as one axis). One can then alter the scale of the second axis (that is, the axis other than the one which is on a probability scale) from linear to any other (logarithmic, reciprocal, square root, etc.) and see if a previously curved line indicating a skewed distribution becomes linear to indicate normality. The slope of the transformed line gives us an estimate of the standard deviation. If... 

Figure 6. Plot on probability paper of cumulative PSD data for a 5OOA Ultrastyragel column. The mean (p = l.TO) and standard deviation (a= O.I2) of the Gaussian PSD were determined graphically.

Instead of taking slopes of the F curve to give the E curve and then determining the spread of this curve, let us use the probability paper method. So, plotting the data on this paper does actually give close to a straight line, as shown in Fig. E13.26. 

Figure 2. Cumulative distribution of instantaneous air concentrations from Figure 1 plotted on log-probability paper jcg = geometric mean, s9 = geometric standard...

Suppose a polydisperse system is investigated experimentally by measuring the number of particles in a set of different classes of diameter or molecular weight. Suppose further that these data are believed to follow a normal distribution function. To test this hypothesis rigorously, the chi-squared test from statistics should be applied. A simple graphical examination of the hypothesis can be conducted by plotting the cumulative distribution data on probability paper as a rapid, preliminary way to evaluate whether the data conform to the requirements of the normal distribution. 

FIG. C.1 A normal, or Gaussian, distribution (a) represented as a frequency function (b) represented as a cumulative function and (c) represented as a cumulative function linearized by plotting on probability paper. 

What is clear without the further aid of statistics is that the methanol concentration is the most important factor. Equally, it is clear that the citric acid concentration is not significant nor are three of the four interactions. Are the methanol concentration main effect and/or the interaction between the methanol and citric acid concentrations significant One way forward is to plot the data from Table 6 on normal probability paper. If all these data are insignificant then they will lie on a straight line. If values are observed that are a long way off the line it is likely that the effects or interactions are significant. 

Any experimental design that is intended to determine the effect of a parameter on a response must be able to differentiate a real effect from normal experimental error. One usual means of doing this determination is to run replicate experiments. The variations observed between the replicates can then be used to estimate the standard deviation of a single observation and hence the standard deviation of the effects. However, in the absence of replicates, other methods are available for ascertaining, at least in a qualitative way, whether an observed effect may be statistically significant. One very useful technique used with the data presented here involves the analysis of the factorial by using half-normal probability paper (19). 

Normal probability paper is obtained by adjusting the vertical in such a way that the plot of P versus X is a straight line. Thus, data that follow a normal probability distribution will produce a straight line when plotted on normal probability paper, as shown in Figure 6. 

Figure 6. Plot of cumulative percent probability versus X plotted on normal probability paper.

The size distributions of the fractions were plotted on log-probability paper as particle diameter (in microns) against cumulative percent of particles smaller than the indicated size. Figure 1 shows such a plot for the Johnie Boy size fractions. Such plots were compared for several samples with similar plots on linear-probability paper. Almost always the data could be described better by a lognormal rather than by a normal distribution law, after proper allowance for the presence of a maximum and a minimum size in each fraction. The parameters of the distributions were determined from the graph the geometric mean as the 50% point (median) and the logarithmic standard deviation as the ratio of the diameters at the 84 and 50% points. 

Very often it is not possible a priori to separate contaminated and uncontaminated soils at the time of sampling. The best that can be done in this situation is to assume the data comprise several overlapping log-normal populations. A plot of percent cumulative frequency versus concentration (either arithmetic or log-transformed values) on probability paper produces a straight line for a normal or log-normal population. Overlapping populations plot as intersecting lines. These are called broken line plots and Tennant and White (1959) and Sinclair (1974) have explained how these composite curves may be partitioned so as to separate out the background population and then estimate its mean and standard deviation. Davies (1983) applied the technique to soils in England and Wales and thereby estimated the upper limits for lead content in uncontaminated soils. 

If we sum the uses/compound, we get a frequency for each medical use. If we plot the number ot different uses against the frequency ot that use on probability paper or log-log plot, we get a distribution curve. The distribution curves for four different subsets ot the data in appear in (9). Each shows that the data form a tairly straight line over two orders ot magnitude. The curves represent an attempt to tit the points with a normal, rather than a log-normal, distribution. It is obvious that the curves do not tit the points and that therefore the medical use distribution function is log-normal rather than normal. My understanding is that this log-normal distribution is typical ot natural text data bases despite the highly specialized character ot the medical use data base ... 

Plot the data from Prob. 1 on log probability paper. Find the line of best fit for these data, and then determine the geometric mean and geometric standard deviation from this line. 

Dimensional analysis, 46 Direction of motion, 75 Disintegration of droplets, 198-200 Dissipation by wind, 68 Distance in cyclones, 118 between molecules, 33 stop, 83, 95, 142 Distributions barometric, 139-140 of charges, 201-207 with coagulation, 312 cumulative, 24,106 histograms for, 15-19 log probability paper for, 24-25 lognormal, 20, 22-24, 27,106-107 mathematical representation of,... 

Use of log probability paper is the simplest way to determine the mean and geometric standard deviation provided the distribution does indeed follow a lognormal shape or at least approximates it. 

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