Probability scale

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. 

The SLIs represent a measure of the likelihood that the operations will succeed or fail, relative to one another. In order to convert the SLI scale to a probability scale, it is necessary to calibrate it. If a reasonably large number of operations in the set being evaluated have known probabilities (for example. 

Figures 62.8, 62.9, 62.10 show the data for generator fan failure plotted on exponential, normal and log normal hazard paper respectively. The exponential plot is a reasonably straight line which indicates that the failure rate is relatively constant over the range of the data. It should be noted that the reason the probability scale on the exponential hazard plot is crossed out is because that is not the proper way to plot data. (This will be discussed later.) The normal plot is curved concave upward which...

For any distribution, the cumulative hazard function and the cumulative distribution junction are connected by a simple relationship. The probability scale for the cumulative distribution function appears on the horizontal axis at the top of hazard paper and is determined from that relationship. Thus, the line fitted to data on hazard paper... 

Suppose, for example, that an estimate based on a Wei-bull fit to the fan data is desired of the fifth percentile of the distribution of time to fan failure. Enter the Weibull plot. Figure 62.6, on the probability scale at the chosen percentage point, 5 per cent. Go vertically down to the fitted line and then horizontally to the time scale where the estimate of the percentile is read and is 14,000 hours. 

An estimate of the probability of failure before some chosen specific time is obtained by the following. Suppose that an estimate is desired of the probability of fan failure before 100,000 hours, based on a Weibull fit to the fan data. Enter the Weibull plot on the vertical time scale at the chosen time, 100,000 hours. Go horizontally to the fitted line and then up to the probability scale where the estimate of the probability of failure is read and is 38 per cent. In other words, an estimated 38 per cent of the fans will fail before they run for 100,000 hours. 

The cumulative percentage points can be plotted on a distorted %-axis (so-called normal probability scale ) that yields a straight line for perfectly ND data. 

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... 

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... 

Author s estimate of status of studies in comparison with potential use by industry expressed on a probability scale extending from 0 to 10., , ... 

To clarify the use of probabilities let us consider the following treatment of illustrative data. A sample that has low concentrations of fingerprint elements has ratios of these elements to zinc that are at the high end of the probability scale. Another sample that has high concentrations of the same elements has ratios that are at the low end of the probability scale as shown by the following two randomly selected Brazil samples ... 

Hazen (1914) developed a special grid for plotting size-frequency data so that the resulting curve is a straight line . This grid consists of a system of coordinates based on a probability scale, that is, a scale based on the probability integral. On this scale, which may be considered as the ordinate of the system, are plotted the cumulative percent oversize or... 

Acute pancreatitis was subsequently discussed (233,234). The main points of debate were the interpretation of attributability and the use of the Naranjo probability scale in a case in which the patient was taking another drug that could not be completely excluded as at least a partial contributor to the acute pancreatitis. 

Restless legs, mumbling, and stuttering have been reported in a patient taking donepezil (56). According to the Naranjo probability scale, the causality was probable, since rechallenge was positive. 

This step involves calibration of the apparatus which will serve as a reference. It consists of analysing the greatest number possible (minimum 50) wines or must samples containing different and accurately known concentrations of each analyte. The concentration points should be uniformly distributed over the probable scale of measure for each analyte. The matrices should mimic as accurately as possible the wines and musts destined for analysis using that particular instrument. For each calibration sample, a measurement is carried out at a maximum number of wavelengths in the infra-red. Multi-linear regression is then carried out on the results which enables the following relationship to be established ... 

Plot of a sieve analysis of a sample of run-of-bank sand is shown in Figure 7.9 by the segmented line labeled stock sand. This sample may or may not meet the required effective size and uniformity coefficient specifications. In order to transform this sand into a usable sand, it must be given some treattnent. The figure shows the cumulative percentages (represented by the normal probability scale on the ordinate) as a function of the increasing size of the sand (represented by the size of separation on the abscissa). 

Probability graph paper. This is useful when one axis is a probability scale. 

When plotted on a log-probability scale (Henry s law diagram) the distributions of elemental concentrations show four patterns (Figure 1) (i) lognormal distribution as for potassium in Thailand (a, D), that can be interpreted as a single source of the element and limited control on its concentration (ii) retention... 

First, for resonant V -V transfer induced by long-range polar forces, the V-V probabilities scale as the squared transition moments of the donor and acceptor molecules and hence should be quite sensitive to changes of these moments in solution. In fact such a mechanism was proposed for HCl in... 

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