Expected frequency

Application A frequency count of workers was tabulated according to the number of defective items that they produced. An unresolved question is whether the observed distribution is a Poisson distribution. That is, do observed and expected frequencies agree within chance variation ... 

Likelihood A measure of the expected frequency with which an event occurs. This may be expressed as a frequency (e.g., events per year), a probability of occurrence during a time interval (e.g., annual probability), or a conditional probability (e.g., probability of occurrence, given that a precursor event has occurred). 

QRA is fundamentally different from many other chemical engineering activities (e.g., chemistry, heat transfer, reaction kinetics) whose basic property data are theoretically deterministic. For example, the physical properties of a substance for a specific application can often be established experimentally. But some of the basic property data used to calculate risk estimates are probabilistic variables with no fixed values. Some of the key elements of risk, such as the statistically expected frequency of an accident and the statistically expected consequences of exposure to a toxic gas, must be determined using these probabilistic variables. QRA is an approach for estimating the risk of chemical operations using the probabilistic information. And it is a fundamentally different approach from those used in many other engineering activities because interpreting the results of a QRA requires an increased sensitivity to uncertainties that arise primarily from the probabilistic character of the data. 

The reasons (i.e., the motivations, concerns, activators, and needs) for considering the use of QRA define the requirements for information. The next question is, can QRA supply the appropriate information to satisfy the need By definition, QRA studies generate numerical estimates of the expected frequency and/or consequence(s) of undesired events. The results of the QRA can be formulated and used on two bases (1) an absolute basis and (2) a relative basis. 

There are a variety of ways to express absolute QRA results. Absolute frequency results are estimates of the statistical likelihood of an accident occurring. Table 3 contains examples of typical statements of absolute frequency estimates. These estimates for complex system failures are usually synthesized using basic equipment failure and operator error data. Depending upon the availability, specificity, and quality of failure data, the estimates may have considerable statistical uncertainty (e.g., factors of 10 or more because of uncertainties in the input data alone). When reporting single-point estimates or best estimates of the expected frequency of rare events (i.e., events not expected to occur within the operating life of a plant), analysts sometimes provide a measure of the sensitivity of the results arising from data uncertainties. 

The expected frequency of having an explosion in the plant is 5 x lO per year. 

If sufficient experience does not exist, you should consider whether the consequence potential (Step 4) or the expected frequency of accidents (Step 5) is great. Consideration of consequence potential should include personnel exposure, public demographics, equipment density, and so forth in relation to the intrinsic hazard posed by the material of concern. In Step 5 you may perceive that the expected frequency of accidents alone is important enough to justify a QRA. However, even though your company may not have much relevant experience with the activity of interest, if the consequence potential of these accidents is not great, you may conclude that the expected frequency of the potential accidents is low enough for you to make your decisions comfortably using qualitative information alone. 

Figures 12 and 13 illustrate two of the more commonly used methods for displaying societal risk results (1) an F-N curve and (2) a risk profile. The F-N curve plots the cumulative frequencies of events causing N or more impacts, with the number of impacts (N) shown on the horizontal axis. With the F-N curve you can easily see the expected frequency of accidents that could harm greater than a specified number of people. F-N curve plots are almost always presented on logarithmic scales because of...

While the F-N curve is a cumulative illustration, the risk profile shows the expected frequency of accidents of a particular category or level of consequence. The diagonal line is a line of constant risk defined such that the product of expected frequency and consequence is a constant at each point along the line. " As the consequences of accidents go up, the expected frequency should go down in order for the risk to remain constant. As the example illustrates, if a portion of the histogram sticks its head up above the line (i.e., a particular type of accident contributes more than its fair share of the risk), then that risk is inconsistent with the risk presented by other accident types. (Note There is no requirement that you use a line of constant risk other more appropriate risk criteria for your application can be easily defined and displayed on the graph.)... 

An event or accident whose expected frequency is very small. The event is not expected to occur during the normal life of a facility or operation... 

Risk is defined as the combination of the expected frequency and consequence of accidents that could occur as a result of an activity. Risk analysis is a formal process of increasing one s understanding of the risk associated with an activity. The process of risk analysis includes answering three questions ... 

Once the mean and standard deviation have been determined, the frequency distribution determined from the PDF can be compared to the original histogram, if one was constructed, by using a scaling factor in the PDF equation. For example, the expected frequency for the Normal distribution is given by ... 

Assume that the system described below exists in a process unit recently purchased by your company. As the manager, the safety of this unit is now your responsibility. You are concerned because your process hazard analysis team identified the potential for an operator error to result in a rupture of the propane condenser. You have commissioned a human reliability analysis (HRA) to estimate the likelihood of the condenser rupturing as the result of such an error and to identify ways to reduce the expected frequency of such ruptures... 

Finally, the HRA analyst would calculate the expected frequency of condenser ruptures as a result of improper isolation. The frequency of condenser tube failures is 0.33 per year (1 every 3 years), and the calculated probability of improper isolation is 0.05. Multiplying these two numbers shows the expected frequency of improper isolation of a failed condenser is 0.017 per year, or about once every 60 years. The manager can use this number to help compare the costs and benefits of improvements proposed as a result of the HRA or other studies. 

In a more quantitative sense, cause-consequence analysis may be viewed as a blend of fault tree end event tree analysis (discussed in tlie two preceding cliapters) for evaluating potential accidents. A major strengtli of cause-consequence analysis is its use as a communication tool. For example, a cause-consequence diagram displays the interrelationships between tlie accident outcomes (consequences) and Uieir basic causes. The method can be used to quantify the expected frequency of occurrence of the consequences if the appropriate chita are available. 

Degrees of Freedom, statistical term for the number of choices that can be made when fixing values of expected frequency leading to the number of independent comparisons that can be made in a sample of observations. 

The higher than expected frequency of alcohol PCP -, and heroin PCP-rel ated deaths may be the result of an interaction of the combined substances. Balster (this volume) anticipated an interaction of the combined substances when he reported "PCP very markedly enhances the effects of classical depressant drugs, including barbiturates and ethanol."... 

The higher than expected frequencies of alcohol PCP - and heroin PCP-related deaths also would have occurred if the combinations were preferred by the users. The motivation may involve the injection of heroin to moderate the adverse effects of PCP, or the use of PCP to ease the pain of heroin withdrawal. Another explanation assumes a stimulant effect of PCP. The use of stimulants, especially cocaine, with heroin is increasingly popular among heroin users (Kozel et al. 1982). 

Computation for Ej On the basis of the specified population, the probability of observing a count in cell j is defined by pj. For a sample of size N, corresponding to N total counts, the expected frequency is given by Ej = Npj. 

The assignment of the 809 and 1560 cm-1 bands of the donor-H complexes were based upon the expected frequencies of the wagging and stretching modes and the 2 1 ratio of intensities (Bergman et al., 1988a). A uniaxial stress study of the vibrational bands (Bergman et al., 1988b) will be reviewed here. Only the As—H complex was studied under stress because the donor-H complexes have nearly identical vibrational spectra for the different donors and are expected to behave similarly under stress. 

Quantitative risk assessment (QRA) The systematic development of numerical estimates of the expected frequency and consequence of potential accidents associated with a facility or an operation. Using consequence and probability analyses and other factors such as population density and expected weather conditions, QRA estimates the fatality rate for a given set of events. 

Figure 23-28 is an example of an individual risk contour plot, which shows the expected frequency of an event causing a specified level of harm at a specified location, regardless of whether anyone is present at that location to suffer that level of harm. 

Epizootic 1. Denoting a temporal pattern of disease occurrence in an animal population in which the disease occurs with a frequency clearly in excess of the expected frequency in that population during a given time interval. 2. An outbreak (epidemic) of disease in an animal population often with the implication that it may also affect human populations. 

The RxC chi-square test can be used to analyze discontinuous (frequency) data as in the Fisher s exact or 2x2 chi-square tests. However, in the RxC test (R = row, C = Column) we wish to compare three or more sets of data. An example would be comparison of the incidence of tumors among mice on three or more oral dosage levels. We can consider the data as positive (tumors) or negative (no tumors). The expected frequency for any box is equal to (row total)(column total)/(A/,otal). 

None of the expected frequency values should be less than 5.0. 

Test is weak with either small sample sizes or when the expected frequency in any cell is less than 5 (this latter limitation can be overcome by pooling combining cells). 

The table now has an expected frequency in parentheses in each cell along with the observed frequency. The calculation (O — E)2/E is performed for each cell and the results summed to give the 2 statistic. 

Now let us examine the distribution and position of disulfides in proteins. The simplest consideration is distribution in the sequence (see Fig. 51), which is apparently quite random, except that there must be at least two residues in between connected half-cystines. Even rather conspicuous patterns such as two consecutive halfcystines in separate disulfides turn out, when the distribution is plotted for the solved structures (Fig. 51), to occur at only about the random expected frequency. The sequence distribution of halfcystines is influenced by the statistics of close contacts in the three-dimensional structures, but apparently there are no strong preferences of the cystines that could influence the three-dimensional structure. 

Kieffer (1982) suggests a set of rules describing expected frequency shifts for different types of vibrational modes upon substitution of a heavy isotope into the mineral structure. Kieffer s (1982) rules for substitution in silicates are as follows. 

Likelihood— An estimate of the expected frequency or probability of the occurrence of an event. 

For graphical evaluation of distribution fit, probability (P-P) plots and quantile (Q-Q) plots are particularly helpful. Sometimes the statistic has been used to quantify the linearity of a P-P plot or Q-Q plot however, in practice it appears that there may be substantial deviation between the observed and expected frequencies, despite an R that would be viewed as large in many statistical contexts. 

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