Cumulative probability plots

Figure 6 shows a cumulative probability plot of both the maximum dally and hourly NO2 averages In cities for the 1980-84 time period. The plotted values can be directly compared to the WHO guideline values of 150/tg/m3 for the maximum 24-hour level and 400/tg/m3 for the maximum 1-hour level. In both cases, about 25% of the cities worldwide exceed the guideline values. Based on these proportions of cites with NO2 concentrations above the short-term guideline values. It Is estimated that approximately 15-20 percent of urban residents In North America and Europe are at Increased risk to short-term high NO2 exposures. 

Based on the data summarized in Table A2.1 and Figure A2.2, we are required to make assumptions to complete our uncertainty analysis. The number of positive samples, within the limit of detection, is low for both ocean and surface waters. In order to develop a probability distribution as well as moments (mean, standard deviation) for our analysis, we must consider some method to represent observations below the LOD. We construct a cumulative probability plot under the assumption that values below the LOD provide an estimate of the cumulative number of sample values above the LOD. This allows us to combine the ocean and freshwater samples so as to construct a probability distribution to fit these observations. This process is illustrated in Figure A2.2. 

Figure 5.5 Cumulative probability plot indicating true detection limits for samples determined by two different analytical methods. Such plots can demonstrate that the real detection hmits are often much lower than those cited by the analyst (the cited detection hmits are marked on the plot). Data for 10,000 samples from part of central and eastern England.

Fig. 7.4 Cumulative probability plots for a cycles and b time to failure for data in Fig. 7.3. Open symbols are for 1 Hz, closed symbols, 50 Hz [15]. With kind permission of John Wiley and sons...

Cumulative probability plots for the frequency of events, under various thin layer thicknesses. 

Frequency of exceedance per year is obtained by multiplying 0.036, tlie mean annual frequency of release, by the cumulative probabilities in Table 21.5.3. Frequency of exceedance plotted against tlie number of people affected produces a risk curve portraying healtli impact in terms of the frequency with which tlie number of people affected exceeds various amounts. Tlie risk curve is... 

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. 

The behavior of the failure rate as a function of time can be gaged from a hazard plot. If data are plotted on exponential hazard paper, the derivative of the cumulative hazard function at some time is the instantaneous failure rate at that time. Since time to failure is plotted as a function of the cumulative hazard, the instantaneous failure rate is actually the reciprocal of the slope of the plotted data, and the slope of the plotted data corresponds to the instantaneous mean time to failure. For the data that are plotted on one of the other hazard papers and that give a curved plot, one can determine from examining the changing slope of the plot whether the tme failure rate is increasing or decreasing relative to the failure rate of the theoretical distribution for the paper. Such information on the behavior of the failure rate cannot be obtained from probability plots. 

If estimated of distribution parameters are desired from data plotted on a hazard paper, then the straight line drawn through the data should be based primarily on a fit to the data points near the center of the distribution the sample is from and not be influenced overly by data points in the tails of the distribution. This is suggested because the smallest and largest times to failure in a sample tend to vary considerably from the true cumulative hazard function, and the middle times tend to lie close to it. Similar comments apply to the probability plotting. 

Ideally, no fewer than 20 failure times, if available, should be plotted from a set of data. Often, in engineering practice there are so few failures that all should be kept in mind so that conclusions drawn from a plot are based on a limited amount of information. Note that if only selected failures from a sample are to be plotted on hazard paper, it is necessary to use all of the failures in the sample to calculate the appropriate cumulative hazard values for the plotting positions. Wrong plotting positions will result if some failures in the data are not included in the cumulative hazard calculations. A similar comment applies to the calculation of plotting positions for probability plotting. 

Figure 1.13. The cumulative probability of the normal distribution. The hatched area corresponds to the difference ACP in the CP plot.

In comparing two distribution functions, a plot of the points whose coordinates are the quantiles qz (pc), qzz(pc) for different values of the cumulative probability pc is a QQ-plot. If zi and zz are identically distributed variables, then the plot of Z -quantiles versus Z2-quantiles will be a straight line with slope 1 and will point toward the origin. 

The essence of a QQ-plot is to plot the ordered sample values against some representative values from a presumed null standard distribution F(°). These representative values are the quantiles of the distribution function F(°) corresponding to a cumulative probability pc, [e.g., (t — 0.5)/M] and are determined by the expected values of the standard order statistics from the reference distribution. Thus, if the configuration of the QQ-plot in Eq. (11.30) is fairly linear, it indicates that the observations ( y(/), i = 1,..., M) have the same distribution function as F(°), even in the tails. 

It is usual to estimate and plot the probability of being event-free, but there will be occasions when interpretation is clearer when the opposite of this, cumulative incidence (or cumulative probability of experiencing the event by that time), is plotted. This is simply obtained as 1 - probability of being event-free. Pocock et al. (2002) discuss issues associated with the interpretation of these plots. These authors point out that interpretation in the conventional type of plot, when the event rates are low, can be exaggerated visually by a break in the y-axis, so take care ... 

Fig. 9.4.3 Normal-probability plot (a) and lognormal probability plot (b) for In particles shown in Figure 9.4.2. The ordinate stands for the cumulative percent of particles, with diameters smaller than d on the abscissa. This follows an error function and should give a straight line if the plot obeys the correspondent distribution, as seen in case (b). (From Ref. 4.)...

Table A2.1 summarizes the surface water concentration data available for making an estimate of the magnitude and range of concentrations of PBLx in fresh and ocean waters. Figure A2.2 provides a probability plot in which the cumulative distribution as reflected in the Z score is plotted against water concentrations for both surface and ocean waters.

The evaluation of the background upper limit value (80 mg/kg) is consistent with the known concentration of Pb in Neapolitan volcanic rocks (average 42.5 mg/kg Paone et al., 2001) and also with the Pb concentration values obtained in areas characterized by similar lithologies and with very limited human impact. Figure 7.9 (A and B) shows Pb histogram and probability plot for the Neapolitan province 982 soil samples. On the probability plot are also reported the background value, the percent cumulative value corresponding to the residential/recreational... 

Darling normality test [27], probability plot and cumulative distribution function (CDF) are eriteria that eould be used to eheek the normality of the data [25, 27 and 28],... 

Fig. 6.10 Weibull plot of the cumulative probability of breakdown for a set of polyethylene film samples as a function of time under an applied field of lOOMVnT1 at room temperature.

In the same way, random values of the other factors can be obtained. These can then be combined to give random values of (DCFRR) and (NPV) and, in turn, used to plot cumulative-probability curves for (DCFRR) and (NPV). The computer may be required to perform some 10,000 to 50,000 calculations. 

In Fig. 9, we plot the cumulative probability distribution for finding water molecules in a cluster of given size for both the membranes at k = 4.4 to 12.8 using all three cut-off distances. Two important asymptotes can be considered in this figure. An aqueous domain that is poorly connected will be composed of many small clusters. Therefore, the cumulative probability will quickly rise to 1.0, indicating that all water molecules are in small clusters. The... 

Fig. 41.5 Example of a normal probability plot. The plotted points are from a small data set where the mean Y = 6.93 and the standard deviation s = 1.895. Note that values corresponding to 0% and 100% cumulative frequency cannot be used. The straight line is that predicted for a normal distribution with...

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