Quantiles

Hazard or event identification provides information on situations or chemicals and their releases tliat can potentially hanii tlie emaromiient, life, or property. Inforniation that is required to identify hazards includes chemical identities, quantilics and location of chemicals in question, chemical properties such as boiling points, ignition temperatures, and to.xicily to hmnans. There arc sci cral nictliods used to identify hazards. The methods that will be discussed later in tliis Part w ill include tlie process checklist and tlie hazard mid operability study (HA20P). 

Figure 1.3. The use of quantiles for displaying data. Three distributions of 100 events each are shown in histogram (left) and in quantile (right) form. The reason for choosing the 17 and the 83 - quantiles is explained in the text.

From a purely practical point of view the range or a quantile can serve as indicator. Quantiles are usually selected to encompass the central 60-90% of an ordered set the influence of extreme values diminishes the smaller this %-value is. No assumptions as to the underlying distribution are made. 

Occasionally there is the need for simultaneous determination of MW, MWD of polymers and identifica-tion/quantilication of additives [38]. This was the case for polymer and additive analysis of SBR/(softeners, flavour agents, stabilisers) (chewing gum) [41]. The many constituents of the SBR portion of the sample were not resolved, since adjacent components were similar in size. It should be stressed, however, that the need for simultaneous determination of the molecular weight of polymers and the identification/quantification of additives is exceptional rather than the rule. The determination of molecular weight distributions by SEC has indicated that oligomer fractions analysed by dissolution and (Soxhlet) extraction methods may differ essentially [42],... 

In the case of finite sample size in analytical practice, the quantiles of Student s f-distribution are used as realistic limits. 

The f-test in this form can only be applied under the condition that the variances of the two sample subsets, sf and sf, do not differ significantly. This has to be checked by the F-test beforehand. The test statistic f has to be compared to the related quantile of the (-distribution h-a,v where v = mx + n2 — 2. 

F exceeds the corresponding quantile of the F-distribution F a>V >V2 if at least one of the means differs significantly from the others. This global statement of variance analysis may be specified in the way to detect which of the mean(s) differ(s) from the others. This can be done by pairwise multiple comparisons (Tukey [1949] Games and Howell [1976] see Sachs [1992]). 

The estimated Fvalue has to be compared with the quantile of the F-distribution, Fi a>v, the tables of which can be found in textbooks of statistics (e.g., Hald [I960] Neave [1981] Dixon and Massey [1983] Graf et al. [1987] Sachs [1992]). The influence of the factor a is significant when F exceeds Ft a>v. In case of unbalanced experiments the different size of measurement series and, therefore, degrees of freedom have to be considered as a result of which both the evaluation scheme and the variance decomposition become more complicated (see Dixon and Massey [1983] Graf et al. [1987]). 

Together with a median it has to be said which type of median has been computed (see the footnote in Sect. 4.1.2) and also which kind of uncertainty (derived from median absolute deviation, mad Xi, or quantiles see Danzer [1989] Huber [1981] Hampel et al. [1986] Rousseeuw and Leroy [1987]). 

In the given formula, o av is the quantile of the respective f-distribution (the degrees of freedom v relates to the number of replicates by which srepeat has been estimated)... 

The second new program allows the user to compare the shapes of molecular weight distributions. For example, if we have the cumulative distribution of hydrodynamic volume for two polymers we can plot the hydrodynamic volume corresponding to the 10th percentile of the distribution for polymer A against the similarly defined hydrodynamic volume for polymer B. Such a plot, made for the entire distribution of both polymers, is called a "quantile... 

Quantile probability plots (QQ-plots) are useful data structure analysis tools originally proposed by Wilk and Gnanadesikan (1968). By means of probability plots they provide a clear summarization and palatable description of data. A variety of application instances have been shown by Gnanadesikan (1977). Durovic and Kovacevic (1995) have successfully implemented QQ-plots, combining them with some ideas from robust statistics (e.g., Huber, 1981) to make a robust Kalman filter. 

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. 

FIGURE 1.8 Probability density function (PDF) (left) and cumulative distribution function (right) of the normal distribution cr2) with mean /a and standard deviation cr. The quantile q defines a probability p. 

Figure 1.8 explains graphically how probabilities and quantiles are defined for a normal distribution. For instance the 1 %-percentile (p = 0.01) of the standard normal distribution is —2.326, and the 99%-percentile (p 0.99) is 2.326 both together define a 98% interval. 

Thus, a cutoff value for the score distance is the 97.5% quantile sjx], o 975- Values of... 

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