Statistics cumulative frequencies

The use of various statistical techniques has been discussed (46) for two situations. For standard air quality networks with an extensive period of record, analysis of residuals, visual inspection of scatter diagrams, and comparison of cumulative frequency distributions are quite useful techniques for assessing model performance. For tracer studies the spatial coverage is better, so that identification of meiximum measured concentrations during each test is more feasible. However, temporal coverage is more limited with a specific number of tests not continuous in time. 

Statistical methods based on histograms, cumulative frequency probability curves (see above), univariate and multivariate data analysis (Miesh, 1981 Sinclair, 1974, 1976, 1991 Stanley, 1987) are widely used to separate geochemical baseline (natural and/or anthropogenic) values from anomalies. 

Fig. 7. Fault assisted top seal leakage, (a) Probability of top seal leakage. Analytical solution for shale beds of constant thickness /, in which identical faults of maximum throw are randomly dispersed. This relationship for probability of seal leakage also holds approximately for seals in which the shale layers and fault throws are each normally distributed about the same mean t. (b) Determination of the throw-cumulative frequency relationship. Faults in a volume of rock, from a map-based statistical analysis of the fault population. Adding 1 to the slope C2 simulates the addition of the third dimension (Gauthier and Lake, 1993). Here a length/Tfnjx fst o 100 1 was used, (c) Determination of the seal risk. Comparing the number of faults required for leakage with the number of faults in the trap volume determines the seal risk. In the example shown, the probability that the seal is breached lies between 50 and 90%, For points in the sealed field, the effect of increasing fault throw on the number of faults needed for breaching is illustrated.

As has been emphasized in this chapter, many statistical tests assume that the data used are drawn from a normal population. One method of testing this assumption, using the chi-squared test, was mentioned in the previous section. Unfortunately, this method can only be used if there are 50 or more data points. It is common in experimental work to have only a small set of data. A simple visual way of seeing whether a set of data is consistent with the assumption of normality is to plot a cumulative frequency curve on special graph paper known as normal probability paper. This method is most easily explained by means of an example. 

One further example of descriptive statistics is the cumulative frequency graph which is described later in the chapter. 

For the discussion of the chaotic behavior statistical methods will be used. The relative cumulative frequency or the probability, respectively, is shown in Figure 7 for four velocity classes, see Table 3. From these data the relative frequency or probability density, respectively, is obtained, see Figure 8. It turns out that the frequency distribution is completely non-Gaussian, and the range characterizing the statistical dispersion is increasing with the relative velocity of the impact, while midrange point and mean value coincide fairly well, see Table 4. 

The fact that the cumulative frequency distribution of CdKc in workers with normal renal function was not statistically different from that found in workers with abnormal kidney function suggests either that renal dysfunction is not related to the absolute cadmium concentration in the kidney cortex, or that renal cortical cadmium de-... 

Table I lists some of the basic mathematical expressions of importance in droplet statistics. The expressions are given in terms of an arbitrary ptb-weighted size distribution. The specific forms are obtained for various integral values of p. For example, the substitution of p = 2 into the equations of Table I yields the cumulative distribution, arithmetic mean, variance, geometric mean, and harmonic mean of the surface-weighted size distribution. Analogous expressions valid for frequencies or mass distributions are obtained by setting p equal to 0 or 3, respectively.

Figure 1 shows a hypothetical tolerance frequency distribution, f(D)dD, along with its corresponding cumulative distribution, P(D). Thus, when the response is quantal in nature, the function P(D) can be thought of as representing the dose-response either for the population as a whole, or for a randomly selected subject. The notion that a tolerance distribution, or dose-response function, could be determined solely from consideration of the statistical characteristics of a study population was introduced independently by Gaddum (2) and Bliss (3). 

The results of these studies are consistent and demonstrate that the frequency of respiratory cancer mortality increased with increasing exposure to radiation (cumulative WLMs). Statistically significant excesses in lung cancer deaths were present after cumulative exposures of less than 50 WLMs in the Czechoslovakian cohort (Sevc et al. 1988) and at cumulative exposures greater than 100 WLMs in the cohorts from the United States and Ontario, Canada (Muller et al. 1985 Samet et al. 1989 Waxweiler et al. 1981). These studies indicate that lung cancer mortality was influenced by the total cumulative radiation exposure, by the age at first exposure, and by the time-course of the exposure accumulation. Most deaths from respiratory cancers occurred 10 or more years after the individual... 

A frequency distribution can be developed from the resulting PW values, and relevant statistics can be computed. In this example the cumulative average PW after 10 trials is 2023 40% of the trials result in a negative PW. The minimum value simulated was — 737 the maximum value simulated was 6761. Of course, if this were an actual application, the number of trials would be much larger, perhaps several thousand or more, and we would have considerably greater confidence in the resulting statistics. 

Figure 5.1 shows that all three graphs used to characterize the relationship between the dose of toxic chemical and the frequency of toxic effect—i.e., the frequency histogram, the cumulative dose-effect curve, and the linear probit plot—put the logarithm of the dose on the x-axis. The reason the probit plot is linear is because of a clever invention a statistically derived scale that represents cumulative-effect frequency on the y-axis. Called probit units, or simply probits, this scale is based on a particular statistic, the standard deviation of the mean. Standard deviations correspond to fixed percentages of a population, and they can therefore be used in place of percentages to represent the fraction of a population that manifests a toxic effect. The mean dose corresponds to a standard deviation of zero because it is located in the exact middle of the bell curve. In terms of standard deviations, a disease frequency of 50% of the population is equivalent to zero standard deviations. One standard deviation below and above the mean corresponds, respectively, to manifestation... 

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