Normal frequency distribution, cumulative distributions with

The quantal dose-response curve is actually a cumulative plot of the normal frequency distribution curve. The frequency distribution curve, in this case relating the minimum protective dose to the frequency with which it occurs in the population, generally is bell shaped. If one graphs the cumulative frequency versus dose, one obtains the sigmoid-shaped curve of Figure 22A. The sigmoid shape is a characteristic of most dose-response curves when the dose is plotted on a geometric, or log, scale. 

These observations were explained in terms of a free-volume treatment that adopts the Grest-Cohen model, in which a system consists of free-volume cells, each having a total hole volume vh. These free-volume cells can be classified as solidlike (n < v c) or liquidlike (w > Vhc), where Vhc is a critical hole volume. Moreover, it is assumed that the free volume associated with a liquidlike cell of the amorphous phase consists of free-volume holes whose size distribution is given by a normal frequency distribution, H vk). This leads to a cumulative distribution function of free-volume hole sizes, r vh), given by... 

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...

Figure 14-15 Cumulative frequency distribution of relative differences for the comparison of drug assays example.The lighter curve indicates the Gaussian cumulative frequency distribution curve. In accordance with the test for normality, a good agreement is observed.

FIGURE 5—4 Frequency distribution curves and quantal concentration-effect and dose-effect curves. A. Frequency distribution curves. An experiment was performed on 100 subjects, and the effective plasma concentration that produced a quantal response was determined for each individual. The number of subjects who required each dose is plotted, giving a log-normal frequency distribution (colored bars). The gray bars demonstrate that the normal frequency distribution, when summated, yields the cumulative frequency distribution—a sigmoidal curve that is a quantal concentration-effect curve. B. Quantal dose-effect curves. Animals were injected with varying doses of sedative-hypnotic, and the responses were determined and plotted. The calculation of the therapeutic index, the ratio of the to the ED q, is an indication of how selective a drug is in producing its desired effects relative to its toxicity. (See text for additional explanation.)... 

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-... 

The median is defined as the diameter for which one-half the total munber of particles are smaller and one-half are larger. The median is also the diameter that divides the frequency distribution curve into equal areas, and the diameter corresponding to a cumulative fraction of 0.5. The mode is the most frequent size, or the diameter associated with the highest point on the frequency function curve. The mode can be determined by. setting the derivative of the frequency function equal to zero and solving for d. For symmetrical distributions such as the normal distribution, the mean, median, and mode will have the same value, which is the diameter of the axis of synunetry. For an asymmetrical or skewed distribution, these quantities will have different values. The median is conunonly used with skewed distributions, because extreme values in the tail have less effect on the median than on the mean. Most aerosol size distributions are skewed, with a long tail to the right, as shown in Fig. 4.4. For such a distribution,... 

In practical terms, the type and token cumulative frequency distributions may be tested for log-normality by plotting these functions on normal probability paper with the logarithm of frequency as the abscissa. When this test was applied to the medical use type and token frequency distributions, the log-normal model was found to describe both distributions very well over two orders of magnitude. Figure 1 shows this relationship for the use type distribution and also for the cluster type distribution. The cluster types involve a different counting than use types, so that two clusters are the same if all their components are Identical. The total number of cluster types is simply the sum of the last column in Table II. 

Figure 3.8. The transformation of a rectangular into a normal distribution. The rectangle at the lower left shows the probability density (idealized observed frequency of events) for a random generator versus x in the range 0 < jc < 1. The curve at the upper left is the cumulative probability CP versus deviation z function introduced in Section 1.2.1. At right, a normal distribution probability density PD is shown. The dotted line marked with an open square indicates the transformation for a random number smaller or equal to 0.5, the dot-dashed line starting from the filled square is for a random number larger than 0.5.

In Fig. 1 the cumulative frequency of the measured mean values for induvidual houses is plotted on a log-normal scale. The aritmetric mean value in our measurements is 160 Bq/nP. Areas with high concentrations are overrepresented in this distribution, (as seen from the figure) and by population weighing the distribution for the municipalities, a population weighted average of 110 Bq/m in the heating season is obtained. 

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.

Figure 2 shows the outcome of a Microsoft Excel simulation of 100 values of the nitrate ion concentration in a single water sample (ppm). It is most unlikely that any material would actually be analyzed 100 times, but the data are used to demonstrate methods by which such results are presented. Each result is given to two places of decimals, all the results lying in the range 0.43-0.57ppm. The Ere-quency column in the spreadsheet shows the number of times that each value 0.43, 0.44, 0.45,. .., 0.57 occurs, and this column along with its neighbor headed Value comprises a frequency table. The data in this table can be plotted as a bar chart, as shown in Figure 2. The shape of this bar chart is approximately the same as the smooth ideal curve for the normal distribution shown in Figure lA. An additional column shows the cumulative frequency values, and when these are plotted as a bar chart the curve obtained looks similar to that in Figure IB. The bar chart, frequency table, and cumulative frequencies are just three of several ways in which replicate experimental data can be presented.

Figure 3.10. Relative and cumulative frequencies as a function of normalized ID (a) Type I, mono-sized particles/voids of 5.35, and a minimum inter-particle/void distance of 1.5 (b) Type II mono-sized particles/voids of 5.35, and a minimum inter-particle/void distance of zero (c) Type III, log-normally sized particles/voids with a mean of 5.57 and a standard deviation of 1.13 (d) Type IV, log-normally sized particles/voids with a mean of 5.91 and a standard deviation of 2.46 and (e) Type V, log-normally sized bi-modal particles/voids with a similar mean particle/void size of 5.74 but different standard deviations of 1.11 and 2.47 and respectively volume fractions of 0.096 and 0.054. The curves represent cumulative Gaussian distribution functions [36]...

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