Areas distribution curve

The total area under the volume and area distribution curves is proportional to the total pore volume and pore area, respectively. By taking the ratio of the graphical area in any interval to the total graph area A, the pore volume or surface area in any interval can be calculated... 

Figure 5 shows the pore area distribution curves of Changchun clay soil before and after consolidation. 

As Figure 5 (a) shows, in horizontal section the porosities of small-area pores (A < 20 pm ) and large-area pores (A > 100 pm ) are greater but the porosity of medium-area pores (20 pm < A < 100 pm2) is small after consolidation, pore areas reduce sharply and the large-area pores almost disappear. The horizontal section samples are dominated by small-area pores after consolidation. Under consolidation pressure, particles contact or overlap with each other, which cause the reducing of pore area. Moreover, the pore area distribution range is smaller. The tendency of pore area distribution curves before and after consohdation are quite alike, which means consolidation pressure is the main factor for the reducing of pore area. 

The pore area distribution curves of vertical section sample are shown in Figure 5 (b). In vertical section the porosity of small-area pores is greater the porosities of medium-area pores and large-area... 

Table 2.26b Areas Under the Normal Distribution Curve from 0 to z 2.122...

Table 2.26a lists the height of an ordinate (Y) as a distance z from the mean, and Table 2.26b the area under the normal curve at a distance z from the mean, expressed as fractions of the total area, 1.000. Returning to Fig. 2.10, we note that 68.27% of the area of the normal distribution curve lies within 1 standard deviation of the center or mean value. Therefore, 31.73% lies outside those limits and 15.86% on each side. Ninety-five percent (actually 95.43%) of the area lies within 2 standard deviations, and 99.73% lies within 3 standard deviations of the mean. Often the last two areas are stated slightly different viz. 95% of the area lies within 1.96cr (approximately 2cr) and 99% lies within approximately 2.5cr. The mean falls at exactly the 50% point for symmetric normal distributions. 

Since significance tests are based on probabilities, their interpretation is naturally subject to error. As we have already seen, significance tests are carried out at a significance level, a, that defines the probability of rejecting a null hypothesis that is true. For example, when a significance test is conducted at a = 0.05, there is a 5% probability that the null hypothesis will be incorrectly rejected. This is known as a type 1 error, and its risk is always equivalent to a. Type 1 errors in two-tailed and one-tailed significance tests are represented by the shaded areas under the probability distribution curves in Figure 4.10. 

Relationship between confidence intervals and results of a significance test, (a) The shaded area under the normal distribution curves shows the apparent confidence intervals for the sample based on fexp. The solid bars in (b) and (c) show the actual confidence intervals that can be explained by indeterminate error using the critical value of (a,v). In part (b) the null hypothesis is rejected and the alternative hypothesis is accepted. In part (c) the null hypothesis is retained. 

For example, the proportion of the area under a normal distribution curve that lies to the right of a deviation of 0.04 is 0.4840, or 48.40%. The area to the left of the deviation is given as 1 - P. Thus, 51.60% of the area under the normal distribution curve lies to the left of a deviation of 0.04. When the deviation is negative, the values in the table give the proportion of the area under the normal distribution curve that lies to the left of z therefore, 48.40% of the area lies to the left, and 51.60% of the area lies to the right of a deviation of -0.04. 

Median Diameter. The median droplet diameter is the diameter that divides the spray into two equal portions by number, length, surface area, or volume. Median diameters may be easily determined from cumulative distribution curves. 

TABLE 3-4 Ordinates and Areas between Abscissa Values -z and +z of the Normal Distribution Curve... 

The opinions of the experts, however obtained, provide a basis for plotting a frequency or probability distribution curve. If the relative Frequency is plotted as ordinate, the total area under the cui ve is unity. The area under the cui ve between two values of the quantity is the probability that a randomly selected value will fall in the range between the two values of the quantity. These probabilities are mere estimates, and their reliability depends on the skill of the forecasters. 

FIGURE 11.23 Power analysis.The desired difference is >2 standard deviation units (X, - / = 8). The sample distribution in panel a is wide and only 67% of the distribution values are > 8. Therefore, with an experimental design that yields the sample distribution shown in panel a will have a power of 67% to attain the desired endpoint. In contrast, the sample distribution shown in panel b is much less broad and 97% of the area under the distribution curve is >8. Therefore, an experimental design yielding the sample distribution shown in panel B will gave a much higher power (97%) to attain the desired end point. One way to decrease the broadness of sample distributions is to increase the sample size. 

If a result is quoted as having an uncertainty of 1 standard deviation, an equivalent statement would be the 68.3% confidence limits are given by Xmean 1 Sjc, the reason being that the area under a normal distribution curve between z = -1.0 to z = 1.0 is 0.683. Now, confidence limits on the 68% level are not very useful for decision making because in one-third of all... 

An area-equivalent Gaussian ( normal ) distribution curve can be superimposed. 

The zeroth moment (k = 0) is simply the area under the distribution curve ... 

The statistics of the normal distribution can be applied to give more information about random-walk diffusion. The area under the normal distribution curve represents a probability. In the present case, the probability that any particular atom will be found in the region between the starting point of the diffusion and a distance of + J = + v/(2/V) on either side of it is approximately 68% (Fig. 5.6b). The probability that any particular atom has diffused further than this distance is given by the total area under the curve minus the shaded area, which is approximately 32%. The probability that the atoms have diffused further than 2 J, that is, 2V(2Dr t) is equal to about 5%. 

Ccs is the constant concentration of the diffusing species at the surface, c0 is the uniform concentration of the diffusing species already present in the solid before the experiment and cx is the concentration of the diffusing species at a position x from the surface after time t has elapsed and D is the (constant) diffusion coefficient of the diffusing species. The function erf [x/2(Dr)1/2] is called the error function. The error function is closely related to the area under the normal distribution curve and differs from it only by scaling. It can be expressed as an integral or by the infinite series erf(x) — 2/y/ [x — yry + ]. The comp-... 

As AD is made smaller, a histogram becomes a frequency distribution curve (Fig. 4.1) that may be used to characterize droplet size distribution if samples are sufficiently large. In addition to the frequency plot, a cumulative distribution plot has also been used to represent droplet size distribution. In this graphical representation (Fig. 4.2), a percentage of the total number, total surface area, total volume, or total mass of droplets below a given size is plotted vs. droplet size. Therefore, it is essentially a plot of the integral of the frequency curve. 

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