Standard deviation of the distribution

The standard deviation of the distribution of means equals cr/N. Since cr is not usually known, its approximation for a finite number of measurements is overcome by the Student t test. It is a measure of error between p and x. The Student t takes into account both the possible variation of the value of x from p on the basis of the expected variance and the reliability of using 5- in... 

In Chaps. 5 and 6 we shall examine the distribution of molecular weights for condensation and addition polymerizations in some detail. For the present, our only concern is how such a distribution of molecular weights is described. The standard parameters used for this purpose are the mean and standard deviation of the distribution. Although these are well-known quantities, many students are familiar with them only as results provided by a calculator. Since statistical considerations play an important role in several aspects of polymer chemistry, it is appropriate to digress into a brief examination of the statistical way of describing a distribution. 

This result shows that the square root of the amount by which the ratio M /M exceeds unity equals the standard deviation of the distribution relative to the number average molecular weight. Thus if a distribution is characterized by M = 10,000 and a = 3000, then M /M = 1.09. Alternatively, if M / n then the standard deviation is 71% of the value of M. This shows that reporting the mean and standard deviation of a distribution or the values of and Mw/Mn gives equivalent information about the distribution. We shall see in a moment that the second alternative is more easily accomplished for samples of polymers. First, however, consider the following example in which we apply some of the equations of this section to some numerical data. 

Equation 3.12 is the best estimate for the standard deviation of the distribution as determined by CA. 

The in equation 4.19 relates to the fact that this is not the true standard deviation, but an estimate to measure the process shift (or drift) in the distribution over the expected duration of production. Equation 4.20 is the best estimate for the standard deviation of the distribution as determined by CA with no process shift. 

The second order central moment is used so frequently that it is very often designated by a special symbol o2 or square root of the variance, o, is usually called the standard deviation of the distribution and taken as a measure of the extent to which the distribution is spread about the mean. Calculation of the variance can often be facilitated by use of the following formula, which relates the mean, the variance, and the second moment of a distribution... 

Mcllvried and Massoth [484] applied essentially the same approach as Hutchinson et al. [483] to both the contracting volume and diffusion-controlled models with normal and log—normal particle size distributions. They produced generalized plots of a against reduced time r (defined by t = kt/p) for various values of the standard deviation of the distribution, a (log—normal distribution) or the dispersion ratio, a/p (normal distribution with mean particle radius, p). 

The normal distribution is generally written as P(x) = (l/ /2TTstandard deviation of the distribution, the whole being normalized so that the area under the curve is equal to 1.0. 

The statistics of the normal distribution can now be applied to give more information about the statistics of random-walk diffusion. It is then found that the mean of the distribution is zero and the variance (the square of the standard deviation) is na2), equal to the mean-square displacement, . The standard deviation of the distribution is then the square root of the mean-square displacement, the root-mean-square displacement, + f . 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 (the root-mean-square displacement) 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 2f is equal to the total area under the curve minus the area under the curve up to 2f. This is found to be equal to about 5%. Some atoms will have gone further than this distance, but the probability that any one particular atom will have done so is very small. 

Figure 4-55 displays the distributions of the computed standard deviations resulting from each fit. We expect these standard deviations to be similar to the standard deviation of the distributions given above. The coincidence is perfect for the y2 fitting, as indicated by white bars and arrows, while the... 

Figure 4-55. Distributions of the calculated standard deviations of the fitted parameters for 1000 experiments. Top panel for n, bottom panel for X2. The white bars represent the %2 fits, the black bars the ssq-fits the arrows represent the standard deviations of the distributions of Figure 4-54.

After deposition on the alumina support using a classical incipient wetness impregnation, only isolated particles are observed by TEM (Fig. 13.28). When compared to the starting PdO particles (mean particle size =1.8 nm), the mean particle size slightly increases to 2.9 nm but the standard deviation of the distribution is kept constant. 

Crystal Size Spread. Figure 3 shows that the size range of the crystals increases as the crystals grow. In Figure 7 the standard deviation of the distribution, cr is plotted against the mean size, L, for run 5. This plot is approximately linear, giving a slope q. 

Therefore the square root of the amount by which the molecular weight ratio exceeds unity measures the standard deviation of the distribution relative to the number average molecular weight. 

Figure 2.10 indicates a gradual approach toward equilibrium by the fact that the concentration tends to be more nearly uniform as time increases. In Appendix C the function z is defined to be b/o, where b is the deviation of a particular value from the mean of a distribution and a is the standard deviation of the distribution. Since the net displacement of a diffusing particle is analogous to b, we may infer that the quantity yJlDt is also analogous to the standard deviation. The point of this is the following. 

In (5) D00 is the median diameter and a is the standard deviation of the distribution. By fitting the experimental R-values, the parameters D0 0 and a can be determined and hence the size distribution of the droplets in the emulsion can be obtained. For microbiological safety aspects Dj 3 is more important. D3>3 is the volume weighted mean droplet diameter and a is the standard deviation of the logarithm of the droplet diameter. The parameter D3 3 is related to the parameter D00 according to ... 

If, e.g., the true parameter cr (standard deviation) of the distribution of a variable x is known, one is able to calculate... 

Clearly these large fluctuations are due to cyclic variations not turbulent fluctuations. The dashed curve is an attempt to remove this cyclic variation effect by using the most probable density value as the mean value of a normal distribution. The standard deviation of the distribution is determined from fitting the data to the side of the new mean that has not been distorted by flame arrival. The reduction of the apparent fluctuations near the flame arrival crank angle is dramatic. Both curves of Figure 5 have had the Poisson statistical fluctuations subtracted. 

Spatial Distribution. As suggested by Gurland (2), a quantitative measure of the degree of homogeneity of particle placement in the matrix is the standard deviation of the distribution of particles in small volumes or areas of the mixture ... 

The standard deviation figures for a here and for OpSCi in Fig. 13 represent the standard deviations of the distributions of the corresponding exponents, and not the errors in determination of their mean value. We also note that the on time distributions are less close to the power-law decays than the off times, partly due to the exponential cutoffs and partly due to varying intensities in the on state (cf. 

---