A The Normal Distribution

Since it is impractical to tabulate PD(x) for various combinations of /a and a, the normal distribution is usually presented in a normalized form where fi = 0 and ff = 1, that is... 

This is quite imposing (and does not need to be memorised ) but illustrates that the distribution is completely characterised by two parameters, the mean (p) and the standard deviation (a). The normal distribution has a symmetrical, bell-shaped curve. Two normal distributions with the same mean (10) but different variances are shown in Figure 21.2. 

Figures (a) The normal distribution with the 5% critical region highlighted. Two normally distributed signals with equal variances overlapping, with the mean of one located at the 5% point of the other (b) - the decision limit overlapping at their 5% points with means separated by 3.3a (c) - the detection limit and their means separated by lOa (d) - the determination limit...

Figure 19.16. Probability curves. A) The normal distribution curve for a = 1.0. B, opposite) Poisson distribution curves for n = 5.0 and IX. = 10.0.

From now on, for the simplicity of presentation, length is measured in units of the Kuhn segment length />. Thus R, L, A, etc. appearing below in this chapter are all dimensionless, i.e., real length divided by b. As Q(L,u,X) we first consider G(R L,u,A), the normalized distribution function of the end-to-end distance R of a d-dimensional Edwards chain. It can be shown that Gb(R Lo,vo) is given to first order in uo by... 

Its shape is shown in Figure 2.2. There is no need to remember this complicated formula, but some of its general properties are important. The curve is symmetrical about yU and the greater the value of a the greater the spread of the curve, as shown in Figure 2.3. More detailed analysis shows that, whatever the values of ju and a, the normal distribution has the following properties. 

These two methods generate random numbers in the normal distribution with zero me< and unit variance. A number (x) generated from this distribution can be related to i counterpart (x ) from another Gaussian distribution with mean (x ) and variance cr using... 

The normal distribution of measurements (or the normal law of error) is the fundamental starting point for analysis of data. When a large number of measurements are made, the individual measurements are not all identical and equal to the accepted value /x, which is the mean of an infinite population or universe of data, but are scattered about /x, owing to random error. If the magnitude of any single measurement is the abscissa and the relative frequencies (i.e., the probability) of occurrence of different-sized measurements are the ordinate, the smooth curve drawn through the points (Fig. 2.10) is the normal or Gaussian distribution curve (also the error curve or probability curve). The term error curve arises when one considers the distribution of errors (x — /x) about the true value. 

The standardized variable (the z statistic) requires only the probability level to be specified. It measures the deviation from the population mean in units of standard deviation. Y is 0.399 for the most probable value, /x. In the absence of any other information, the normal distribution is assumed to apply whenever repetitive measurements are made on a sample, or a similar measurement is made on different samples. 

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. 

To predict the properties of a population on the basis of a sample, it is necessary to know something about the population s expected distribution around its central value. The distribution of a population can be represented by plotting the frequency of occurrence of individual values as a function of the values themselves. Such plots are called prohahility distrihutions. Unfortunately, we are rarely able to calculate the exact probability distribution for a chemical system. In fact, the probability distribution can take any shape, depending on the nature of the chemical system being investigated. Fortunately many chemical systems display one of several common probability distributions. Two of these distributions, the binomial distribution and the normal distribution, are discussed next. 

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. 

Interpreting Control Charts The purpose of a control chart is to determine if a system is in statistical control. This determination is made by examining the location of individual points in relation to the warning limits and the control limits, and the distribution of the points around the central line. If we assume that the data are normally distributed, then the probability of finding a point at any distance from the mean value can be determined from the normal distribution curve. The upper and lower control limits for a property control chart, for example, are set to +3S, which, if S is a good approximation for O, includes 99.74% of the data. The probability that a point will fall outside the UCL or LCL, therefore, is only 0.26%. The... 

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. 

From Example A6.2 we know that after 100 steps of the countercurrent extraction, solute A is normally distributed about tube 90 with a standard deviation of 3. To determine the fraction of solute in tubes 85-99, we use the single-sided normal distribution in Appendix lA to determine the fraction of solute in tubes 0-84 and in tube 100. The fraction of solute A in tube 100 is determined by calculating the deviation z (see Chapter 4)... 

Ohm s law the statement that the current moving through a circuit is proportional to the applied potential and inversely proportional to the circuit s resistance (E = iR). (p. 463) on-column injection the direct injection of thermally unstable samples onto a capillary column, (p. 568) one-taUed significance test significance test in which the null hypothesis is rejected for values at only one end of the normal distribution, (p. 84)... 

We can imagine measuring experimental curves equivalent to those in Fig. 9.11 by, say, scanning the length of the diffusion apparatus by some optical method for analysis after a known diffusion time. Such results are then interpreted by rewriting Eq. (9.85) in the form of the normal distribution function, P(z) dz. This is accomplished by defining a parameter z such that... 

This shows that Schlieren optics provide a means for directly monitoring concentration gradients. The value of the diffusion coefficient which is consistent with the variation of dn/dx with x and t can be determined from the normal distribution function. Methods that avoid the difficulty associated with locating the inflection point have been developed, and it can be shown that the area under a Schlieren peak divided by its maximum height equals (47rDt). Since there are no unknown proportionality factors in this expression, D can be determined from Schlieren spectra measured at known times. 

To determine R(/) for the normal distribution, a standard normal variate must be calculated by the following formula ... 

Example 3 illustrated the use of the normal distribution as a model for time-to-failure. The normal distribution has an increasing ha2ard function which means that the product is experiencing wearout. In applying the normal to a specific situation, the fact must be considered that this model allows values of the random variable that are less than 2ero whereas obviously a life less than 2ero is not possible. This problem does not arise from a practical standpoint as long a.s fija > 4.0. 

A convenient approximate limit based on the normal distribution given by... 

A remarkable property of the normal distribution is that, almost regardless of the distribution of x, sample averages x will approach the gaussian distribution as n gets large. Even for relatively small values of n, of about 10, the approximation in most cases is quite close. For example, sample averages of size 10 from the uniform distribution will have essentially a gaussian distribution. 

Also, in many apphcations involving count data, the normal distribution can be used as a close approximation. In particular, the approximation is quite close for the binomial distribution within certain guidelines. 

Many distributions occurring in business situations are not symmetric but skewed, and the normal distribution cui ve is not a good fit. However, when data are based on estimates of future trends, the accuracy of the normal approximation is usually acceptable. This is particularly the case as the number of component variables Xi, Xo, etc., in Eq. (9-74) increases. Although distributions of the individual variables (xi, Xo, etc.) may be skewed, the distribution of the property or variable c tends to approach the normal distribution. 

The probabihty-density function for the normal distribution cui ve calculated from Eq. (9-95) by using the values of a, b, and c obtained in Example 10 is also compared with precise values in Table 9-10. In such symmetrical cases the best fit is to be expected when the median or 50 percentile Xm is used in conjunction with the lower quartile or 25 percentile Xl or with the upper quartile or 75 percentile X[j. These statistics are frequently quoted, and determination of values of a, b, and c by using Xm with Xl and with Xu is an indication of the symmetry of the cui ve. When the agreement is reasonable, the mean v ues of o so determined should be used to calculate the corresponding value of a. 

When a distribufion of particle sizes which must be collected is present, the aclual size distribution must be converted to a mass distribution by aerodynamic size. Frequently the distribution can be represented or approximated by a log-normal distribution (a straight line on a log-log plot of cumulative mass percent of particles versus diameter) wmich can be characterized by the mass median particle diameter dp5o and the standard statistical deviation of particles from the median [Pg.1428]

In practice, we can compute K as follows [19,23]. We start with a set of trajectories at the transition state q = q. The momenta have initial conditions distributed according to the normalized distribution functions... 

Step 1. From a histogram of the data, partition the data into N components, each roughly corresponding to a mode of the data distribution. This defines the Cj. Set the parameters for prior distributions on the 6 parameters that are conjugate to the likelihoods. For the normal distribution the priors are defined in Eq. (15), so the full prior for the n components is... 

Because each side chain can be identifiably assigned to a particular component, the mixture coefficients and the normal distribution parameters can be detennined separately. 

Produet toleranee ean signifieantly influenee produet variability. Unfortunately, we have dilfieulty in finding the exaet relationship between them. An approximate relationship ean be found from the proeess eapability index, a quality metrie interrelated to manufaeturing eost and toleranee (Lin et al., 1997). The random manner by whieh the inherent inaeeuraeies within a manufaeturing proeess are generated produees a pattern of variation for the dimension whieh resembles the Normal distribution (Chase and Parkinson, 1991 Mansoor, 1963) and therefore proeess eapability indiees, whieh are based on the Normal distribution, are suitable for use. See Appendix IV for a detailed diseussion of proeess eapability indiees. 

The random manner by whieh the inherent inaeeuraeies within the proeess are generated produees a pattern of variation for the dimension that resembles the Normal distribution, as diseussed in Chapter 2. As a first supposition then in the optimization of a toleranee staek with number of eomponents, it is assumed that eaeh eomponent follows a Normal distribution, therefore giving an assembly toleranee with a Normal distribution. It is also a good approximation that if the number of eomponents in the staek is greater than 5, then the final assembly eharae-teristie will form a Normal distribution regardless of the individual eomponent distributions due to the central limit theorem (Misehke, 1980). 

Data that is not evenly distributed is better represented by a skewed distribution such as the Lognormal or Weibull distribution. The empirically based Weibull distribution is frequently used to model engineering distributions because it is flexible (Rice, 1997). For example, the Weibull distribution can be used to replace the Normal distribution. Like the Lognormal, the 2-parameter Weibull distribution also has a zero threshold. But with increasing numbers of parameters, statistical models are more flexible as to the distributions that they may represent, and so the 3-parameter Weibull, which includes a minimum expected value, is very adaptable in modelling many types of data. A 3-parameter Lognormal is also available as discussed in Bury (1999). 

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