Normal distribution theory

Normal distribution theory can be used to test whether a particular sample value is consistent with other values or with our past experience. If the mean p and the variance are known, then we can determine how deviant an observed value x,-, appears to be by calculating the statistic Z = (x,- - p)/cr and comparing this with the table ofstandard normal deviates. Suppose that one of the values for our QC specimen was 170ng/ml. Past experience has led us to believe that the results for this QC specimen are normally distributed with p = 207.6 ng/ml and cr = 14.1. Is the... 

In the analyses of blood specimens from subjects participating in bioavailability studies, the FDA instructs laboratories to include quality control specimens (QC) at each of three known concentrations (low, mid, and high). The QC specimens are processed in duplicate with each batch of subject specimens. The acceptance criteria for the batch, based on the results of these QC specimens, is that at least four of the six values must fall within a specified range about their nominal concentrations. In addition, no more than one value at each of the three QC concentration levels can be outside its acceptance range. Combining binomial and normal distribution theory, we can estimate the number of batch runs we expect to reject because of random error. 

Parametric methods, i.e. those based on normal distribution theory, are recommended for the analysis of log-transformed bioequivaience measures. The general approach is to construct a 90% confidence interval for the quantity pT-pR and to reach a conclusion of pharmacokinetic equivalence if this confidence interval is within the stated limits. The nature of parametric confidence intervals means that this is equivalent to carrying out two one-sided tests of the hypothesis at the 5% level of significance 10, 77). The antilogs of the confidence limits obtained constitute the 90% confidence interval for the ratio of the geometric means between the multisource and comparator products. 

Normal distribution theory states that the chances of falling outside these control limits is 0.27%. If the subgroup ranges or standard deviations all fall within the upper and lower control limits, then there is evidence that the process variability is stable. It is unlikely that any point would fall beyond the control limits purely due to chance causes. Likewise, if the chart for subgroup averages shows all the points within the control limits, then the process is stable with... 

A value of Cp = 1.33 would indicate that the distribution of the product characteristics covers 75% of the tolerance. This would be sufficient to assume that the process is capable of producing an adequate proportion to specification. The numbers of failures falling out of specification for various values of Cp and Cp can be determined from Standard Normal Distribution (SND) theory (see an example later for how to determine the failure in parts-per-million or ppm). For example, at Cp = 1.33, the expected number of failures is 64 ppm in total. 

The stochastic tools used here differ considerably from those used in other fields of application, e.g., the investigation of measurements of physical data. For example, in this article normal distributions do not appear. On the other hand random sums, invented in actuary theory, are important. In the first theoretical part we start with random demand and end with conditional random service which is the basic quantity that should be used to decide how much of a product one should produce in a given period of time. 

Some modifications of the Gaussian distribution have been made. The log normal applies when the logarithm follows a normal distribution, but it does not seem to be useful for RTD representation. The Gram-Char 1 ier series (Kendall, Advanced Theory of Statistics, vol. 1, 1958) is an infinite series... 

Wu, Ruff and Faethl249 made an extensive review of previous theories and correlations for droplet size after primary breakup, and performed an experimental study of primary breakup in the nearnozzle region for various relative velocities and various liquid properties. Their experimental measurements revealed that the droplet size distribution after primary breakup and prior to any secondary breakup satisfies Simmons universal root-normal distribution 264]. In this distribution, a straight line can be generated by plotting (Z)/MMD)°5 vs. cumulative volume of droplets on a normal-probability scale, where MMD is the mass median diameter of droplets. The slope of the straight line is specified by the ratio... 

In signal detection theory such mistakes are considered to be the logical consequence of the fact that the normal distribution of the sensation caused by the noise alone and that of the sensations caused by signal plus noise overlap to a considerable degree (see figure 1). 

The x is best when the differences between theory and experiment are normally distributed and when the variance a is correctly estimated. The optimum has a value close to unity, y smaller than 1 indicates an overestimation of the variance. On the other hand, a is not needed for the evaluation of R-factor, which measures the residual difference in percentage. 

The disadvantage of R-factor is that the same R factor value may not indicate the same level of fit depending on the noise in the experimental data. The other difference is that the R-factor is based on an exponential distribution of differences. This makes the R factor a more robust GOT against possible large differences between theory and experiment. The exponential distribution has a long tail compared to the normal distribution. R-factor is used extensively in crystallographic methods. 

Bias The systematic or persistent distortion of an estimate from the true value. From sampling theory, bias is a characteristic of the sample estimator of the sufficient statistics for the distribution of interest. Therefore, bias is not a function of the data, but of the method for estimating the population statistics. For example, the method for calculating the sample mean of a normal distribution is an unbiased estimator of the true but unknown population mean. Statistical bias is not a Bayesian concept, because Bayes theorem does not relay on the long-term frequency expections of sample estimators. 

The wavefunction and its square are known as gaussian or bell curves they occur in probability theory as the normal distribution. This function, together with three higher-energy solutions for the harmonic oscillator, is shown in Fig. 3.5. 

Initially it will be assumed that the variation of the measurement around the true batch potency follows a normal distribution. This assumption means that if the same batch were repeatedly assayed, the data values would be distributed in a symmetric bell-shaped curve as in Fig. 5A. Most values would be clustered near the center (true potency), with some extreme values lying farther away. In theory, 68.2% of the data values would be found between p - a and p + c, 95.4% of the values would be between p - 2a and p + 2a, and 99.7% of the values would be within the range p - 3a to p + 3a. 

Statistical estimation uses sample data to obtain the best possible estimate of population parameters. The p value of the Binomial distribution, the p value in Poison s distribution, or the p and a values in the normal distribution are called parameters. Accordingly, to stress it once again, the part of mathematical statistics dealing with parameter distribution estimate of the probabilities of population, based on sample statistics, is called estimation theory. In addition, estimation furnishes a quantitative measure of the probable error involved in the estimate. As a result, the engineer not only has made the best use of this data, but he has a numerical estimate of the accuracy of these results. 

Of particular interest in this study is the nature of the non-aromatic structures in the three main maceral groups. It should be noted that the exinites in both the coals separated by float-sink are 90% sporinite. It has been theorized that small molecules, especially the aliphatics, are fairly mobile at some period during the formation of coal (5,6). The studies which support this theory were done on coals that are very rich in exinites and some contained alginite. Two of the coals chosen in the present work (PSOC 828 and 1103) have a more normal distribution of macerals and yet the pyrolysis results indicate that migration of molecules from the exinites to vitrinite and then incorporation into the macromolecular structure might have occurred. 

Figure 3.33. The normalized distributions of ionization products over their separation. The dashed-dotted line relates to the simplest integral theory (IET) and the dashed line, to its modified version (MET). The thick line represents the same distribution calculated with DET/UT. The ionization rate was assumed to be exponential, Wj(r) = Wc exp -[2(r - a)]// (Wc = 103ns, l = 1 A,D = 10 7 cm2/s, and c = 10 3 M. From Ref. [133].

Figure 3.99. The normalized distribution of ions produced by the biexcitonic ionization, according to Markovian theory (thick line) and non-Markovian UT distributions at different excitation lifetimes (dashed lines) (1)td = 1()6 ns, (2) zD — 100ns,(3)td — 10 ns. (From Ref. 275.)...

In the parameter estimation theory it is generally assumed that the experimental errors arc normally distributed with zero mean and a constant variance parameter values can then be estimated by max-... 

Two models are available for interpreting attenuation spectra as a PSD in suspensions with chemically distinct, dispersed phases using the extended coupled phase theory.68 Both models assume that the attenuation spectrum of a mixture is composed of a superposition of component spectra. In the multiphase model, the PSD is represented as the sum of two log-normal distributions with the same standard deviation, that is, a bimodal distribution. The appearance of multiple solutions is avoided by setting a common standard deviation to the mean size of each distribution. This may be a poor assumption for the PSD (see section 11.3.2). The effective medium model assumes that only one target phase of a multidisperse system needs to be determined, while all other phases contribute to a homogeneous system, the so-called effective medium. Although not complicated by the possibility of multiple solutions, this model requires additional measurements to determine the density, viscosity, and acoustic attenuation of the effective medium. The attenuation spectrum of the effective medium is modeled via a polynomial fit, while the target phase is assumed to have a log-normal PSD.68 This model allows the PSD for mixtures of more than two phases to be determined. 

Otto, E., et al. (1999). Log-normal size distribution theory of Brownian aerosol coagulation for the entire particle size range Part 11—Analytical solution using Dahneke s coagulation kernel. J. Aerosol Science. 30, 1, 17-34. 

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