Statistics bell-shaped distribution

The proof that these expressions are equivalent to Eq. (1.35) under suitable conditions is found in statistics textbooks. We shall have occasion to use the Poisson approximation to the binomial in discussing crystallization of polymers in Chap. 4, and the distribution of molecular weights of certain polymers in Chap. 6. The normal distribution is the familiar bell-shaped distribution that is known in academic circles as the curve. We shall use it in discussing diffusion in Chap. 9. 

Finally, time-resolved spectroscopy with femtosecond pulses was recently carried out by Gale and coworkers on a similar HD0 D20 sample (125). Due to the notably wider bandwidth of the applied IR pulses in the latter investigations, no details on reshaping of the transient spectra in dependence of the excitation frequency were accessible. A time-dependent position of the peak position of the induced sample bleaching was interpreted in terms of a shift within the statistical distribution of OH frequencies with a time constant of 1 ps. However, because only the parallel signal of the induced sample transmission was detected, the measured dynamics corresponds to a superposition of vibrational, reorientational, and structural relaxation. The data are interpreted by the help of a model of with random (bell-shaped) distribution of OH oscillators, quite different from the results of other groups. 

Normal Distribution is a continuous probability distribution that is useful in characterizing a large variety of types of data. It is a symmetric, bell-shaped distribution, completely defined by its mean and standard deviation and is commonly used to calculate probabilities of events that tend to occur around a mean value and trail off with decreasing likelihood. Different statistical tests are used and compared the y 2 test, the W Shapiro-Wilks test and the Z-score for asymmetry. If one of the p-values is smaller than 5%, the hypothesis (Ho) (normal distribution of the population of the sample) is rejected. If the p-value is greater than 5% then we prefer to accept the normality of the distribution. The normality of distribution allows us to analyse data through statistical procedures like ANOVA. In the absence of normality it is necessary to use nonparametric tests that compare medians rather than means. 

The Mann-Whitney test statistic is the nonparametric analog of the Student s /-test and is used to compare data from two groups [9]. Unlike the parametric Student s f-test which assumes a normal bell-shaped distribution, the Mann-Whitney statistic requires only that the sample data collected are randomly selected. 

Because of its mathematical properties, the standard deviation a is almost exclusively used to measure the dispersion of the partiele size distribution. When the skewed particle size distribution shown in Fig. 9 is replotted using the logarithm of the particle size, the skewed curve is transformed into a symmetrical bellshaped curve as shown in Fig. 10. This transformation is of great significance and importance in that a symmetrical bell-shaped distribution is amenable to all the statistical procedures developed for the normal or gaussian distribution. 

The vibrational population of the CO decreases monotonically with a slight but distinct drop in population near v = 12, as the vibrational energy of the CO increases. This is in sharp contrast to the bell-shaped distribution observed in the 0 + CS reaction (23-25), which could be accounted for by an impulsive model (26) indicating the possibility of the absence of a significant well in the triplet OCS intermediate. In the 0 + CF reaction, however, the observed CO population was found to lie close to that predicted by a simple statistical model (20). taking the total available reaction energy, Etot 126 -I- 2.5RT — 128 kcal/mole. On the basis of this model, the relative vibrational population of CO can be estimated from the following simple expression (20) ... 

The random-walk model of diffusion can also be applied to derive the shape of the bell-shaped concentration profile characteristic of bulk diffusion. As in the previous section, a planar layer of N tracer atoms is the starting point. Each atom diffuses from the interface by a random walk of n steps in a direction perpendicular to the interface. As mentioned (see footnote 5) the statistics are well known and described by the binomial distribution (Fig. S5.5a-S5.5c). At large values of N, this discrete distribution can be approximated by a continuous function, the Gaussian distribution curve7 with a form ... 

An RTD curve, for instance, can be represented in algebraic form in more than one way and for different purposes. The characteristic bell shape of many RTDs is evident in the real examples of Figure 5.4. Such shapes invite comparison with some well-known statistical distributions and representation of the RTD by their equations. Or a realistic mechanism may be postulated, such as a network of reactor elements and a type of flow pattern, and the parameters of that mechanism evaluated from a measured RTD. 

Many distributions obtained in experimental and observational work are found to have a more or less bell-shaped probability curve. These distributions are described by the normal or gaussian distribution shown in Fig. 2. This theoretical distribution is extremely important in statistics, and its use is not limited to data which are exactly, or very nearly normal. 

Deviations that arise probabilistically and have two characteristics (a) the magnitude of these errors is more typically small, and (b) positive and negative deviations of the same magnitude tend to occur with the same frequency. Random error is normally distributed, and the bell-shaped curve for frequency of occurrence versus magnitude of error is centered at the true value of the measured parameter. See Statistics (A Primer)... 

The normal, or Gaussian, distribution occupies a central place in statistics and measurement. Its familiar bell-shaped curve (the probability density function or pdf, figure 2.1) allows one to calculate the probability of finding a result in a particular range. The x-axis is the value of the variable under consideration, and the y-axis is the value of the pdf. 

The characteristic bell shape of many RTDs can be fit to well-known statistical distributions. Hahn and Shapiro (Statistical Models in Engineering, Wiley, 1967) discuss many of the standard distributions and conditions for their use. The most useful distributions are the gamma (or Erlang) and the gaussian together with its Gram-Charlier extension. These distributions are represented by only a few parameters that can be used to determine, for instance, the mean and the variance. 

Gaussian function A highly useful function named after mathematician Carl Friedrich Gauss. The familiar bell-shaped function is symmetric and has the property that its integral is 1. In statistics, a Gaussian distribution is called a normal distribution and has the familiar parameters mean ((j.) and standard deviation (a) ... 

Statistical analyses of continuous data which distribute according to a bell-shaped curve (e.g., adult body... 

The representation of this equation for anything greater than two variates is difficult to visualize, but the bivariate form (m = 2) serves to illustrate the general case. The exponential term in Equation (26) is of the form x Ax and is known as a quadratic form of a matrix product (Appendix A). Although the mathematical details associated with the quadratic form are not important for us here, one important property is that they have a well known geometric interpretation. All quadratic forms that occur in chemometrics and statistical data analysis expand to produce a quadratic smface that is a closed ellipse. Just as the univariate normal distribution appears bell-shaped, so the bivariate normal distribution is elliptical. 

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