Normal or Gaussian distribution

If the data set is Puly nomial and the enor in y is random about known values of a , residuals will be distr ibuted about the regression line according to a normal or Gaussian distribution. If the dishibution is anything else, one of the initial hypotheses has failed. Either the enor dishibution is not random about the shaight line or y =f x) is not linear. 

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 most commonly encountered probability distribution is the normal, or Gaussian, distribution. A normal distribution is characterized by a true mean, p, and variance, O, which are estimated using X and s. Since the area between any two limits of a normal distribution is well defined, the construction and evaluation of significance tests are straightforward. 

Mathematical Models for Distribution Curves Mathematical models have been developed to fit the various distribution cur ves. It is most unlikely that any frequency distribution cur ve obtained in practice will exactly fit a cur ve plotted from any of these mathematical models. Nevertheless, the approximations are extremely useful, particularly in view of the inherent inaccuracies of practical data. The most common are the binomial, Poisson, and normal, or gaussian, distributions. 

A well-known statistieal distribution is the normal or Gaussian distribution and is expressed by... 

If a large number of replicate readings, at least 50, are taken of a continuous variable, e.g. a titrimetric end-point, the results attained will usually be distributed about the mean in a roughly symmetrical manner. The mathematical model that best satisfies such a distribution of random errors is called the Normal (or Gaussian) distribution. This is a bell-shaped curve that is symmetrical about the mean as shown in Fig. 4.1. 

The normal or Gaussian distribution a bell-shaped frequency profile defined by the function... 

Statistical testing of model adequacy and significance of parameter estimates is a very important part of kinetic modelling. Only those models with a positive evaluation in statistical analysis should be applied in reactor scale-up. The statistical analysis presented below is restricted to linear regression and normal or Gaussian distribution of experimental errors. If the experimental error has a zero mean, constant variance and is independently distributed, its variance can be evaluated by dividing SSres by the number of degrees of freedom, i.e. 

In particle size analysis it is important to define three terms. The three important measures of central tendency or averages, the mean, the median, and the mode are depicted in Figure 2.4. The mode, it may be pointed out, is the most common value of the frequency distribution, i.e., it corresponds to the highest point of the frequency curve. The distribution shown in Figure 2.4 (A) is a normal or Gaussian distribution. In this case, the mean, the median and the mode are found to fie in exactly the same position. The distribution shown in Figure 2.4 (B) is bimodal. In this case, the mean diameter is almost exactly halfway between the two distributions as shown. It may be noted that there are no particles which are of this mean size The median diameter lies 1% into the higher of the two distri-... 

Various theoretical distribution functions have been proposed, such as normal or Gaussian distribution and the log-normal distribution. The simplest case is... 

The best known statistical distribution is the Normal or Gaussian distribution whose equation may be written... 

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. 

In a situation whereby a large number of replicate readings, not less than 5 0, are observed of a titrimetric equivalence point (continuous variable), the results thus generated shall normally be distributed around the mean in a more or less symmetrical fashion. Thus, the mathematical model which not only fits into but also satisfies such a distribution of random errors is termed as the Normal or Gaussian distribution curve. It is a bell-shaped curve which is noted to be symmetrical about the mean as depicted in Figure 3.2. 

A set of replicate measurements is said to show a normal or Gaussian distribution if it shows a symmetrical distribution about the mean value. 

If subsequent analyses of the bulk sample deviate by more than a predetermined amount, the whole batch of results is rejected. Results are thus only accepted if they fall between specified values of s above and below the mean, where Is includes 68%, 2s includes 95% (the normally accepted value), and 3s includes 99.7% of results. The scatter of results usually assumes a symmetrical normal or Gaussian distribution about the mean, as shown in Figs 12.1 and 12.2. 

Fig. 12.2. A normal or Gaussian distribution of results with % population enclosed by various standard deviation values.

The normal or Gaussian distribution was in fact first discovered by de Moivre, a French mathematician, in 1733. Gauss came upon it somewhat later, just after 1800, but from a completely different start point. Nonetheless, it is Gauss who has his name attached to this distribution. 

A very important probability distribution is the normal or Gaussian distribution (after the German mathematician, Karl Friedrich Gauss, 1777-1855). The normal distribution has the same value for the mean, median and mode. The equation describing this distribution (the probability density function)... 

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. 

Figure 2.1. The standardized normal or Gaussian distribution. The shaded area as a fraction as the entire area under the curve is the probability of a result between Xj and X2.

Thus the concentration ratio c/c0 is seen to be described at all times as a function of the single parameter z- The function P(z) defined by Equation (61) is the normal or Gaussian distribution function, Equation (C.10). Example 2.5 considers how the concentration profile of the diffusing species changes with time according to the normal distribution function. 

The most familiar of such functions is the normal, or Gaussian, distribution function ... 

FIG. C.1 A normal, or Gaussian, distribution (a) represented as a frequency function (b) represented as a cumulative function and (c) represented as a cumulative function linearized by plotting on probability paper. 

A curve with the shape given by Eq. 18-2 is called a normal (or Gaussian) distribution. Usually it is denoted as p (x) where x is the spatial coordinate and a is the standard deviation which characterizes the width of the distribution along the x-axis. The mathematical definition and properties of the normal distribution are presented in Box 18.2. 

The one-dimensional normal (or Gaussian) distribution along the jc-axis is defined by... 

A further simplification of the parent binomial distribution occurs when the number of successes is relatively large, that is, we get more than about 30 counts in a measurement. Then, the binomial distribution can be represented as a normal or Gaussian distribution. Here we write... 

The most common reason for a lack of repeatability is the existence of random fluctuations in the environment surrounding the instrument xit possibly in its power supply x2, and also in its input signal x due to random variations in the operation of a device upstream of the instrument in question. These random fluctuations frequently display a normal or Gaussian distribution. The output of a device in response to such random fluctuations may then be expressed as ... 

Most systems of fine particles have the log-normal type of particle size distribution. That is, with the logarithm of the particle size, the particle size distribution follows the normal or Gaussian distribution in semilog scales. Therefore, the density function for the log-normal distribution can be expressed by... 

Example 1.2 A coarsely ground sample of com kernel is analyzed for size distribution, as given in Table El.3. Plot the density function curves for (1) normal or Gaussian distribution, (2) log-normal distribution, and (3) Rosin-Rammler distribution. Compare these distributions with the frequency distribution histogram based on the data and identify the distribution which best fits the data. 

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