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Normalized variable gaussian

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. [Pg.136]

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. [Pg.79]

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. [Pg.26]

For continuous variables, the most common pdf is the normal or Gaussian distribution, which is written as... [Pg.348]

Both the binomial and Poisson distributions apply to discrete variables, whereas most of the random variables involved in experiments are continuous. In addition, the use of discrete distributions necessitates the use of long or infinite series for the calculation of such parameters as the mean and the standard deviation (see Eqs. 2.47, 2.48, 2.52, 2.53). It would be desirable, therefore, to have a pdf that applies to continuous variables. Such a distribution is the normal or Gaussian distribution. [Pg.39]

As shown in Figure 2.1, the daily average temperature in May appears to be uniformly distributed around a central point situated at about 13 °C, which happens to equal its mean temperature. This bell-shaped curve is common to all processes in which the variability (variance) in the data is random follows a normal, or Gaussian distribution. The continuous mathematical function that characterizes this is of an exponential form ... [Pg.26]

The normal (or Gaussian) distribution is a continuous probability distribution that is often used as a first approximation to describe real-valued random variables that tend to cluster around a single mean value. The graph of the associated probability density function, which is bell shaped, is known as the Gaussian function or bell curve. See Fig. 9.3. [Pg.248]

The normal (or Gaussian) probability distribution plays a central role in both the theory and application of statistics. It was introduced in Section 21.2.1. For probability calculations, it is convenient to use the standard normal distribution, 1) which has a mean of zero and a variance of one. Suppose that a random variable X is normally distributed with a mean x and variance dx-Then, the corresponding standard normal variable Z is... [Pg.505]

This section discusses a class of methods known as the first-order reliability methods to compute the probability of failure of structural systems. These methods are based on the first-order Taylor s series expansion of the performance function G(X). The first-method, known as the first-order second-moment (FOSM) method, focuses on approximating the mean and standard deviation of G and uses this information to compute Pf. Then, the FOSM method is extended to the advanced FOSM method in two steps first, the methodology is developed for the case where all the variables in X are Gaussian (normal) and, second, the methodology is extended to the general case of non-normal variables. [Pg.3651]

Characterization of Chance Occurrences To deal with a broad area of statistical apphcations, it is necessary to charac terize the way in which random variables will varv by chance alone. The basic-foundation for this characteristic is laid through a density called the gaussian, or normal, distribution. [Pg.488]

The central limit theorem thus states the remarkable fact that the distribution function of the normalized sum of identically distributed, statistically independent random variables approaches the gaussian distribution function as the number of summands approaches infinity—... [Pg.157]

Figure 1.8. Schematic frequency distributions for some independent (reaction input or control) resp. dependent (reaction output) variables to show how non-Gaussian distributions can obtain for a large population of reactions (i.e., all batches of one product in 5 years), while approximate normal distributions are found for repeat measurements on one single batch. For example, the gray areas correspond to the process parameters for a given run, while the histograms give the distribution of repeat determinations on one (several) sample(s) from this run. Because of the huge costs associated with individual production batches, the number of data points measured under closely controlled conditions, i.e., validation runs, is miniscule. Distributions must be estimated from historical data, which typically suffers from ever-changing parameter combinations, such as reagent batches, operators, impurity profiles, etc. Figure 1.8. Schematic frequency distributions for some independent (reaction input or control) resp. dependent (reaction output) variables to show how non-Gaussian distributions can obtain for a large population of reactions (i.e., all batches of one product in 5 years), while approximate normal distributions are found for repeat measurements on one single batch. For example, the gray areas correspond to the process parameters for a given run, while the histograms give the distribution of repeat determinations on one (several) sample(s) from this run. Because of the huge costs associated with individual production batches, the number of data points measured under closely controlled conditions, i.e., validation runs, is miniscule. Distributions must be estimated from historical data, which typically suffers from ever-changing parameter combinations, such as reagent batches, operators, impurity profiles, etc.
Figure 16.6 Calibration of the radiocarbon ages of the Cortona and Santa Croce frocks the software used[83] is OxCal v.3.10. Radiocarbon age is represented on the y axis as a random variable normally distributed experimental error of radiocarbon age is taken as the sigma of the Gaussian distribution. Calibration of the radiocarbon agegivesa distribution of probability that can no longer be described by a well defined mathematical form it is displayed in the graph as a dark area on the x axis... Figure 16.6 Calibration of the radiocarbon ages of the Cortona and Santa Croce frocks the software used[83] is OxCal v.3.10. Radiocarbon age is represented on the y axis as a random variable normally distributed experimental error of radiocarbon age is taken as the sigma of the Gaussian distribution. Calibration of the radiocarbon agegivesa distribution of probability that can no longer be described by a well defined mathematical form it is displayed in the graph as a dark area on the x axis...
A normal (gaussian) probability density function in one centered and standardized variable X reads... [Pg.205]

Fig. 14 A comparison of different approaches to describe the charge carrier mobility in a Gaussian-type hopping system as a function of the normalized concentration of the charge carriers, (a) Full curves are the result of effective medium calculations [100] while symbols are computer simulations [96], (b) Full curves are calculated using the variable range hopping concept [101], symbols are the computer simulations. From [100] with permission. Copyright (2007) by the American Institute of Physics... Fig. 14 A comparison of different approaches to describe the charge carrier mobility in a Gaussian-type hopping system as a function of the normalized concentration of the charge carriers, (a) Full curves are the result of effective medium calculations [100] while symbols are computer simulations [96], (b) Full curves are calculated using the variable range hopping concept [101], symbols are the computer simulations. From [100] with permission. Copyright (2007) by the American Institute of Physics...
Thus the mean and variance of a gamma distribution are sufficient to determine its two parameters, a and /3. Note that the coefficient of variation (standard deviation divided by the mean) is equal to the square root of 1/or. The most probable value of k (the mode) occurs at (a - l)//3 if oc > 1, and as a T oo the gamma distribution itself becomes the gaussian or normal distribution for the variable, /8k.13... [Pg.147]

Particle size, like other variables in nature, tends to follow well-defined mathematical laws in its distribution. This is not only of theoretical interest since data manipulation is made much easier if the distribution can be described by a mathematical law. Experimental data tends to follow the Normal law or Gaussian frequency distribution in many areas of statistics and statistical physics. However, the log-normal law is more frequently found with particulate systems. These laws suffer the disadvantage that they do not permit a maximum or minimum size and so, whilst fitting real distributions in the middle of the distribution, fail at each of the tails. [Pg.96]


See other pages where Normalized variable gaussian is mentioned: [Pg.243]    [Pg.243]    [Pg.57]    [Pg.106]    [Pg.94]    [Pg.2484]    [Pg.331]    [Pg.68]    [Pg.206]    [Pg.1058]    [Pg.599]    [Pg.313]    [Pg.41]    [Pg.218]    [Pg.124]    [Pg.15]    [Pg.345]    [Pg.170]    [Pg.196]    [Pg.309]    [Pg.310]    [Pg.266]    [Pg.349]    [Pg.62]    [Pg.264]    [Pg.709]    [Pg.185]    [Pg.5]    [Pg.115]   
See also in sourсe #XX -- [ Pg.245 , Pg.246 ]




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Normalized variables

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