Gaussian distributions single variable

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.

The majority of statistical tests, and those most widely employed in analytical science, assume that observed data follow a normal distribution. The normal, sometimes referred to as Gaussian, distribution function is the most important distribution for continuous data because of its wide range of practical application. Most measurements of physical characteristics, with their associated random errors and natural variations, can be approximated by the normal distribution. The well known shape of this function is illustrated in Figure 1. As shown, it is referred to as the normal probability curve. The mathematical model describing the normal distribution function with a single measured variable, x, is given by Equation (1). 

Our discussions so far have been limited to assuming a normal, Gaussian distribution to describe the spread of observed data. Before proceeding to extend this analysis to multivariate measurements, it is worthwhile pointing out that other continuous distributions are important in spectroscopy. One distribution which is similar, but unrelated, to the Gaussian function is the Lorentzian distribution. Sometimes called the Cauchy function, the Lorentzian distribution is appropriate when describing resonance behaviour, and it is commonly encountered in emission and absorption spectroscopies. This distribution for a single variable, x, is defined by... 

Instead, we mean here the use of experimental data that can be expected to lie on a smooth curve but fail to do so as the result of measurement uncertainties. Whenever the data are equidistant (i.e., taken at constant increments of the independent variable) and the errors are random and follow a single Gaussian distribution, the least-squares method is appropriate, convenient, and readily implemented on a spreadsheet. In section 3.3 we already encountered this procedure, which is based on least-squares fitting of the data to a polynomial, and uses so-called convoluting integers. This method is, in fact, quite old, and goes back to work by Sheppard (Proc. 5th... 

Moment analysis provides the means to determine a model independent characteristic of the absorption rate or dissolution rate (Riegelman and Collier 1980). A single value characterising the rate of the entire dissolution or absorption process is obtained, which is called the mean absorption or dissolution time (MAT and MDT, respectively). These parameters can be determined without any assumptions regarding absorption or disposition pharmacokinetics, apart from the general prerequisites of linear pharmacokinetics and absence of intraindividual variability described above. MAT/MDT can be interpreted as the most probable time for a molecule to become absorbed/dissolved, based on a normal Gaussian distribution. 

A Gaussian distribution is a type of probability distribution that characterizes the distribution of a randomly distributed variable about a single mean (average) value. This is the most commonly encountered type of distribution in statistical analysis, so it is good to be at least a little familiar with it. Provided one knows that the probability distribution for a variable is Gaussian, the distribution for that variable can be completely characterized by its mean and standard deviation, which has the shape of a bell curve ... 

Here we summarize some useful properties of the Gaussian distribution function. First we consider the case of a single variable... 

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. 

Gaussian errors with standard deviations 1, 4, 4, 3, and 1, respectively. The performance of maximum likelihood, single-scale Bayesian, and multiscale Bayesian rectification are compared by Monte-Carlo simulation with 500 realizations of 2048 measurements for each variable. The prior probability distribution is assumed to be Gaussian for the single-scale and multiscale Bayesian methods. The normalized mean-square error of approximation is computed as,... 

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