Gaussian distribution curve

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. 

Column Efficiency. Under ideal conditions the profile of a solute band resembles that given by a Gaussian distribution curve (Fig. 11.1). The efficiency of a chromatographic system is expressed by the effective plate number defined from the chromatogram of a single band. 

The number of corrosion spots increases with time, but the maximum penetration rate remains roughly constant locally and with time. The penetration rate corresponds to a Gaussian distribution curve [18]. 

The distribution of diffusing atoms represented by the Gaussian distribution curve ... 

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. 

The absorption curves given by coal macerals approached the horizontal (magnetic field strength) axis more slowly than a Gaussian distribution curve. Shape analysis (16) showed that over much of the curve, the form closely approximated a Lorentzian distribution curve, but both positive and negative deviations were found in the wings of the curves (that is, in various examples, the curves approached the axis either somewhat more or somewhat less rapidly... 

Free diffusion columns are arranged to be sufficiently long for the initial concentrations at the extreme ends of the cell to remain unaltered during the course of the experiment. For a monodispersed system under these conditions the concentration gradient curves (Figure 2.4b) can be shown, by solving Fick s equations, to take the shape of Gaussian distribution curves represented by the expression... 

This can be compared to the expression for the area under the tail of the gaussian-distribution curve... 

From a table of areas under the gaussian-distribution curve (Table 26-1) corresponding to the variable... 

Equation (23-24) is an example of a Poisson distribution, characteristic of a continuous-flow process, in contrast to the binomial-distribution characteristic of a batch process [Equation (23-13)]. Both approximate the gaussian-distribution curve when the number of stages becomes large. ... 

On the basis of our assumption of a large number of reactions, it can be assumed that the distribution of reactions can be described by a Gaussian distribution curve such that... 

The Gaussian distribution curve assumes that an infinite number of measurements of X, have been made. The maximum of the Gaussian curve occurs atx = /r, the true value of the parameter we are measuring. So, for an infinite number of measurements, the population mean is the true value x,. We assume that any measurements we make are a subset of the Gaussian distribution. As the number of measurements, N, increases, the difference between x and /x tends toward zero. For N greater than 20 to 30 or so, the sample mean rapidly approaches the population mean. For 25 or more replicate measurements, the true value is approximated very well by the experimental mean value. Unfortunately, even 20 measurements of a real sample are not usually possible. Statistics allows us to express the random error associated with the difference between the population mean fi and the mean of a small subset of the population, x. The random error for the mean of a small subset is equal tox — fi. 

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