Curve, Gaussian

Figure C3.5.7. Possible modes of vibrational wavepacket (smootli Gaussian curve) motion for a highly vibrationally excited diatomic molecule produced by photodissociation of a linear triatomic such as Hglj, from [8].

Statistically, a similar Indication of precision could be achieved by utilising the 95% probability level if the results fell on a "Gaussian" curve, viz., the confidence would lie within two standard deviations of the mean. R 2 x SD = 56.3 24.8... 

Normal Distribution of Observations Many types of data follow what is called the gaussian, or bell-shaped, curve this is especially true of averages. Basically, the gaussian curve is a purely mathematical function which has very specif properties. However, owing to some mathematically intractable aspects primary use of the function is restricted to tabulated values. 

The peak width (w) is the distance between each side of a peak measured at 0.6065 of the peak height. The peak width measured at this height is equivalent to two standard deviations (2o) of the Gaussian curve and, thus, has significance when... 

Gaussian curve and, thus, also has significance when dealing with chromatography theory. 

The peak width at the points of inflexion of the elution curve is twice the standard deviation of the Poisson or Gaussian curve and thus, from equation (8), the variance (the square of the standard deviation) will be equal to (n), the total number of plates in the column. 

Levenspiel [3] has given a family of Gaussian curves, and the equations representing these curves are given in Table 8-7. 

When an experimental value is obtained numerous times, the individual values will symmetrically cluster around the mean value with a scatter that depends on the number of replications made. If a very large number of replications are made (i.e., >2,000), the distribution of the values will take on the form of a Gaussian curve. It is useful to examine some of the features of this curve since it forms the basis of a large portion of the statistical tools used in this chapter. The Gaussian curve for a particular population of N values (denoted x ) will be centered along the abscissal axis on the mean value where the mean (r ) is given by... 

Fig. 10-3. Experimental proof that x-ray emission speetrography and radioactivity both conform to the unique Gaussian fluctuation curve based on N alone. Crosses = data of Rutherford and Geiger circles = x-ray emission data solid line = theoretical Gaussian curve. (Liebhafsky, Pfeiffer, and Zemany, Anal. Chem., 27, 1257.)...

These data are plotted in Figure 10-3 about the Gaussian curve for which the standard deviation is the square root of the mean. The data of Rutherford and Geiger, which were obtained by counting alpha-particles, are plotted about the same curve. In the figure, both sets of data fit the Gaussian about equally.well. 

For the usual accurate analytical method, the mean f is assumed identical with the true value, and observed errors are attributed to an indefinitely large number of small causes operating at random. The standard deviation, s, depends upon these small causes and may assume any value mean and standard deviation are wholly independent, so that an infinite number of distribution curves is conceivable. As we have seen, x-ray emission spectrography considered as a random process differs sharply from such a usual case. Under ideal conditions, the individual counts must lie upon the unique Gaussian curve for which the standard deviation is the square root of the mean. This unique Gaussian is a fluctuation curve, not an error curve in the strictest sense there is no true value of N such as that presumably corresponding to a of Section 10.1—there is only a most probable value N. 

Fig. 21.8. Frequency histograms of single-channel conductances for (A) SENS and (B) LEVR parasites. Gaussian curves were fitted to each distribution using the maximum likelihood procedure. The peaks for the SENS isolate were 21.4 + 2.3 pS (8% area) labelled G25 33.0 + 4.8 pS (31% area) labelled G35 38.1 + 1.2 pS (19% area) labelled G40 and 44.3 + 2.2 pS (42% area) labelled G45. The peaks for the LEVR isolate were 25.2 4.5 pS (21% area) labelled G25 41.2 1.7 pS (49% area) labelled G40 and 46.7 1.1 pS (30% area) labelled G45.

Finite resolution and partial volume effects. Although this can occur in other areas of imaging such as MRS, it is particularly an issue for SPECT and PET because of the finite resolution of the imaging instruments. Resolution is typically imaged as the response of the detector crystal and associated electron to the point or line source. These peak in the center and fall off from a point source, for example, in shapes that simulate Gaussian curves. These are measures of the ability to resolve two points, e.g. two structures in a brain. Because brain structures, in particular, are often smaller than the FWHM for PET or SPECT, the radioactivity measured in these areas is underestimated both by its small size (known as the partial volume effect), but also spillover from adjacent radioactivity... 

The recovery of /(C) is an ill-conditioned problem and it is reasonable to take a Gaussian distribution. Moreover, when probes experience two distinct environments (as expected for instance for gel/fluid heterogeneity), a sum of two Gaussian curves should be adequate for data analysis3 ... 

We start the chapter with a few simpler applications Beer-Lambert s law and Gaussian curves. Light absorption measurements of solutions are most commonly used for the investigation of many chemical processes. A good understanding of Beer-Lambert s law and in particular the application of the very elegant matrix notation, is useful for the methods developed later in the... 

Gaussian profiles are also utilized to approximate peak shapes observed in different types of spectroscopy. Again, we need to stress that the actual molecular processes behind a spectroscopically observed transition are very complex and do not strictly follow Gaussian curves. However, here too, Gaussian curves can serve as useful approximations. 

A Gaussian curve is characterised by peak position, peak height and peak width. Commonly, the half width is used, i.e. the width at half peak height. We accommodate this convention. In statistics, the Gaussian distribution is usually normalised to unit integral, however, this is not useful in the present context. 

The following function gauss. m creates a Gaussian curve with a given width and centre and a peak maximum of one, which is more convenient for our purposes. 

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