Gaussian distribution peak value

Fig. 2. Plasma levels of MMP-9 in healthy patients. Note the non Gaussian distribution of values with a small second peak of normal values.

In a situation under stable conditions with a wind speed of 4 m s and tr, = 0.12 radians, the wind is bloviring directly toward a receptor 1 km from the source. How much must the wind shift, in degrees, to reduce the concentration to 10% of its previous value (At 2.15o- from the peak, the Gaussian distribution is 0.1 of the value at the peak.)... 

The gaussian distribution is a good example of a case where the mean and standard derivation are good measures of the center of the distribution and its spread about the center . This is indicated by an inspection of Fig. 3-3, which shows that the mean gives the location of the central peak of the density, and the standard deviation is the distance from the mean where the density has fallen to e 112 = 0.607 its peak value. Another indication that the standard deviation is a good measure of spread in this case is that 68% of the probability under the density function is located within one standard deviation of the mean. A similar discussion can be given for the Poisson distribution. The details are left as an exercise. 

Figure 6. Shown is the correlation between the liquid s fragility and the exponent p of the stretched exponential relaxations, as predicted by the RFOT theory, superimposed on the measured values in many liquids taken from the compilation of Bohmer et al. [50]. The dashed line assumed a simple gaussian distribution with the width mentioned in the text. The solid line takes into account the existence of the highest barrier by replacing the barrier distribution to the right of the most probable value by a narrow peak of the same area the peak is located at that most probable value. Taken from Ref. [45] with permission.

Eq. 17.42 is the expression of the resolution for CE in electrophoretic terms. However, the application of this expression for the calculation of Rs in practice is limited because of D,. The diffusion coefficient of different compounds in different media is not always available. Therefore, the resolution is frequently calculated with an expression that employs the width of the peaks obtained in an electropherogram. This way of working results in resolution values that are more realistic as all possible variances are considered (not only longitudinal diffusion in Eq. 17.42). Assuming that the migrating zones have a Gaussian distribution, the resolution can be expressed as follows ... 

If the peak is not symmetrical, different values will be calculated for n because the width measurements will not follow the predicted Gaussian distribution. In general, for asymmetrical peaks, n increases the higher up on the peak the width is measured. 

The most common performance indicator of a column is a dimensionless, theoretical plate count number, N. This number is also referred to as an efficiency value for the column. There is a tendency to equate the column efficiency value with the quality of a column. However, it is important to remember that the column efficiency is only part of the quality of a column. The calculation of theoretical plates is commonly based on a Gaussian model for peak shape because the chromatographic peak is assumed to result from the spreading of a population of sample molecules resulting in a Gaussian distribution of sample concentrations in the mobile and stationary phases. The general formula for calculating column efficiency is... 

Fig. 6.5. Peak spreading strongly affects enrichment ratio at fixed probability of retention. The coefficient of variance CV is equal to the ratio of the standard deviation to the mean, and is a measure of peak breadth. For example, in both curves shown in Fig. 6.3 the CV is 1.0. The enrichment ratio was calculated for a situation in which mutant fluorescence intensity was double wild-type fluorescence intensity, the mutant was initially present at 1 in 106 cells, and the probability of retention was fixed at 95 %. The logarithmic fluorescence intensity was assumed to follow a Gaussian distribution. Fixing the probability of retention defines the cutoff fluorescence value for screening at a given CV. Enrichment ratio drops precipitously with increasing CV, as the mutant and wild-type fluorescence distributions begin to overlap. At a CV of 0.2, the enrichment factor is 600. However, at a CV of 0.4, the enrichment factor has dropped to 3 Clearly, every effort should be expended to minimize peak spreading and subsequent overlap of the mutant and wild-type fluorescence distributions.

Figure2.1 Relationships between the widths of a gaussian distribution and the value of a at various fractions of the peak maximum (C/Cmax)-...

Since we are not able to find the true value for any parameter, we often make do with the average of all of the experimental data measured for that parameter, and consider this as the most probable value." Measured values, which necessarily contain experimental errors, should lie in a random manner on either side of this most probable value as expressed by the normal or Gaussian distribution. This distribution i. a bell-shaped curve that represents the number of measurements N that have a specific value x (which deviates from the mean or most probable value Xq by an amount x - Xo, representative of the error). Obviously the smaller the value of x - Xo, the higher the probability that the quantity being measured lies near the most likely value xq, which is at the top of the peak. A plot of N against x, shown in Figure 10.1, is called a Gaussian distribution or error curve, expressed mathematically as ... 

These practical problems have led to the search for simpler and less expensive approaches. The indirect method has become rather popular.It is based on the observation that most analysis results produced in the clinical laboratory seem to be normal, Figure 16-1 shows one example from the author s laboratory. As can be seen, the values of the serum sodium concentration have a distribution with a preponderant central peak and a shape not too far from that of the Gaussian distribution. The underlying assumption of the indirect method is that this peak is composed mainly of normal values. The advocates of the method therefore claim that it is possible to estimate the normal interval If we extract the distribution of normal values from this part of the distribution. Normal limits determined by the indirect method on the basis of the distribution shown in Figure 16-1 would, however, obviously be biased compared with the shown health-associated reference limits. (Note that the term normal is here used intentionally to distinguish between the concepts of normal values and reference values.)... 

Figure 5.23 shows the results of numerical evaluation of the function under the above integral at certain temperature for several different p(EA). As could be expected, the difference between and the EA value at the peak of the curve critically depends on the behavior of p(EA) in the vicinity of ax. The characteristics of the curves are listed in Table 5.4. Distributions are peaking at EA values smaller than the appropriate E et because of the asymmetry of the curves. Notice that FWHM of Gaussian distribution is 2.355[Pg.167]

Real chromatograms (Fig. 2.6-3) take into account the thermodynamic influences as well as the kinetics of mass transfer and fluid distribution. A rectangular concentration profile of the solute at the entrance of the column soon changes into a bell-shape Gaussian distribution, if the isotherm is linear. Figure 2.7a shows this distribution and some characteristic values, which will be referred to in subsequent chapters. With mass transfer resistance or nonlinear isotherms the peaks become asym-... 

Kurtosis is a statistical measure for describing whether a distribution of values is flatter (platykurtic) or more peaked (leptokurtic) than the Gaussian distribution. 

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