Gaussian curves parameters

FIA peaks are not Gaussian curves, so that the parameters above do not describe the peak shape in full—particularly the trailing portion, which is peculiar to this type of recording and distinguishes it from the transient signals typical of other analytical techniques. 

Light bulb lifetimes, and the corresponding Gaussian curve, are characterized by two parameters. The arithmetic mean, x—also called the average—is the sum of the measured values divided by n, the number of measurements ... 

Polymer gel Polymer cohesive energy density P (cal cm ) Deviation from geometric mean mixing rule parameter z Non-Gaussian elasticity parameter, N Curve fit crosslink density (10s mol cm-3) Experimental crosslink density (10s mol cm-3)... 

FIGURE 6.7 Comparison between the smoothed exp. and the fitted tr( )/ of SO2. Upper part the smoothed exp. cr( )/ of SO2. On this upper part, the fitted curve is not superimposed on the exp. curve because these two curves are too close. The two components of this XS have been fitted simultaneously the strong component is described with Formula (29) (4 parameters) and the weak component with a Gaussian (3 parameters). The fitted parameters are given in Table 6.4. Lower part the fit residual which is typically 1% of the max amplitude. Note that the typical experimental uncertainty is significantly larger than 1%. Then the residual in not necessarily due to the model but may be mostly due to the experimental XS. 

This is a continuous function for the experimental variables, which is used as a convenient mathematical idealisation to describe the distribution of finite numbers of results. The factor 1 /(ay/lji) is a constant such that the total area under the probability distribution curve is unity. The mean value is given by p and the variance by a2. The variance in the Gaussian distribution corresponds to the standard deviation s in Eqn. 8.3. Figure 8-3 illustrates the Gaussian distribution calculated with the same parameters used to obtain the Poisson distribution in Figure 8-2, i.e. a mean of 40 and a standard deviation of V40. It can be seen that the two distributions are similar, and that the Poisson distribution is very dosely approximated by the continuous Gaussian curve. 

Figure 6-4a shows two Gaussian curves in which we plot the relative frequency y of various deviations from the mean versus the deviation from the mean. As shown in the margin, curves such as these can be described by an equation that contains just two parameters, the population mean p. and the population standard deviation a. The term parameter refers to quantities such as pu and a that define a population or distribution. This is in contrast to quantities such as the data values x that are variables. The term statistic refers to an estimate of a parameter that is made from a sample of data, as discussed below. The sample mean and the sample standard deviation are examples of statistics that estimate parameters p. and a, respectively. 

O Spreadsheet Summary Chapter 15 of Applications of Microsoft si Excel in Analytical Chemistry begins with an exercise treating the resolution of overlapped Gaussian peaks. The overlapped chromatogram, the response, is modeled as the sum of Gaussian curves. Initial estimates are made for the model parameters. Excel calculates the residuals, the difference between the response and the model, and the sum of the squares of the residuals. Excel s Solver is then used to minimize the sum of the squares of the residuals, while displaying the results of each iteration. 

Due to the mass balance the area Eq. 6.148 is identical to that of the Gaussian peak. Parameters tg, ag and tEMg are determined by curve fitting and thus the values of the moments are directly available. 

Figure 1.3 Chromatographic peaks, (a) The concept of retention time. The hold-up time is the retention time of an unretained compound in the column (the time it took to make the trip through the column) (b) Anatomy of an ideal peak (c) Significance of the three basic parameters and a summary of the features of a Gaussian curve (d) An example of a real chromatogram showing that while travelling along the column, each analyte is assumed to present a Gaussian distribution of concentration.

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. 

In this case the reproducibility model of Aq is simply a normal probability distribution function p(Aq) under the null hypothesis, with mean = 0 and standard deviation = 7. The parameter for testing Hq is the P-value, or, as we call it here, the similarity index (SI), for our simple library search system defined as the integral of the reproducibility function, in this case a symmetrical Gaussian curve ... 

It can be recognized that the quantity [2.Ni. (Vjjj -I- K. v )] in the denominator of the exponential in Equation [3.14] is the full peak width AVj measured at the inflection points (see Equation [3.19]). But this specific peak width is, for a Gaussian curve, twice the standard deviation parameter Oy = ( Vi)/2 (recall that V, the volume of mobile phase that has eluted, is a continuous variable). Then Equation [3.14] can be rewritten as ... 

The parameters were obtained by performing a least squares fitting of a Gaussian curve of the desired functional form. The fitting function is an exponential with exponent equal to the relevant potential/2RT. For the bonds, the fitting function was... 

The normal distribution is a symmetrical bell-shaped curve referred to in statistics as a Gaussian curve. It is a two-parameter function, one parameter is the mean, Xa which due to the symmetry of the curve coincides with the mode and median, and the other is the standard deviation a, which is a measure of the width of the distribution. The normal distribution of particle size is given by... 

The determination of these parameters is shown in Figure 13.2. We see that the chromatogram is a bell-shaped band or Gaussian curve, which is characterized by... 

A study was made to determine whether any of the parameters describing the polarity of liquids relate to the swelling of nitrile rubber vulcanisates. A Gaussian curve... 

This is the most widely used measure of precision and is a parameter of the normal error or Gaussian curve (Topic Bl, Fig. 4). Figure 1 shows two curves for the frequency distribution of two theoretical sets of data, each having an infinite number of values and known as a statistical population. 

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