Normal error curve equation

Note that Laplace (not Gauss) first derived the equation for the Gaussian (normal) error curves, which need not be normal in the sense that they normally apply to errors encountered in practice (text above). 

If the right side of this equation is plotted versus dimensionless time for various values of the group Q)JuL (the reciprocal Peclet number), the types of curves shown in Figure 11.8 are obtained. The skewness of the curve increases with 3) JuL and, for small values of this parameter, the shape approaches that of a normal error curve. In physical terms this implies that when 3JuL is small, the shape of the axial concentration profile does not change... 

For small values of the dispersion parameter one may take advantage of the fact that equation 11.1.37 takes the shape of a normal error curve. This implies that for a step function input a plot of (C — Cq)/(Cq — Co) or F(t)... 

Indeterminate errors arise from the unpredictable minor inaccuracies of the individual manipulations in a procedure. A degree of uncertainty is introduced into the result which can be assessed only by statistical tests. The deviations of a number of measurements from the mean of the measurements should show a symmetrical or Gaussian distribution about that mean. Figure 2.2 represents this graphically and is known as a normal error curve. The general equation for such a curve is... 

Equation 2.54 has the form of the normal error curve and from the geometrical properties of the curve we can show that... 

The statistical parameters generated in the process of fitting the data to the equation are also used to determine the significance of the equation. A common criterion is to retain coefficients if their two-tailed probability is less than 0.05 P(2-tail) < 0.05. A two-tailed probability smaller than 0.05 means that the deviation from the true value lies in the positive or negative regions of the normal error curve corresponding to less than 5% of the area. It... 

Figure a 1-5 shows a series of five normal error curves. In each, the relative frequency is plotted as a function of the quantity i (Equation al -15). which is the deviation... 

The method of standard additions can be used to check the validity of an external standardization when matrix matching is not feasible. To do this, a normal calibration curve of Sjtand versus Cs is constructed, and the value of k is determined from its slope. A standard additions calibration curve is then constructed using equation 5.6, plotting the data as shown in Figure 5.7(b). The slope of this standard additions calibration curve gives an independent determination of k. If the two values of k are identical, then any difference between the sample s matrix and that of the external standards can be ignored. When the values of k are different, a proportional determinate error is introduced if the normal calibration curve is used. 

Gaussian distribution A symmetrical bell-shaped curve described by the equation y = Aexp(—The value of x is the deviation of a variable from its mean value. The variance of such measurements (the square of the e.s.d.) is fl/2. In many kinds of experiments, repeated measurements follow such a Gaussian or normal error distribution. 

Equation (10) is the basic elution curve equation it is a Poisson function, but when n is large, the function approximates to a normal error function or Gaussian function. In practical chromatography systems, n is always greater than 100 and, thus, all chromatographic peaks will be Gaussian or nearly Gaussian in shape. 

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). 

The accuracy of the CCSD(T) method for strongly bound molecules is illustrated in Figure 5. This figure provides a statistical analysis of the errors in the computed values for a representative group of molecules. The curves represent the normal error distributions for three different methods commonly used to solve the electronic Schrodinger equation second-order Mpller-Plesset perturbation theory (MP2), coupled cluster theory with single and double excitations, and... 

The corresponding tracer experiment in a sand column is shown in Fig. 3.18. The complementary error function, or erfc, is a tabulated function selected values are given in Table 3.4. Note that erfc is equal to 1 — erf, where erf, the error function, is obtained by integration of the normal (Gaussian) curve. It also can be obtained from the equation... 

Fig. 3 shows the experimentally obtained normalized RVD and the curve obtained by modeling it with the normalized form of equation (3). The peak of the experimental curve is slightly lower than that of the modeled curve because the absorbance of the optical probe reached its detection limit. The same argument explains why the peak was flatter than what would be expected. Furthermore, for a fixed screw design, in spite of varying flow rates and screw speeds, all the modeled curves superimposed within reasonable error as shown in Fig. 4. This further corroborates the earlier work of Elkouss et al [8]. For the resulting master curve ( (v)),...

At low temperatures, using the original function/(T ) could lead to greater error. In Tables 4.11 and 4.12, the results obtained by the Soave method are compared with fitted curves published by the DIPPR for hexane and hexadecane. Note that the differences are less than 5% between the normal boiling point and the critical point but that they are greater at low temperature. The original form of the Soave equation should be used with caution when the vapor pressure of the components is less than 0.1 bar. In these conditions, it leads to underestimating the values for equilibrium coefficients for these components. 

Fig. 4.3. (A) Composite multispecies benthic foraminiferal Mg/Ca records from three deep-sea sites DSDP Site 573, ODP Site 926, and ODP Site 689. (B) Species-adjusted Mg/Ca data. Error bars represent standard deviations of the means where more than one species was present in a sample. The smoothed curve through the data represents a 15% weighted average. (C) Mg temperature record obtained by applying a Mg calibration to the record in (B). Broken line indicates temperatures calculated from the record assuming an ice-free world. Blue areas indicate periods of substantial ice-sheet growth determined from the S 0 record in conjunction with the Mg temperature. (D) Cenozoic composite benthic foraminiferal S 0 record based on Atlantic cores and normalized to Cibicidoides spp. Vertical dashed line indicates probable existence of ice sheets as estimated by (2). 3w, seawater S 0. (E) Estimated variation in 8 0 composition of seawater, a measure of global ice volume, calculated by substituting Mg temperatures and benthic 8 0 data into the 8 0 paleotemperature equation (Lear et al., 2000).

The normal distribution (bell-shaped Gaussian curve) is the best mathematical model to represent random, or indeterminate, errors (equation (21.13) and Fig. 21.2) ... 

Equation 7-18 relates the volume of added M+ to [M+], [X-], and the constants V7, C, and CjJ,. To use Equation 7-18 in a spreadsheet, enter values of pM and compute corresponding values ofVM, as shown in Figure 7-10 for the iodide titration of Figure 7-7. This is backward from the way you normally calculate a titration curve in which VM would be input and pM would be output. Column C of Figure 7-10 is calculated with the formula [M+l = 10 pm, and column D is given by [X-] = k sp/[M+]. Column E is calculated from Equation 7-18. The first input value of pM (15.08) was selected by trial and error to produce a small V You can start wherever you like. If your initial value of pM is before the true starting point, then VM in column E will be negative. In practice, you will want more points than we have shown so that you can plot an accurate titration curve. 

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