Normal error curve experiment

Ccs is the constant concentration of the diffusing species at the surface, c0 is the uniform concentration of the diffusing species already present in the solid before the experiment and cx is the concentration of the diffusing species at a position x from the surface after time t has elapsed and D is the (constant) diffusion coefficient of the diffusing species. The function erf [x/2(Dr)1/2] is called the error function. The error function is closely related to the area under the normal distribution curve and differs from it only by scaling. It can be expressed as an integral or by the infinite series erf(x) — 2/y/ [x — yry + ]. The comp-... 

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. 

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... 

Figure 12. Extent of dissolution and re-precipitation between aqueous Fe(III) and hematite at 98°C calculated using Fe-enriched tracers. A. Percent Fe exchanged (F values) as calculated for the two enriched- Fe tracer experiments in parts B and C. Large diamonds reflect F values calculated from isotopic compositions of the solution. Small circles reflect F values calculated from isotopic compositions of hematite, which have larger errors due to the relatively small shifts in isotopic composition of the solid (see parts B and C). Curves show third-order rate laws that are fit to the data from the solutions. B. Tracer experiment using Fe-enriched hematite, and isotopically normal Fe(lll). C. Identical experiment as in part B, except that solution Fe(lll) is enriched in Te, and initial hematite had normal isotope compositions. Data from Skulan et al. (2002).

The ELS detector was previously also referred to as a mass detector, pointing to the fact that the response is (mainly) determined by the mass of the sample rather than by its chemical structure. Van der Meeren et al., though, demonstrated that the ELSD calibration curves of phospholipid classes were also dependent on the fatty acid composition (52). The dependence on the fatty acid composition is, however, completely different in nature and much less pronounced than for UV detection. The reason for this behavior is to be found in the partial resolution of molecular species, even during normal-phase chromatography. Thus, the peak shape depends not only on the chromatographic system but also on the fatty acid composition and molecular species distribution of the PL sample (47). Because it was shown before, based on both theoretical considerations and practical experiments, that the ELS detector response is generally inversely proportional to peak width (62,104), it follows that the molecular species distribution of the PL standards used should be similar to the sample components to be quantified. It was shown that up to 20% error may be induced if an inappropriate standard is used (52). 

Model I linear regression is suitable for experiments where a dependent variable Y varies with an error-free independent variable X and the mean (expected) value of Y is given by a -f bX. This might occur where you have carefully controlled the independent variable and it can therefore be assumed to have zero error (e.g. a calibration curve). Errors can be calculated for estimates of a and b and predicted values of Y. The Y values should be normally distributed and the variance of Y constant at all values of... 

Numerical evaluation of the Kassel integral permitted a comparison between theoretical and experimental fall-off behaviour . With an average molecular diameter of 5.5 A the calculated rate coefficient-azoethane pressure curve showed the best agreement with experiment at an effective number of oscillators of 18, somewhat less than half of the maximum 2N— 6. Because of the complexity of the reaction the experimental curve is probably in error, rendering comparison unreliable. Similar calculations for azomethane using the earlier uninhibited kinetic data showed best agreement with experiments at a molecular diameter of 4.7 A and an effective number of oscillators of 12, one half of the total normal modes of vibrations. 

For flames that exhibit the parametric instability, the velocity at which the exponential growth of velocity fluctuations started for each experiment was noted. These critical velocities are shown in Fig. 7.5, normalized by the laminar-flame speeds reported in [13]. All points shown on this plot represent the ensemble average of measurements from five experiments, and the error bars indicate the standard deviation about the mean value. The other curve on this plot was calculated using the analytical model of a premixed flame under the influence of an oscillating gravitational field by Bychkov [17], ris described above. Each point represents the smallest normalized acoustic velocity at the most unstable reduced wave number that resulted in the parametric instability. The experimental results show the same trend as the theoretical model mixtures with an equivalence ratio of 0.9, which require the smallest normalized acoustic velocity to trigger the parametric instability while flames on either side require larger values. 

The error Aviv in the determination reflects the accuracy of measurement of the vibration frequency (see. Fig. III.7, curve 2). Normally, the vibration frequency is fixed by the generator, and these vibrations are reproduced by the dust-covered surface. The error Ayly reflects the accuracy of measurement of the vibration amplitude. In our experiments, the combined error Av/v + Ayly was never greater than 5%. Hence, as with the centrifuging technique, the principal error here can be attributed to particle size variation. 

Figure 8.7 Fictitious results of replicate measurements of a quantity x, generated using an appropriate random number generator. The histograms show the fractions of measured values of x that fall within each bin. The bin width is chosen in each case so that several of the central bins contain some measured values, (a) results of 100 simulated experiments, (b) results of 1000 simulated experiments (the dashed curve represents the theoretical normal distribution o-(x) that was used to generate the plotted data by using it as a weighting function for a random number generator). Reproduced from Taylor, An Introduction to Error Analysis, University Science Books (1982), with permission.

The normal distribution was first introduced by de Moivre, who approximated binomial distributions for large n (Figure D.3). His work was extended by Laplace, who used the normal distribution in the error analysis in experiments conducted. Legendre came up with the method of least squares. Gauss, by 1809, justified the normal distribution for experimental errors. The name bell curve was coined by Gallon and Lexis. 

Precise measurements have small random errors (uncertainties) and are reproducible in repeated trials of experiments accurate measurements have small systematic errors and give a result close to the true or accepted value. If a large number of measurements are made of the same quantity, the readings will all lie on a normal or Gaussian distribution curve as shown in Figure 11.16. 

Fig. 12. Experiment with one channel polarizers Normalized coincidence rate as a function of the relative polarizers orientation. Indicated errors are 1 standard deviation. The solid curve is not a fit to the data but the prediction by Quantum Mechanics.

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