Deviation, average quadratic

Although the quadratic equations for retention (In k) as a function of mobile phase composition (experimental data, they are inadequate to describe retention within experimental error. For binary mixtures the standard deviation between the quadratic equation (eqn.3.38) and experimental data is typically between 5 and 10% (depending on the solute) [322], For ternary systems average deviations of 10 to 20% are typical [324]. However, the inclusion of additional (higher order) terms at will is not an attractive way to improve the description of the experimental data. We will discuss this more fully in chapter 5 (section 5.5). 

Ishlhara and West (2) reported enthalpy data from 2177 to 2355 K and reported a quadratic enthalpy equation for the liquid. This equation yields a temperature dependence for C which is unreasonably large. Therefore we have fit their enthalpy data to a linear equation which fits their measured points with an average deviation of 0.15 kcal mol" and a maximum deviation of 0.35 kcal mol". Based on the scatter in the data we believe higher fits are unjustified and adopt the resulting constant value of C =... 

Plots for 11 isoelectronic sequences having more than three data, or having three centered on IZ1-Z2I = zero, were prepared, one of which is shown in Fig. 8.3. Six of the plots have six data points, and three more have five points (one in the center and two duplicated on each side) the remainder have three non-redundant points. The average standard deviation of fitting for these is 2.595°. Figure 8.4 has the largest scatter around its trend line (a = 3.951°), so it is used as an example of what the ( — 1) standard deviations look like. There is no theoretical basis for using quadratic trend lines they are used for sake of simplicity. 

FIGURE 20.8 Calibration of the pH measurement. R, the ratio of tetramethylrhodamine and fluorescein fluorescence, versus the pH for liposomes constructed with internal pH values ranging from 3 to 7. The average ratio and standard deviation is shown with a marker and bars at 7 pH values. The calibration curve represents a quadratic model and was determined by multiple regression analysis. (Reprinted from Chen, Y. and Arriaga, E. A., Anal. Chem., 78, 821, 2006. Copyright 2006. With permission from American Chemical Society.)... 

In this section, Bayesian analysis is performed to identify the uncertain coefficients of the quadratic function. The effective temperature T is assumed to be different from the measured values since there is measurement noise in the data acquisition process and the temperature in different parts of the building could also be non-uniform. The difference is assumed Gaussian with zero mean and variance a. In this study, this standard deviation is taken to be aj = 0.5°C since the difference between the indoor and outdoor temperature measurement was around 1 °C and the average value was used as the measured temperature T for the wth day. On the other hand, the squared fundamental frequency is identified by the Bayesian spectral density approach to be presented in Chapter 3. Therefore, the uncertain parameters include the coefficients of thequadratic function and the effective temperatures 0 = [l>o, b, b2, T, Ti,7 ], where iV is the number of data points. The data include the measurements of the temperature and the identified squared fundamental frequencies X> = [li,. .., j, 2> >... 

There are few substances where it is worth to fit all the coefficients. In most cases, a quadratic polynomial is sufficient. Due to the difficulties described above, data points in the critical area are of ten left out in the regression nevertheless, average deviations of more than 1. .. 2% must often be accepted, which is caused by the experimental uncertainty of caloric measurements. For many components, heat capacity data for temperatures above the normal boiling point do not exist. In these cases, only a linear function is justified for the correlation, and the extrapolation to high temperatures becomes arbitrary. It can help to generate additional artificial data points with an estimation method. High-precision data can be correlated with the following PPDS equation... 

The first aspect taken into account is the length of each bond. The average length of covalent bonds in a molecule can be determined sufficiently exactly by spectroscopic measurements. The deviation from this ideal value appears quadratically in the potential ... 

The results were fitted using six experimental results from the first eight experiments and one result from the average value of the last three center point experiments. Comparing the measured values with the predicted values using model 1, the maximum deviation was calculated as 32%. The deviation for the center point was calculated as 4% in the worst case. The new quadratic model represents the experimentally observed behavior better than the first model. However, an exact fit can also be achieved using aU 11 experiments. In the second model, the equation for the response surface can be expressed as... 

Values of a or may be determined in different ways. Since proton transfer in the transition state is expected to vary with the strength of the Bronsted acid or base, a and p should in general represent tangents to a curved log /c-log K plot. For small variations in pK systematic deviations from linearity are usually too small to be observed and only an average value of a or can be obtained, but for larger changes the curvature may be sufficient to evaluate a quadratic term in the dependence (equation 33) ... 

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