Error quotient

For a simple quotient and large values of N, the rule13 for the combination of errors leads to... 

Finally, a measure of lack of fit using a PCs can be defined using the sum of the squared errors (SSE) from the test set, flSSETEST = Latest 2 (prediction sum of squares). Here, 2 stands for the sum of squared matrix elements. This measure can be related to the overall sum of squares of the data from the test set, SStest = -Xtest 2- The quotient of both measures is between 0 and 1. Subtraction from 1 gives a measure of the quality of fit or explained variance for a fixed number of a PCs ... 

A useful trial variational function is the eigenfunction of the operator L for the parabolic barrier which has the form of an error function. The variational parameters are the location of the barrier top and the barrier frequency. The parabolic barrierpotential corresponds to an infinite barrier height. The derivation of finite barrier corrections for cubic and quartic potentials may be found in Refs. 44,45,100. Finite barrier corrections for two dimensional systems have been derived with the aid of the Rayleigh quotient in Ref 101. Thus far though, the... 

For multiplication and division, first convert all uncertainties into percent relative uncertainties. Then calculate the error of the product or quotient as follows ... 

We can find a value of x that satisfies this equation by trial-and-error guessing with the spreadsheet in Figure 6-9. In column A, enter the chemical species and, in column B, enter the initial concentrations. Cell B8, which we will not use, contains the value of the equilibrium constant just as a reminder. In cell B11 we guess a value for x. We know that x cannot exceed the initial concentration of IO7, so we guess x = 0.001. Cells C4 C7 give the final concentrations computed from initial concentrations and the guessed value of x. Cell Cl 1 computes the reaction quotient, Q, from the final concentrations in cells C4 C7. 

Experience shows that the activity coefficients on this scale stay near unity (usually within experimental error) as long as the concentrations of the reactants are kept low, say less than 10% of the concentrations of the medium ions. The activity ( concentration) of several ions, notably H+, can be determined conveniently and accurately by means of e.m.f. methods, either with or without a liquid junction. In the latter case the liquid junction potential is small (mainly a function of [H+] ) and easily corrected for (3). The equilibrium constant for any reaction, on the ionic medium scale, may then be defined as the limiting value for the concentration quotient ... 

This rule is an approximation to a more exact statement that the fractional error of a product or quotient is the square root of the sum of the squares of the fractional errors in the numbers being multiplied or divided. 

This quotient is the critical quantity for the practical application of an analytical method. In addition, the relative standard deviation of an analyte result sometimes must not exceed a specified maximum. These two requirements lead to the. .. pragmatic setting of so-called limits of quantitation. Since such limits depend on the particular value observed, they cannot be regarded as meaningful characteristics of the method [MULLER et al., 1994], In brief conclusion, the analyst is well advised not to report determinations below this limit, because they will probably be biased owing to unacceptably high random errors. 

These considerations illustrate why it is easy to mistake lack of agreement between calculated and experimental values of kn, due to the assumption of an incorrect reaction mechanism, for a medium effect. If the model of a reaction to which the simple equilibrium theory is applied is in error, the solvent isotope effect expression (56) will contain some incorrect factors of the form (1 — n + mf)). Suppose, for example, that the expression (48) or (49)—applicable to a reaction of A-l mechanism—is used in conjunction with experimental data for an A-2 mechanism. Analysis of the results should lead to the conclusion that a factor of the form (1—n + mf>)2 (cf. equation (50)) has been omitted from the required theoretical equation. However, alternatively the conclusion might be drawn that equation (100) ought to have been used in place of (48), and the lack of agreement would then be ascribed to the presence of the factor Y. But Fg is itself a quotient of transfer... 

By minimizing the error between b ) and the b simultaneously, which are respectively the intercept and the slope of the graph, one can find the best fit for the calculated data points. In (18.3), w, is the weight of the point on the line determined from the error bars in each isothermal-isobaric simulation. For the propagation of error, and in particular, by using the uncertainty in sums and differences and the uncertainty in products and quotients rules, the intersection of the lines, x, can be shown to be ... 

In plant cell cultures, shake flask culture is an indispensable stage of cultivation. Investigations in a shake flask are very essential and critical to bioprocess scale-up and optimization. We have developed a simple and convenient technique based on the principle of the Warburg manometric method to measure 02 uptake rate (OUR) and C02 evolution rate (CER) of suspended cells in a shake flask culture. This technique has been successfully applied to suspension cultures of Panax notoginseng cells, and some important bioprocess parameters, such as OUR, CER, respiratory quotient (RQ), SOUR and specific CER (SCER), were quantitatively obtained [99]. As long as the environment temperature is strictly controlled to within an error of 0.1 °C, the measuring system is accurate and reproducible, is easy to operate, is economical, and is also able to treat many samples simultaneously. 

Many of the mistakes that the children made in the grouping by the quotient condition took the form of failing to distinguish the divisor from the quotient and giving the divisor as the quotient. So, when there were 4 recipients, and 3 boxes (the quotient) each with 4 items (the divisor), a common error was to state that each recipient would be given 4 items. 

Roseler et al., 1993). However the spectrally multiplexing procedure requires a photometric technique to be applied. To circumvent an absolute calibration of the intensity scale and to avoid experimental errors due to different optical paths and particularly, different irradiated areas, the evaluation should be based on quotients of intensities which were obtained under identical conditions except the polarization states. 

Analysis of the shape of error surfaces. To conclude this section, we consider a more quantitative approach to error estimation. The first step is to estimate the accuracy of the individual data points this can either be done by analysis of the variability of replicate measurements, or from the variation of the fitted result. From that, one can assess the shape of the error surface in the region of the minimum. The procedure is straightforward the square root of the error, defined as the SSD, is taken as a measure of the quality of the fit. A maximum allowed error is defined which depends on the reliability of the individual points, for example, 30% more than with the best fit, if the points are scattered by about 30%. Then each variable (not the SSD as before) is minimised and also maximised. A further condition is imposed that the sum of errors squared (SSD) should not increase by more than the fraction defined above. This method allows good estimates to be made of the different accuracy of the component variables, and also enables accuracy to be estimated reliably even in complex analyses. Finally, it reveals whether parameters are correlated. This is an important matter since it happens often, and in some extreme cases where parameters are tightly correlated it leads to situations where individual constants are effectively not defined at all, merely their products or quotients. Correlations can also occur between global and local parameters. 

Again using an example from Chapter 7, which involves calculation of a quotient, the fractional error will be given by... 

The dew-point pressure is the pressure at which the above summation is equal to one. This pressure is again found by a trial-and-error process. Here again the composition of the infinitesimal amount of liquid at the dew point may be computed by carrying out the summation at the dew-point pressure. The individual quotients obtained in this manner represent the composition of the liquid at the dew point. 

The truncation error ex is estimated as the contribution of the second-derivative term in Eq. (6.B-2) to the difference quotient in Eq. (6.B-3). Show that this gives... 

With Hr defined as above, the rounding error in the difference quotient is dominated by the subtraction in the numerator of Eq. (6.B-3). GREGPLUS uses the estimate 6mS 6 ) for the rounding error of each numerator term, thereby obtaining the root-mean-square estimate... 

More exactly, on the ionic medium scale, the equilibrium constant may be defined as the limiting value (as the solution composition approaches that of the pure ionic medium) for the equilibrium concentration quotient L. In usual measurements at low reactant concentrations, the deviaticins of L from are usually smaller than the experimental errors hence it is preferable to set L = rather than to extrapolate. 

Compared to = 85.0, this is reasonably good agreement given the nature of the calculation. We can check to see, by trial and error, if a better answer could be obtained. Because the value is low for the concentrations we calculated, we can choose to alter x slightly so that this ratio becomes larger. If we let x = 0.0190, the concentrations of NO and SO3 are increased to 0.0390 and 0.0490, and the concentrations of SO2 and NOj are decreased to 0.0010 and 0.0200 (the stoichiometry of the reaction is maintained by calculating the concentrations in this fashion). Then the quotient becomes 91.0, which is further from the value for... 

The derivatives which appear in (2.236) are replaced by difference quotients, whereby a discretisation error has to be taken into account... 

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