Error quotient

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

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 

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 

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

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

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. 

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

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. 

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. 

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  

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. 

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. 

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 

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  

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 

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