Standard errors more generally

Small standard errors tell us that we are in a reliable sampling situation where the repeat mean values are very likely to be closely bunched together a large standard error tells us we are in an unreliable situation where the mean values are varying considerably. 

It is not possible at this stage to say precisely what we mean by small and large in this context, we need the concept of the confidence interval to be able to say more in this regard and we will cover this topic in the next chapter. For the moment just look upon the standard error as an informal measure of precision high values mean low precision and vice versa. Further if the standard error is small, it is likely that our estimate x is close to the true mean, p,. If the standard error is large, however, there is no guarantee that we will be close to the true mean. 

The standard error concept can be extended in relation to any statistic (i.e. quantity calculated from the data). 

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Alternatively, the experimental error can be given a particular value for each reaction of the series, or for each temperature, based on statistical evaluation of the respective kinetic experiment. The rate constants are then taken with different weights in further calculations (205,206). Although this procedure seems to be more exact and more profoundly based, it cannot be quite generally recommended. It should first be statistically proven by the F test (204) that the standard errors in fact differ because of the small number of measurements, it can seldom be done on a significant level. In addition, all reactions of the series are a priori of the same importance, and it is always a... 

Thus, when a property of the sample (which exists as a large volume of material) is to be measured, there usually will be differences between the analytical data derived from application of the test methods to a gross lot or gross consignment and the data from the sample lot. This difference (the sampling error) has a frequency distribution with a mean value and a variance. Variance is a statistical term defined as the mean square of errors the square root of the variance is more generally known as the standard deviation or the standard error of sampling. 

More generally, whatever statistic we are interested in, there is always a formula that allows us to calculate its standard error. The formulas change but their interpretation always remains the same a small standard error is indicative of high precision, high reliability. Conversely a large standard error means that the observed value of the statistic is an unreliable estimate of the true (population) value. It is also always the case that the standard error is an estimate of the standard deviation of the list of repeat values of the statistic that we would get were we to repeat the sampling process, a measure of the inherent sampling variability. 

On the basis of his experience in the engineering field, Taguchi recommended the use of the s/n ratio with a logarithmic transformation, as it is useful for most practical situations whenever the standard error of the response is a function of the response value. Although in general the above three equations are used widely, more than 70 ratios were defined by Taguchi [11]. 

When faced with a choice between two assays you may compare either their SDs, or their standard errors for any fixed n, to determine which assay is most useful. The more precise assay is generally preferred if biases are similar, assuming costs and other practical considerations are comparable as well. In such circumstances, an assay with SD = v = 0.09 is much preferable to one with r = 0.36 similarly, a triplicate assay procedure with SE = 0.02 is greatly preferable to another triplicate procedure with SE = 0.07. 

Duplicate or triplicate experiments, each involving twelve or more separate measurements have generally given the mean rate coefficient with a standard error of 0-2 — 0-4% of its value without difficulty. However, the same experimental accuracy leads to a greater error in second-order rate coefficients. 

It is not generally necessary to avoid correlations, although techniques exist for reducing correlation coefficients by changing the structure of the data analysis. It is more important to reduce the standard errors of interesting quantities than to reduce the correlations. 

The data cited are taken from the original papers and therefore sometimes lack conformity, especially in representation of experimental errors . The latter are given by standard deviations in some papers, by two or three times this value in others, and sometimes have been calculated from more general considerations. 

A stochastic dynamics simulation requires a value to be assigned to the collision frequency friction coefficient 7. For simple particles such as spheres this can be related to the diffusion constant in the fluid. For the simulation of a rigid molecule it may be possible to derive 7 via the diffusion coefficient from a standard molecular dynamics situation. In the more general case we require the friction coefficient of each atom. For simple molecules such as butane the friction coefficient can be considered to be the same for all atoms. The optimal value for 7 can be determined by trial and error, performing a stochastic dynamics simulation for different values of 7 and comparing the results with those from experiment (where available) or from standard molecular dynamics simulations. For large molecules the atomic friction coefficient is considered to depend upon the degree to which each atom is in contact with the solvent and is usually taken to be proportional to the accessible surface area of the atom (as defined in Section 1.5). 

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