Practice of Kinetics (Comprehensive Chemical Kinetics, Volume 1)

374 TREATMENT OF EXPERIMENTAL DATAThen, if each value of a can be written as afa(u) where a(a), the standard deviationof a, is independent of the value of u, the weight of each value of f(u) decreasesrapidly as u decreases in accordance with the equation2w=aOn the other hand, if the relative error in a remains constant so that the value of.(a) in the expression u+a(a) is proportional to a, the weight of each value off(a) is the same irrespective of the value of u; in this casew = 1for all aIt is clear that the two methods of weighting differ greatly and in consequence therate coefficient calculated from a given set of data on the basis of the first methodis not exactly the same as that calculated using the second method.This simple example illustrates the very real problems which arise when it isdecided to weight individual observations in computing some type of averagequantity; it is for this reason that the problem is often avoided by allocating equalweights to all observations irrespective of the improbability of this being justifiedin the case of most non-linear functions of a. Be that as it may, we should mentionthat the above criterion for deciding the weights, viz.is not the only one used in practice. Often, weighting is carried out on the basiswhich is rather less severe than that based on the reciprocal of c'{f(uij)}. Thesetwo methods of weighting are sometimes described as weighting according to (a)the reciprocals of the squares of the error bars or (b) the reciprocals of the errorbars themselves. Error bars are used in the graphical representation of data toindicate the precision with which each point is-known when only one of the variablesis subject to error. As far as the author is aware, there is no single conventionemployed by all scientific workers for the construction of error bars. Assumingthat the error is in the quantity plotted on the ordinate, the most common methodused is to construct a line parallel to the ordinate of length twice the standarddeviationt such that its mid-point is coincident with the value in question. In thist If the point in question is the mean of several determinations, the length of the error bar shouldbe equal to twice the standard deviation of the mean or the standard error as it is more often called.