Practice of Kinetics (Comprehensive Chemical Kinetics, Volume 1)

372 TREATMENT OF EXPERIMENTAL DATAwhere R, the number of ai values making up the population, is extremely large.The variance of the population is a measure of the scatter of the observed valuesof a at the time (t- to) about the population mean. An alternative measure of thewidth of the distribution is the square root of the variance, {a2(a)}*, or moresimply a(a); this quantity is termed the standard deoiution of the distribution andhas the advantage that it characterizes the width of the distribution on the samescale as a.Now, if, as a result of performing a large number of experiments, we observeda random scatter of the atj values about some mean E,, it follows that the valuesof f(a,) would also be scattered about some mean value. For purely randomerrors, it can be shown that the weight which should be assigned to each value off(ai,).in our previous formulae is inversely proportional to the variance of thecorresponding population of f(a,), a2{f(aij)}; i.e.The constant of proportionality, Q', may be regarded as the variance of a set ofobservations of unit weight. If eqn. (58) giving the best value of the rate coefficientis examined carefully, it will be seen that the presence of a constant multiplier inthe weighting factors makes no difference at all to the calculated value of k;; thesame statement can be made about eqns. (59) and (63) giving the values of Ziand &). Therefore, for purely computational purposes, we omit the constantof proportionality, a2, from eqn. (64) and writeObviously the greater the value of the variance of f(a,), the less is the weight givento that point. Provided that the random error in f(aij) originates from the randomerrors of observation of alj and provided that these are small so that u2(ail) issmall, we can writeThe values of i3f(a)/aa follow at 6nce from the rate equation under test; for ordersa and b