Practice of Kinetics (Comprehensive Chemical Kinetics, Volume 1)

376 TREATMENT OF EXPERIMENTAL DATAFirst, consider k;. It can be shown that the line fitted through the points (qj, y,,)passes through the weighted means of the values of xi, and ytj, X, and y,, and thati=nrwherez, =andThus every value of yIj, that is, f(alj), contributes to the average value of k; inaccordance with (1) a weighting factor dependent on the random errors of measurementof a and (2) a second weighting factor dependent on its distance from theweighted mean of the xi, values, Furthermore, the value of f(ao) does not appearin the equation for k; since we have deliberately avoided constraining the linethrough the point {to, f(a,)}.Second, consider the quantity s(k;). This is an estimate of the standard errorof k; based on (n,- 2) degrees of freedom. We use the quantity standard error todescribe the width of the distribution of k; values which we would observe if wewere able to make a vast number of separate determinations; we use the termstandard error rather than standard deviation to remind us that the distributionunder consideration is a distribution of mean values, each of which is itself derivedfrom a population. The t m (n,-2) degrees of freedom signifies that, out of ourn, experimental pairs of observations, only (n,-2) are available to give us anestimate of the precision of the measurement, the other two having been 'lost' infixing the two parameters, I, and k;, of our fitted line. Now, clearly we can neverobtain a vast number of determinations of a statistic such as k; in order to obtaina value for its standard error; necessarily, the number of observations which we canmake is limited and so we cannot do any better than make an estimate of what thewidth of the distribution would be for a vast population. This estimate is extremelyuseful for it enables us to calculate from the sample mean, i.e. kj, the limitsbetween which the population mean is likely to lie. Suppose we designate thepopulation mean by the symbol, K;. We now introduce the statistic, t,