Correlations of Sums or Differences

AFrER calculating thc correlations between several series of values, it frequently happens that we want the correlations given by some of the series added together ; or, what comes to the same thing, we want the correlations of the average of gome of the series.Suppose, for instance, that we have tested a number of children on t w o separate occasions, We arc bound to work out the two sets of results scparatcly, in order to ascertain how far they differ from one another, and whether the differences indicate a change of experimental conditions or may reasonably be ascribed to mere ' chance.'But having done this, we next generally desire the result of taking the mean of both occasions.Or again, suppose that we have measured the correlations of the accuracy and also of the speed of any performance ; it is almost always interesting to regard the performance as a whole, allotting marks partly for speed and partly for accuracy.Or once more, supposing that we have found out the correlations between a nurnber of experimental tests and position in school, we may wish to learn how far the school position correlates with all the tests pooled together.Scarcely less important than the correlations between sums are those between differences.When, for instance, children have been tested twice, our chief interest might be in their improvement; we wish to get the correlations of this improvement, that is, of the remainder obtained by subtracting the first result from the second.But the calculations involved in obtaining the corrclations for tllc sums or the differences are generally laborious.Before we can start working out the further coefficients required, we have to make the necessary additions or subtractions of the original values.And before commencing even this operation, we are obliged to reduce these values to suitable proportions to one another; if, as is usually the caee, we * Under such circumstances, (8a) and (9) will sometimes yield values which are impossible, not lying between + 1 and -1.This does not show that the equation is wrong, but only that the case of which they treat-an infinite number of a's and b's presenting on an average the same sized inter-correlations as the a's aud b's observedcannot occur with the given values (usually, owing merely to their errors of sampling).This impossibility can be traced to the logical requirement that, whatever series 1, 8, and 3 may denote, r, must necessarily lie between the limits : r12r13=&=d1rlaa-r12r,92rr.(see Yule, ibid.p. 246).The converse fact is worth remembering: whenever ( 6 u ) aud (9) --~