500 Theory of the Average. [CHAP, xix, 



should find that this at once tends to fill up the valley; and 

 if we went on long enough, that is, if we kept on taking 

 averages of sufficiently large numbers, a peak would begin to 

 arise in the centre. In fact the familiar single binomial 

 curve would begin to make its appearance. 



36. The question then at once suggests itself, ought we 

 to do this? Shall we give the average free play to perform 

 its allotted function of thus crowding things up towards the 

 centre? To answer this question we must introduce a dis 

 tinction. If that peculiar double-peaked curve had been, as 

 it conceivably might, a true error-curve, that is, if it had 

 represented the divergences actually made in aiming at the 

 real centre, the result would be just what we should want. 

 It would furnish an instance of the advantages to be gained 

 by taking averages even in circumstances which were origin 

 ally unfavourable. It is not difficult to suggest an appro 

 priate illustration. Suppose a man firing at a mark from 

 some sheltered spot, but such that the range crossed a broad 

 exposed valley up or down which a strong wind was generally 

 blowing. If the shotmarks were observed we should find 

 them clustering about two centres to the right and left of 

 the bullseye. And if the results were plotted out in a curve 

 they would yield such a double-peaked curve as we have 

 described. But if the winds were equally strong and preva 

 lent in opposite directions, we should find that the averaging 

 process redressed the consequent disturbance. 



If however the curve represented, as it is decidedly more 

 likely to do, some outcome of natural phenomena in which 

 there was, so to say, a real double aim on the part of nature, 

 it would be otherwise. Take, for instance, the results of 

 measuring a large number of people who belonged to two 

 very heterogeneous races. The curve of facility would here 

 be of the kind indicated on p. 45, and if the numbers of the 



