482 Theory of the Average. [CHAP. xix. 



B, C, D, is a minimum. But we might have selected a point 

 such that the cubes, or the fourth powers, or any higher 

 powers should be a minimum. These would all yield curves 

 resembling in a general way the dotted line in our figure. 

 Of course there would be insuperable practical objections to 

 any such courses as these; for the labour of calculation 

 would be enormous, and the results so far from being better 

 would be worse than those afforded by the employment of 

 the ordinary average. But so far as concerns the general 

 principle of dealing with discordant and erroneous results, it 

 must be remembered that the familiar average is but one 

 out of innumerable possible resources, all of which would 

 yield the same sort of help. 



17. Once more. We saw that a resort to the average 

 had the effect of humping up our curve more towards the 

 centre, expressive of the fact that the errors of averages are 

 of a better, i.e. smaller kind. But it must be noticed that 

 exactly the same characteristics will follow, as a general rule, 

 from any other such mode of dealing with the individual 

 errors. No strict proof of this fact can be given here, but a 

 reference to one of the familiar results of taking combina 

 tions of things will show whence this tendency arises. Ex 

 treme results, as yielded by an average of any kind, can only 

 be got in one way, viz. by repetitions of extremes in the 

 individuals from which the averages were obtained. But 

 intermediate results can be got at in two ways, viz. either by 

 intermediate individuals, or by combinations of individuals 

 in opposite directions. In the case of the Binomial Law 

 of Error this tendency to thicken towards the centre was 

 already strongly predominant in the individual values before 

 we took them in hand for our average ; but owing to this 

 characteristic of combinations we may lay it down (broadly 

 speaking) that any sort of average applied to any sort of law 



