SECT. 7.] Theory of the Average. 471 



out a diagram showing the relative frequency of each com 

 bination, we could do so without appealing to experience. 

 We could draw the appropriate binomial curve from the 

 generating conditions given in the statement of the problem. 

 But now consider the results of firing at a target con 

 sisting of a long and narrow strip, of which one point is 

 marked as the centre of aim 1 . Here (assuming that there 

 are no causes at work to produce permanent bias) we know 

 that this centre will correspond to the mean value. And we 

 know also, in a general way, that the dispersion on each side 

 of this will follow a binomial law. But if we attempted to 

 plot out the proportions, as in the preceding case, by erecting 

 ordinates which should represent each degree of frequency 

 as we receded further from the mean, we should find that 

 we could not do so. Fresh data must be given or inferred. 

 A good marksman and a bad marksman will both distribute 

 their shot according to the same general law; but the 

 rapidity with which the shots thin off as we recede from the 

 centre will be different in the two cases. Another constant 

 is demanded before the curve of frequency could be correctly 

 traced out. 



7. (2) The second division, to be next considered, 

 corresponds for all logical purposes to the first. It com 

 prises the cases in which though we have no a priori know 

 ledge as to the situation about which the values will tend to 

 cluster in the long run, yet we have sufficient experience at 

 hand to assign it with practical certainty. Consider for 

 instance the tables of human stature. These are often very 

 extensive, including tens or hundreds of thousands. In such 

 cases the mean or central value is determinable with just as 



i The only reason for supposing ing errors in two dimensions, would 

 thisexceptionalshapeistosecuresim- yield slightly more compile 

 plicity. The ordinary target, allow- suits. 



