Stafixt iral In ve«ti(jat Ions 



413 



from being exhausted, indeed its treasures are hardly yet touchetl; and 

 probably will not l)e until some lalx)riou8 German breaks by accident into its 

 vein of wealth'. 



Admitted that Galton did not see the whole liearing of his conception, yet 

 as pioneer he blazed the track. His methotl of approach was simple as usual, 

 (I rough but iidecjuate appro.ximation for the purpose he had in hand. lie 

 took ius hi.s average distributii)ii of n individuals the normal curve divided 

 into It equal areas, and he took as the chai'acter value of his individuals that 

 corri'.spoiidiiig to the areal bisector of each of these e((ual areas. He might 

 have taken the mean.s of those equal areas. Neither is in accordance with 

 the more accurate method of approach which results from a fuller applica- 

 tion of the theory of probability, liut the ditference is not very important 

 unless the number of individuals be small; thus for a sjimple of 100 we find: 



I 



The ratios of first to second prizes given by the three methods are 

 respectively : Median 74-6 to 25-4, Mean 766 to 23-4, Actual 73-6 to 26-5. 



Thus the tluee methods do not lead to widely different results as far as the 

 ])rize ratio is concerned ; the first prize may, as Galton observes, be adetjuately 

 taken as three times the second. The value for 10 competitors is 7 TO to 

 281 and for 1000 is 741 to 25'9. Hence the greater the number of com- 

 petitors the more accurate is Galton's ratio of 3 to 1. But, as he remarks, the 

 number in most competitions is limited by the fact that many individuals 

 know it would be futde to enter, so that it is reasonable to proportion the 

 prizes on numbere far larger than are actually entered'. Galton's inspiration 

 certainly divined that a better result could be obtained by using the median 

 rather than the mean in this case ! 



As far as I know no attention has been paid to Galton's ratio in settling 

 the monetary value of prizes in athletic competitions', agricultural shows and 

 scliolastic awards. Its importance is ol)scured by the tact that it is merely 

 an average value, and we are only to-day discovering how the system of 

 differences — for example, between the first and second individuals — is 



hut really uudenionsti-ahly. What lie did not write was that the applications of a trutli are 

 rarely fully recognised hy its first discoverer. It is just because the exceptional mind does not 

 and cannot exhaust the utility of its own creations that lesser minds value its teachings and 

 recognise a niaater-liuildcr. 



' A prophecy which has come unexpectedly true! See von Bortkiewicz's recent paper on 

 the range or distance between first and last individuals in a series. 



- The tii-st and second hoi*s«\s in the Derby are not merely the best out of many horses that 

 actually run, but nut of the very much larger number of potential runners. 



^ I am informed that the 3 : 1 ratio is in use in at least one public school, but the origin of 

 it is obscure. 



