158 REPORT — 1889. 



as in tlie general field of modern science the received inductive methods 

 transcend the simple enumeration of the ancients. Now, the Calculus of 

 Probabilities teaches that the best answer to our question will not be 

 obtained by taking that which on the face of the evidence seems to be 

 manifested.^ 



The need of this caution is illustrated by the annexed statistics. 

 Looking at these three groups of statistics you might conclude that the 

 first one, designated A, emanated from and, if prolonged, would converge 

 to 30, as that number is the one most frequently repeated. It might 

 similarly be inferred that B in the same sense represents 38. With 

 regard to C, there might be more hesitation, since no one place or figure 

 preponderates. If, however, we double the size of our compartments 

 and consider which is the fullest of these enlarged places, that distinction 

 will be found to belong to 57-58. Accordingly 57*5 might seem the type 

 represented by this group. 



But in fact all these groups appertain to the same series, each figure 

 in all of them being formed in the same way, namely, by the addition of 

 ten digits taken at random from mathematical tables. If this series 

 were indefinitely prolonged the figure most frequently repeated would 

 be 45, a figure which in two of the groups does not even occur once. 

 A much better approximation to the greatest ordinate of the complete 

 series is obtained by taking an average other than the greatest ordinate 

 of each set of samples. For instance, the median — or figure which has 

 as many of the given observations above it as below it — is for A 42, that 

 being the fourteenth figure in the group of twenty-seven. Similarly the 

 median of B is 44 ; of C 50. The median of the whole set, numbering- 

 eighty-one, is 45 ; whereas the greatest ordinate is prima facie 88, or 

 perhaps 57*5.''' 



' Well does Dr. Venn say in the context of the passage cited from his Cainhndge 

 Antho-opometry : ' Any successful appeal to this [the point of maximum frequency] 

 requires far more extended statistics than those at our disposal.' Yet he has 520 

 returns before him 1 



- See Jovrnal Royal Statistical Society, June 1888, where it is attempted to meet 

 the difficulty presented by such ambiguity. The method there recommended is to 

 rearrange the statistics in larger groups defined by a new ' degree ' or ' unit ' which 

 is some multiple of the given one (that is, of unity in our example). The unit to be 

 adopted is the smallest interval which will bring out the one-headed character of the 

 curve ; in the cases above instanced generally 6 or 7. Now, we may begin this opera- 

 tion not only from either extremity of the given discontinuous curve (as stated in the 

 paper referred to), but also with equal plausibility from any intermediate point. 

 There are thus about as many systems as the new degree is greater than the old one ; 

 in the cases before us usually six or seven. The apex of any of these arrangements 

 giving an equally plausible solution, it is proper to take the Mean of them all. I 

 have performed this operation on each batch of twenty-seven figures (given in the 

 text), and on the united eighty-one, with results in each case differing very little from 

 the Arithmetic Mean, which is the best answer that can be extracted from these data. 



Professor Unwin, to whom this problem has been submitted, recommends forming 

 a derived curve by joining the tops of each pair of adjacent ordinates in the given 

 discontinuous curve ; and continuing this process of graphical derivation until we 

 reach a smooth (one-headed) curve. He has been so kind as to subject to this treat- 

 ment the eighty- one figures above given, and after eight repetitions of the process 

 finds for the eighth derived curve one whose greatest ordinate is 4.S— a very 

 respectable approximation, when we consider that what may be called the real point 

 is 45 ; that the result given by the Arithmetic Mean, which is here the best solution, 

 is 45-2 ; and that the probable error to which even that best solution is liable is 1-4. 



These processes are, however, very troublesome. Still, in doubtful cases, it may 

 be well to check the Median by recurring to first principles and ascertaining the 

 whereabouts at least of the Greatest Ordinate. 



