44 Mr. F. Galton on Statistics by Intercomparison, 



Now, if we apply these results to observed facts, we shall 

 rarely find that the series has been due to any large number of 

 equal elements. Thus, in the stature of man the probable error, 



Tfl 



, is about 30, which makes it impossible that it can be 



q — m r 



looked upon as due to the effect of more than 200 equally small 



elements. On consideration, however, it will appear that in 



certain cases the number may be less, even considerably less, 



than the tabular value, though it can never exceed it. As an 



illustration of the principle upon which this conclusion depends, 



Tfl 



we may consider what the value of would be in the case 



J q — m 



of a wall built of 17 courses of stone, each stone being 3 inches 

 thick, and subject to a mean error in excess or deficiency of one 

 fifth of an inch. Obviously the mean height m of the wall 

 would be 3 x 17 inches ; and its probable error q — m would be 

 very small, being derived from a binomial ogive of 17 elements, 

 each of the value of only one fifth of an inch. Now we saw from 

 our previous calculation that this would be eight fifths, or 1'6 



inch, which would give the value to of r-^, or about 321 : 



° q—m l'b 



consequently we should be greatly misled if, after finding by 

 observation the value of that fraction, and turning to the Table 

 and seeing there that it corresponded to more than 200 equal 

 elements, we should conclude that that was the number of courses 

 of stones. The Table can only be trusted to say that the num- 

 ber of courses certainly does not exceed that number; but it may 

 be less than that. 



The difficulty we have next to consider is that which I first 

 mentioned, but have intentionally postponed. It is due to the 

 presence of influences of extraordinary magnitude, as Aspect in 

 the size of fruit. These influences must be divided into more than 

 two phases, each differing by the same constant amount from 

 the next one, and that difference must not be greater than exists 

 between the opposite phases of the " small" alternatives. If we 

 had to divide an influence into three phases, we should call them 

 " large," " moderate," and " small ; " if into four, they would 

 be " very large," " moderately large," " moderately small," and 

 c< very small," and so on. Any objects (say, fruit) which are liable 

 to an influence so large as to make it necessary to divide it into 

 three phases, really consist of three series generically different 

 which are entangled together, and ought theoretically to be sepa- 

 rated. If there had been two influences of three phases, there 

 would be nine such series, and so on. In short, the fruit, of 

 which we may be considering some hundred or a few thousand 



