42 Mr. F. Gal ton on Statistics by Intercompar 





In binomial ogive 

 of 17 elements. 



In exponential ogive, 



or in binomial ogive 



of 999 elements. 



The mean 



500 



250 



71 



16 



500 



250 



82 



17 



The mean +1 unit probable error... 

 „ +2 units 



,, +3 units 





The closeness of the resemblance is striking. It rapidly in- 

 creases and extends in its range as the number of elements in 

 the binomial increases ; there need therefore be no hesitation in 

 recognizing the fact that a binomial of, say, 30 elements or up- 

 wards is just as conformable to ordinary statistical observation 

 as is the exponential. If one agrees, the other does, because 

 they agree with one another. 



The fewest number of elements that suffice to form a binomial 

 having the above-mentioned conformity is a criterion of the 

 meaning of the word "small," which was lately employed, because 

 each of those elements would be just entitled to rank as small. 



I obtain the value of any one of them in an ogive by protract- 

 ing the series and noticing how many grades are included in the 

 interval q — m. It will be found that in a binomial of 17 ele- 

 ments #— m is equal to eight fifths of one grade. Thence I con- 

 clude that in any generic series an influence the range of whose 

 mean effects in the two alternatives of above and below average 

 is not greater than, say, one half of the probable error of the 

 series, is entitled to be considered " small." 



I now proceed to show how a medley of small and minute 

 causes may, as a first approximation to the truth, be looked upon 

 as an aggregate of a moderate number of " small " and equal 

 influences. In doing this, we may accept without hesitation, 

 the usual assumption that all small, and a fortiori all minute 

 influences, may be dealt with as simple alternatives of excess or 

 deficiency — the values of this excess and deficiency being the 

 mean of all the values in each of these two phases. The way in 

 which I propose to build up the fictitious groups may be exactly 

 illustrated by a game of odd and even, in which it might be 

 agreed that the 'predominance of " heads " in a throw of three 

 fourpenny pieces, shall count the same as the simple " head " of a 

 shilling. The three fourpenny pieces may fall all heads, 2 heads 

 and 1 tail, 1 head and 2 tails, or all tails — the relative frequency 

 of these events being, as is well known, 1, 3, 3, 1. But by our 

 hypothesis we need not concern ourselves about these minute 

 peculiarities; the question for us is simply the alternative one, 

 are the "heads" in a majority or not? We may therefore treat 

 a ternary system of the third order of smallness exactly as a 

 simple alternative of the first order of smallness. Or, again, 



