318 Mr. F. Y. Edge worth on the Law of Error 



that the phenomenon observed is thus generated ; there are 

 taken at random n elements, each of which may have either 

 the value zero or a, but not with equal probability ; the chance 

 of zero being p, the chance of a (say greater) q. The proba- 

 bility that the sum of n such elements should have the value 

 na would of course be given by the binomial law. Now we 

 have to determine, upon the hypothesis made, both n and a. 

 For a first equation we have (1) na — l. A second equation 

 is afforded by the probable error of the observed group ; for, 

 as will appear presently, any binomial locus is accurately re- 

 presented by the ancillary probability-curve up to the probable 

 error, and indeed as far as what Mr. Galton calls the decile. 

 Therefore we may equate the observed probable error to *475c, 

 where c is the modulus of the ancillary probability-curve. 

 But the modulus of this curve is in terms of n and a \/ 2pqn a, 

 p and q are given when M, the point of maximum ordinate, 

 is observed ; for p : q : : OM : ML. Thus then we are given 

 the value of v ' n a ; let it = k. From this equation combined 



k 2 I 2 



with the first we have a = -j, n= -'yr 



It may be objected to this reasoning that one of the data, 

 the distance between extreme limits, I, is apt to be very inac- 

 curate. It will be found, however, that the error introduced is 

 on the safe side, causing the evidence in favour of law to 

 appear less than it really is. 



It may be assumed then that a large proportion of unsym- 



metrical cases are reducible to the general form of Bernouilli's 



theorem. In discussing this form, let us again take Poisson as 



our guide. By parity of reasoning we shall still find Poisson's* 



formula (15) a sufficiently good approximation, provided that 



k 

 the powers of — y= constitute a descending series. But the 



further reductions by which we descend from k to the Pois- 

 sonian u, the Todhunterian t, must now be employed with 

 great caution ; in fact the formula given in most of the de- 

 rived authorities is confessedly applicable only to the body, 

 not the extremity, of the binomial-curve. Applied to the 

 latter it gives a value which is neither accurate to the extent, 

 nor in the degree, which might be supposed. It is accurate 

 over a range of divergence, not so much of the order \//jl, 

 but \/2pq/JL — a serious correction when the asymmetry is very 

 great. And the degree of accuracy is measured not by the 



* Art. 77. 



