Mr. F. Y. Edgeworth's Problems in Probabilities. 375 

 And the probability of an observation falling inside (Q + z) is 



i+#(Q) + f<W.)]s 



z, as before, being small. Hence the probability of the error 

 z being committed is proportioned to 



b +£ ~ + F-^rJ X L ¥ *~ f » J ■ 



a [Wf«] ar-^C 



NOW , /rv . n -C47702 n w 



V*7TC \7TC \TTC 



Whence, as the law of facility for the error of the Quartile 

 point, we have -i-7« ?2 



where J is taken so that \ydz between extreme limits =1. 



Thus we have found the error incident to both extremities 

 of the line which we assume as the " probable error " of the 

 curve under consideration. And it appears at first sight that 

 we might calculate the error in the length of the line by adding 

 together the Modulus-squared for each of the two errors on 

 which the error of the line depends. There occurs, however, 

 the difficulty that these errors are not independent *. A drift 

 which shifts the Median in either direction is apt to shift the 

 Quartile in the same direction, and vice versa. In view of 

 this difficulty, I see no way but to be contented with the sum 

 of the Moduli-squared as a superior limit to the sought Modu- 

 lus-squared of error ; and with the smaller of them as an 

 inferior limit. The superior limit of the Modulus-squared. 



(for the error of the Quartile) is then nearly — . The (supe- 



n 



rior limit of the) Modulus for the same error is \f -c. The 



error incurred in determining the Modulus (of the given set 



of observations) follows from the proposition that the Modulus 



is 2*097 x probable-error. The Modulus for the error in the 



determination of the Modulus of the given observations is 



c 

 found to be < 3*7 — -7= ; where for c we may put the apparent 



Modulus (in the example above given, 2,000,000). 

 * The same difficulty attends the use of the " Octiles " and "Deciles." 



