Exercises in the Calculation of Errors, 99 



(a) One source of error is at the origin or centre from 

 which are measured the deviations, x, y, &c, which form 

 the data for determining the required coefficients. The 

 error in the determination of the centre will depend upon 

 the method of determining it. For the purpose in hand I 

 recommend the sort of method which Mr. Galtonhas pursued: 

 the use of "percentiles," together with some process of 

 " smoothing " *. The simplest variety of this generic prin- 

 ciple is to use the observed quartiles and median in order to 

 determine the most probable values of those points consistent 

 with the condition that the median should be midway between 

 the two quartiles. Thus, if q ly q 2 , ni be the observed quartiles 

 and median respectively, and the sought ones Q x , Q 2 , M ; we 

 have to determine the latter, under the condition that the ex- 

 pression 



^i(Qi~^i) 2 + KM-m) 2 +^ 2 (Q 2 -^) 2 + 2X(Q 1 + Q 2 -2M) 



should be a maximum ; where "K is an undetermined factor, 

 w is the weight pertaining to the determination of the median 

 by putting the observed median (in) for the real one (M) ; 

 and iv ly w 2 are the corresponding weights for the quartiles 

 respectively. Whence 



Qi=2i ; 0,2=92 ; M = mH — . 



And, to determine X, we have 



Qi + Qa =2m-\ . 



* a w x w % w 



Now the error committed in taking the observed as the 



real median has for modulus \/2L. y as Laplace has proved. 



V An 



And, as the present writer following his method has reasoned, 



the error committed in taking each observed quartile for the 



real one has for modulus \/-2L- f. Accordingly — , the 



* Proc. Hoy. Soc. vol. xlv. p. 140. 



t Phil. Mag. 1886, vol. xxh. p. 375. 



The scruples expressed in the passage referred to are groundless. 

 Laplace's method is justified by the presumption that a variable, such as 

 the position of the median, depending on a number of independent agencies, 

 obeys the law of error (see Phil. Mag. Nov. and Dec. 1892). Also the 

 displacements of the two quartiles are independent of each other and of 

 that of the median ; as follows from the theory that each displacement is 



of the order -r= (with reference to unit of modulus). This consideration 



shows how many percentiles — quartiles^ octiles, &c. — ought to be utilized 

 in order to employ the generic principle to the most advantage. At least 



H2 



