100 Prof. F. Y. Edgeworth's Exercises 



correction of m, 



_ gi + q 2 — 2m 

 3-2 ; 

 the corrected value of m 



__ l-2m + gi + g 2 

 3-2 ; 



and the modulus of the error incident to this determination 



= 4/l-44w-r2 + 27r-rl-7-i-3-2 \/rc = '77-f- s/n. 



Thus by the use of the principal percentiles, the median and the 

 quartiles, with a simple process of smoothing, a result is obtained 

 which is better than that combination of observations which has 

 been thought to be the best, viz., the Arithmetic mean ; for which 



1 / '77\ 

 the modulus of error is — —[ > — t=)- 



\n\ v n' 



The error which has been attributed to the determination 

 of the centre affects of course each deviation which is measured 

 from that centre as origin. But the influence which the error 

 of the observed deviations x and y has upon the coefficient of 

 correlation p 12 cannot be estimated, until the method of com- 

 bining the former in order to determine the latter has been 

 assigned. The methods which present themselves may be 

 classified as (1) the most accurate, (2) the more convenient ; 

 each introducing an error additional to those which have been 

 indicated under the heading (a) . _ 



{/3) (1) Regarding each assigned, or " subject'''* x di- 

 vided into the associated, or " relative," y, as affording an 



observation-equation— =pi 2) we see that the best combina- 

 tion of these data is obtained by affecting each observation 

 —with a weight inversely proportional to its modulus-squared. 



Now by hypothesis every y, whatever the x with which it is 

 associated, has for the modulus of its fluctuation \Zl-—p 2 l2 ^. 



as many as suffer independent displacements may with advantage be 

 admitted ; probably at least the octiles and deciles in general. 



It is important to observe that the principle may be extended to H dis- 

 cordant" observations which do not range under a single probability- 

 curve ; in which case the m>'s are to be determined according to first 

 principles, from the height of the ordinate in each neighbourhood (Phil. 

 Mag. loc. cit.). 



* See Galton, Proc. Roy. Soc. 1888, p. 140. 



t See the formula on p. 98 above. 



