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VIII. Exercises in the Calculation of Errors, 

 By Professor F. Y. Edgeworth, M.A., D.C.L.* 



AN attempt to estimate and reduce the errors in the calcu- 

 lation of correlations between organs | may prove not 

 only useful to those who cultivate that branch of exact biology, 

 but also interesting to the less specialized student of Proba- 

 bilities, in virtue of some precepts of general application. 



As shown in former papers, the calculation of correlations 

 consists of three stages. We begin with the coefficients called 

 by Mr. Galton r, and by the present writer p 12 , /o 13 , &c; which 

 determine the correlation between the members of each pair 

 of organs ; for the probability of any two deviations os and y 

 (each measured from the corresponding average, in units of 

 the proper modulus) being associated is equal to a constant 

 raised to a power denoted by the following expression : 



^i-Pu ^1-PIi ^~Ph 



The conclusion sought is the quantic of the second degree 

 which forms the exponent of the expression for the proba- 

 bility that particular values of all the organs should be asso- 

 ciated ; which in the case of three variables is of the form 



ax 2 + by 2 + cz 2 + 2fyz + 2gxz + 2hxy. 



This result affords answers to questions like the follow- 

 ing : — (1) Given the values of one or more of the variables, 

 what are the values of the remaining variables which are 

 most probably associated with the given ones ? (2) Given the 

 values of one or more of the variables, what is the dispersion 

 of each of the other variables about its most probable value X ? 



Intermediate between the first and last stage of the calcu- 

 lation is the determination of the proportionate values of the 

 coefficients a, b, c, /, g, &c. ; by which we are able to answer 

 questions of the first kind, but not of the second § . 



I propose to estimate for each of these three computations 

 the error to which it is liable ; that is the extent to which 

 the results obtained from a given number of specimens are 

 likely to differ from the results which would be obtained from 

 an indefinitely large number of specimens. 



I. The ^-coefficients which come first are each liable tc 

 errors of two or three kinds : — 



* Communicated by the Author. 



f See papers by the present writer in Phil. Mao;., Aug 1 ., Nov., Dec. 

 1892, and Jan. 1893. ' 



% See Phil. Mag. Jan. 1893. § Ihid. 



