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LIII. On Lines and Planes of Closest Fit to Systems of Points 

 in Space. By Karl Pearson, F.R.S., University College, 

 Pond on "*. 



(1) TN many physical, statistical, and biological investi- 

 A gations it is desirable to represent a system of 

 points in plane, three, or higher dimensioned space by the 

 " best-fitting " straight line or plane. Analytically this 

 consists in taking 



y — a^ + a^v, or z=a Q + a l x + b l y } 

 or z = a -f- ci\X 1 + a. 2 x 2 + a z x s + . . . + a n x n , 

 where y, x, 2, x u a\ 2 , . . . x n are variables, and determining the 

 " best'"'' values for the constants a , a h l> h a , a l5 a 2 , a 3 , . . . a n 

 in relation to the observed corresponding values of the 

 variables. In nearly all the cases dealt with in the text-books 

 of least squares, the variables on the right of our equations 

 are treated as the independent, those on the left as the de- 

 pendent variables. The result of this treatment is that we 

 get one straight line or plane if we treat some one variable as 

 independent, and a quite different one if we treat another 

 variable as the independent variable. There is no paradox 

 about this ; it is, in fact, an easily understood and most im- 

 portant feature of the theory of a system of correlated 

 variables. The most probable value of y for a given value 

 of x, say, is not given by the same relation as the most pro- 

 bable value of x tor a given value of y. Or, to take a concrete 

 example, the most probable stature of a man with a given 

 length of leg/ being s, the most probable length of leg for a 

 man of stature s will not be I. The " best-fitting " lines and 

 planes for the cases of z up to n variables for a correlated 

 system are given in my memoir on regression |. They 

 depend upon a determination of the means, standard-devia- 

 tions, and correlation-coefficients of the system. In such 

 cases the values of the independent variables are supposed to 

 be accurately known, and the probable value of the dependent 

 variable is ascertained. 



(2) In many cases of physics and biology, however, the 

 " independent variable is subject to just as much deviation 

 or error as the " dependent " variable. We do not, for 

 example, know x accurately and then proceed to find y, but 

 both x and y are found by experiment or observation. We 

 observe x and y and seek for a unique functional relation 

 between them. Men of given stature may have a variety 



* Communicated by the Author. 



t Phil. Trans, vol. clxxxvii. A. pp. 301 et seq. 



