2 On the Si/stematic Fitthoj of Ciirres 



i{ f(x) he an algebraical cxprcssion in iiitcf^er powci-s of x, \vc rcducc the fitting 

 of curves of thi.s type tu the theory of panibola-Htting with which wc shall be 

 occupied later. 



A very intercsting case of such work arises in dealing witli frequency 

 distributions, which wc suppose to bc normal or approxiniately normal, but of 

 which only a portion of the distribution can bo known or observed. For example, 

 the marks of cjxndidates in a competitive examiuation, wherein candidutes below 

 a ccrtain grade have been rejectcd by a preliminary examinatiou, or are cast out 

 without placing. Or again, the statures of the soldiera in a rcgiment with 

 a minimum admissible height. 



Clearly in such cases as these we have to fit a curve 



givcn a certain nuniber only of values of y and x. 



The method of least Squares or that of moments would enable us theoretically 

 to determine y,. « and b and so to find the constants of the best fitting normal 

 distribution. But with tho curve in its above form tlic e(|uations, especially in the 

 case of least scpiares, become unmauagcable. If, however, we write y = e'', wc find 

 the problem reduces to fitting 



Y=a'a^ + b'x + c, 

 wherc Y is known for a ccrtain ränge of values of x. 



As far ;is I know the first attempt to determine the constants of a normal 

 curve when only a portion of the distribution is known w:is made by Mr Francis 

 Galton in his memoir on the speed distribution of Americjvn Trotting Horses*. 

 The American record contains only horses which oun trot a mile in less than 

 a given number of seconds. Henco assuming the distribution to be normal we 

 obtain oniy a portion of the frequency distribution, i.e. the number of horses that 

 can trot a mile in each number of seconds less than this maximum. 



Taking a normal curve 



Mr Galton has determined the position of the mode, i.e. the valuc of h, only 

 by inspection of the plottcd figures. It seemed worth wiiile to compare his 

 rcsults with what we sliould get by fitting curves 



Y= a'x' + h'x + c' 

 to the logarithms of his frequency data, using the method of fitting parabolas of 

 the second onier discussed on ]>. 14 below-f. 



It seems well to briefly indicate the process used. The curve for the year 

 1893 was determined by mc, tho.se for 1802, 18f)4 and 18!).") I owe entircly to the 

 encrgy of Mr Leslic Bramlcy-Moorc. 



* li. S. Proc. Vol. 02, p. 310. 



+ Tho curve bcing parnbolio the methods of moments and of Icast Squares are now sensibty identical 

 in result, altbough uot alike in their stages. 



