334 Mr. F. Y. Edgeworth on the 



It may be observed that this test of symmetry does not 

 postulate that the curve under observation is a Probability- 

 curve. The method is particularly convenient when, as in the 

 case of measurements of human stature, we know beforehand 

 by a copious induction the value of the mean-square-of-error. 



II, We have next to test whether any other facility-curve 

 fits symmetrical groups of observations better than the Pro- 

 bability-curve. We need not for this purpose fay on an 

 indefinite number of curves. We are limited by the con- 

 dition that the substitute for the Probability-curve must be 

 an equally simple form. It must not have more than one 

 parameter ; for, of course, by multiplying parameters we can 

 make any species of curve fit any data. Nor must the vari- 

 able be involved in a very complicated manner; for we require 

 a serviceable as well as a faithful representative of the data. 



These considerations appear to limit the field of competition 

 to a few rival curves, some algebraic and some exponential. The 

 chief algebraic competitors are the Right Line and the Para- 

 bola. The exponential curves are related to the Probability- 

 curve, in that, where the latter involves the variable abscissa in 

 the second power, the former involve it in the first and third 

 power respectively. We might call them the First and Third 

 Exponentials. Affecting each of the functions so designated 

 with a proper factor (rendering the integral between extreme 

 limits equal to unity), we have as the ordinates of the four 

 rival curves the following : — 



3 /. ^ 2 \ l'._± 1 -f 3 . 





e c 3 ; - 



6'iV cJ ' 4c 2 V c 2 V ' 2c 3 > *l-7859...c 4 



Each of these curves possesses one parameter, which is to be 

 determined from the data. For this purpose we should adopt 

 some simple and uniform method which may give no advan- 

 tage to the Probability-curve. A method which well fulfils 



these conditions is to equate the observed mean error!— ) 



with that function of the parameter which ought to be equal 

 to that datum, if the curve corresponded to the actual group- 

 ing. We have thus, putting e for the observed mean- error, 

 the following equations for the parameters : — 



Cx—de; c 2 =fe; c 3 = e; c 4 = fl'9784 . . . e. 

 To these may be added the equation for the parameter (the 



* =|r(|). f =rG)-f-r(|). 



