502 Mr. F. Y. Edgeworth on the Determination 



form is that of the probability-curve. For, in determining 

 what member of this family our facility-curve is, we at the 

 same time find its mean-square-of-error. Anthropometrical 

 statistics exemplify this case. It is known by a copious 

 induction that the height-measurements of a homogeneous 

 population correspond to a probability-curve, or at least a 

 Binomial (to use Mr. Gralton's phrase) not materially different 

 from that form. Take, for instance, Dr. Baxter's statistics 

 referred to above and cited in full below. Prof. Unwin, to 

 whom I submitted these figures, has by an elegant method 

 found for the reduced curve, 



^ = 4770 (1-074) -«*. 

 Identifying the right-hand member with the expression 



1 _£? 



where N is the total number of observations, namely 315,620, 

 and c is the sought modulus, we find c = 3*7 (inches), the same 

 value which Signor Perozzo found for the ten Italian pro- 

 vinces, and which I have found for many other groups. 

 Similarly, if we were given the mortality of a stationary 

 population, not for each year of age but for periods of five 

 years, as in the example given below, we might first construct 

 a continuous curve from these imperfect data according to the 

 hypothesis of Grompertz, and then find the radius of gyration 

 for that curve. 



When we have no previous knowledge of the form of the 

 curve, the operation of quadrature becomes indeterminate. 

 In this case it is proposed to proceed as follows : — What is 

 required for the purpose in hand — the use of the Theory of 

 Errors as an aid to Induction by the Elimination of Chance — 

 is not so much an exact determination of the modulus, as a 

 superior, but not very wide, limit. For, by using a modulus 

 larger than the real one, we shall be on the safe side ; we 

 shall be underrating the evidence in favour of what is usually 

 the demonstrandum, the proposition that some observed 4cart 

 (as in the example cited from Laplace at the beginning of 

 this paper) is due to law rather than chance. And, on the 

 other hand, if our assumed constant is not very much larger 

 than the real one, we shall not require a materially larger 

 number of observations to get up to the same degree of 

 evidence. In short, we do not want to conduct our reasoning 

 along the very edge of fallacy, but rather to keep at a safe 

 distance, without, however, having to make a laborious 

 detour. 



