172 



On the 1 Normal Curve ' of Statistics. 



It is shown that, whatever the laws of distribution may be, the 

 correlation-solid of the distributions of L and M is the same as that 

 of the distributions of ZL + mM and Z'L + m'M, where Z, m, Z', m 

 are any constants whatever. 



If L and M are distributed "normally," and the distributions are 

 independent, the correlation-solid will be a normal solid. Hence it 

 follows that the distribution of ZL + mM is also normal. 



Galton's definition of normal correlation is adopted; his "coeffi- 

 cient of correlation " being: therefore cos D. It is shown that the 

 correlation -solid of two normally correlated distributions is a normal 

 solid, and, therefore, if the distributions of L and of M are normally 

 correlated, the values of ZL + mM are normally distributed, and the 

 distributions of ZL + mM and of Z'L + m'M are normally correlated. 



The value of D, in a case of normal correlation, can be obtained 

 without calculating the means, mean squares, and mean product. If 

 we find the medians L x and M l5 and form a table of double classifica- 

 tion, thus : — 





Below L x . 



Above Lx. 





P 



Q 





Q 



P 



then D = —*L x 180°. 

 P + Q 



If we know the proportions of individuals for which L exceeds 

 values X and X', and the proportions for which M exceeds values Y 

 and Y', we can, for any particular value of D, construct an area 

 representing the proportion of individuals for which L lies between 

 X and X', and M between Y and Y'. The simplest case is that in 

 which the distributions of L and of M are classified in the same way, 

 e.g., according to the " decile " method. The proportions of indi- 

 viduals falling into the 100 classes corresponding to a double decile 

 classification are obtained by constructing a certain figure, which is 

 the same whatever the value of D may be, and moving the figure 

 through a distance equal to D/360° of its whole length. The diagram 

 so obtained contains 100 areas, representing the proportions in the 

 100 classes in question. 



The definitions of independence and of normal correlation are ex- 

 tended to any number of distributions, and ifc is shown that if the 

 distributions of L, M, N". . . . are normal, and either independent or 

 correlated, the values of ZL + mM + riN +. . . . are normally distri- 

 buted. 



