OF ATTRIBUTES IX STATISTICS. 



267 



and the number of independent frequencies of order 2m in n variable* is e<|iml to the 

 sum of the first m + I terms of this series. 



Tlit- number of independent frequencies of order 2m -j- 1 is of course equal to the 

 number of independent frequencies of order 2m. 



Tin -so rules give the numbers in Table IV. bel<>\\. 



TABLK IV. Complete Equality of Contnuiai 



14. In the ' Phil. Trans.' for 1898 a very striking theorem was given by Mr. W. 

 F. SHEPPARD,* expressing the frequencies like (AB), (Aft), &c., in quadrants of the 

 normal surface in terms of the coefficient of correlation r. Our equations like (2) in 

 1 1 , on p. 263 above, enable us at once to extend the use of this theorem to the case 

 of three variables. Let us take two examples from the case of Heredity on the 

 assumption of Galton's Law. 



(1) If the father and grandfather of a man are both above the average as regards 

 any one character, what is the chance that he will be above the average ? 



The following are the correlation coefficients : 



Son and father + '3000 



Son and grandfather 4- '1500 



Father and grandfather + -300Q 



Mr. SHEPPARD'S theorem then gives the following for the frequencies per 10,000 

 above and below average . 



Son. 

 above below 



Son. 

 above belov 



5 



. 



u -r 

 J 



8986 



2015 





 - 



2740 



2960 



.2 2015 

 8 



2985 



II 



3 8 



2260 



2740 



The first scheme holding for father and grandfather as well as son and lather. 



'Phil. Trans.,' A, vol. 192, p. 101. 

 2 M 2 



