Law of Error and Correlated Averages. 435 



as before. By parity the number above (71-5) 2 is calculated to 

 be 102. 



With regard to the lower limb, in order to predict the 

 number below65'5 2 ,\v enave ^57-38-4290-25 = 367*13= 727 

 modulus, which corresponds to 



l(l-(-6914 + -7 x -0067)) x 1000 = i(l--6961) x 1000 = 152 ; 



a result which is wider than the former result from the true 

 figure 147 by only an unit. For the number below (64'5) 2 

 there is found by parity a result not much worse than that 

 which was obtained from the primary observations. 



I have applied a similar test to the group which is formed 

 by cubing the original observations. The results of both 

 verifications are embodied in the annexed Table * : — 



Below 

 645 in. 



Below 

 65*5 in. 



Above 

 70 5 in. 



Above 

 71-5 in. 



Obsfirvprl 



72 



147 



184 



104 







I 



Is 

 o 



6 



From the original 1 

 observations J 



From the squares 1 

 thereof J 



From the cubes 



79 



82 

 86 



151 



152 

 151 



190 



190 

 191 



104 



102 



98 





This verification might, I think, have been predicted from 

 the circumstance that the Arithmetic means of the squares and 

 cubes differ by very little from the respective medians, 68' 2 2 

 and 68*2 3 (the square root of the one Arithmetic mean being 

 68*2, the cube root of the other 68"3, each correct to the first 

 decimal). Now the distortion to be apprehended is the 

 unsvmmetrical extension of the upper, and shrinking of the 

 lower limb ; but this cannot be considerable, while the 

 Median and Arithmetic mean are nearly coincident. 



Accordingly we may add to the verifications above re- 

 corded other instances in which the consilience between the 

 Median and Arithmetic mean is preserved. Thus in the case 

 of observations on the height of adult males recorded by the 

 Anthropometric Committee of the British Association (Report 

 of the Brit. Assoc. 1883), the Arithmetic mean of the primary 

 observations (expressed as per-milles, and upon the under- 



* I have to thank Mrs. Bryant, D.Sc, of the North London Collegiate 

 School, for having worked the greater nart of the arithmetical examples in 

 this and the preceding paper (Phil. Mag. Aug. 1892). 



