Law of Error and Correlated Averages. 437 



Tin's dislocation is exhibited in the annexed diagram (fig. 1) ? 

 where 13'AB is the original curve, u(3 what it 'becomes 

 when each of the observations (measured from the centre 12) 

 is squared. 



11' for F(H) be put the cube or any odd power of (x — X), 

 then the original probability-curve under which the values of 

 x (and x — X) ranged will be transformed to a curve sym- 

 metrical on both sides of X, but not a probability-curve. 



These exceptions arise when F is a function of the deviation 

 ot an observation from its average value. The rule is fulfilled 

 when F is a simple continuous function of the observations 

 themselves measured, as is usual with concrete quantities, 

 from an origin below the least possible value — as we measure 

 human heights, or death-rates, or other statistics, from zero ; 

 and not from 67^ inches, or *2 per cent., or whatever the 

 average may be. If any such function of an observation is 

 substituted for the observation itself in a group obeying the 

 law of error, we may expect that the transformed group will 

 also obey that law. 



We have here the explanation of incidents which must 

 have puzzled many students of Probabilities : why Mr. Galton 

 should have found the Arithmetic and Geometric means of 

 observations to give sensibly identical results (Proc. Roy. 

 Soc. 1886) ; how Quetelet could be justified in affirming that 

 holh, weights and heights of men obey the law of error 

 (Anthropometric) ; supposing, as is plausible, that the weight 

 of a man is apt to be proportioned to the square of his height. 



In fine we have here an answer to the objection which has 

 been made to Quetelet's doctrine of the Mean Man by Cournot 

 and other high authorities on Statistics. The objection is 

 thus stated by the eminent Prof. Westergaard {'lheorie der 

 Statistik, p. 189) :— 



" Suppose we had measured for a number of men three lines 

 of the body which make a right-angled triangle, and we wished to 

 determine the corresponding triangle of the Average-man. Then 

 it may be shown [es zeigt sich dann so fort] that the three 

 averages do not make a right-angled triangle. Call the sides 

 a v b lt c v a 2 , 6 2 , c 2 &c, c v c 2 &c. being the hypothenuses ; then the 

 sides of tho average-triangle are 



-Sa, l$b, and- $c = -$>s/tf + o\ 

 n n n n 



One should have accordingly 



(^)' 2 + (26) 2 = (2:v / ^ + ^)/ 

 But this can only occur — except by accident — [in der Hegel nur 

 Phil Mag. S. 5. Vol. U. No. 210. Nov. 1802. 2 H 



