436 



Prof. F. Y. Edgeworth on the 



standing * that each number of men,*?, g. 155, entered against a 

 certain height, e. g.Ql inches, means that there were 155 men 

 between 66*5 and 67*5 inches) is 67*023. The Arithmetic 

 mean of the squares is 4498*671, very nearly the square of 

 the primary Arithmetic mean, viz. 4492*08. Also the Arith- 

 metic mean of the cubes is not far from the cube of the 

 Arithmetic mean. 



Similar verifications would no doubt be obtained, if we 

 employed, for F, any other simple function, e. g. c log H or 

 ce—H. The limits within which the rale may be expected 

 to hold will appear, if we consider an exception. Let 



H = 0-X), and F(X)=(#-X) 2 = f 2 

 (employing the same notation as on p. 431). If the varying 

 values of f range under a probability-curve, then the squares 

 of these measurements will not range under such a curve. 

 The operation of squaring will cause the negative limb of the 



Fi-.l 



O £1 



original curve to be screwed round to the positive side ; and, 

 in addition to this displacement, there will be the distortion 



1 -* 2 

 caused by substituting, for the error-function , — e c2 •> the 



1 —4. 



function a .- -- * ci T • 

 2 \x s/irc 



* A misunderstanding : for as I have learnt, since the above was 

 printed, the entry 155 men against 66 inches means that there were 155 

 men above 66 inches ; but our argument is not affected by pushing up the 

 whole set of measurements en bloc half an inch higher. 



"J" In general substituting for x, in the error-function, <j)—i(a:), and 



multiplying by -r- (f>— l (#) ; where <f)(z) is the new value of an observa- 

 tion which originally measured x — or rather something between x and 

 x + dx. 



