. The London, Edinburgh and Dublin philosophical magazine and journal of science. the greater nart of the arithmetical examples inthis and the preceding paper (Phil. Mag. Aug. 1892). 436 Prof. F. Y. Edgeworth on the standing * that each number of men,*?, g. 155, entered against acertain height, e. inches, means that there were 155 menbetween 66*5 and 67*5 inches) is 67*023. The Arithmeticmean of the squares is 4498*671, very nearly the square ofthe primary Arithmetic mean, viz. 4492*08. Also the Arith-metic mean of the cubes is not far from the cube of theArithmetic mean. Similar verifica
. The London, Edinburgh and Dublin philosophical magazine and journal of science. the greater nart of the arithmetical examples inthis and the preceding paper (Phil. Mag. Aug. 1892). 436 Prof. F. Y. Edgeworth on the standing * that each number of men,*?, g. 155, entered against acertain height, e. inches, means that there were 155 menbetween 66*5 and 67*5 inches) is 67*023. The Arithmeticmean of the squares is 4498*671, very nearly the square ofthe primary Arithmetic mean, viz. 4492*08. Also the Arith-metic mean of the cubes is not far from the cube of theArithmetic mean. Similar verifications would no doubt be obtained, if weemployed, for F, any other simple function, e. g. c log H orce—H. The limits within which the rale may be expectedto hold will appear, if we consider an exception. Let H = 0-X), and F(X)=(#-X)2 = f2(employing the same notation as on p. 431). If the varyingvalues of f range under a probability-curve, then the squaresof these measurements will not range under such a operation of squaring will cause the negative limb of the O £1 original curve to be screwed round to the positive side ; and,in addition to this displacement, there will be the distortion 1 -*2caused 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 wasprinted, the entry 155 men against 66 inches means that there were 155men above 66 inches ; but our argument is not affected by pushing up thewhole set of measurements en bloc half an inch higher. J In general substituting for x, in the error-function, —l (#); where <f)(z) is the new value of an observa-tion which originally measured x— or rather something between x andx+dx. Law of Error and Correlated Averages. 437 Tins dislocation is exhibited in the annexed diagram (fig. 1)?where 13AB is the original curve, u(3 what it becomeswhen each of the observations (measured from the centre 12)is
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