368 Mr. F. Y. Edgeworth on the 
shall mcake an error of £ in taking the mean as the true value is 
/» 00 
Jo 
of the form 
*/-<*>'(! 
Vdh 
(SaJ+nf)- 
^r, (say w even). 
The errors committed by taking the mean as the real value are 
ranged under the above* sub -exponential expression. Whence 
it appears that what may be called the measure of the pro- 
bable error is the sum of the squares of the apparent errors 
divided by n } not, as some might seem to imply, by (n — 1); 
although this modulus, as I think it may be called with pro- 
priety, is not to be multiplied by the usual factor '476, but by 
the length of the abscissa which halves the half-area of the 
curve just indicated. 
What is frequently said in favour of the expression 
2 _vt — l _zj_ ( wnere £ i s the mean of the observations) instead 
f 2 — — — -^ in the present and similar problems, namely, 
that the latter expression, the sum of squares of apparent 
errors, is certainly less than the sum of squares of real errors, 
appears to be true but not pertinent. For what have we to do 
with the sum of squares of real errors, except as a mark of 
one or other of the quaesita above proposed? — (1) The law 
according to which the observations diverged, the mo- 
dulus of the probability-curve, from the groups constituted by 
which the observations are regarded as random selections ; 
(2) the errors incurred by taking the mean as the real value 
It may be observed, too, that the most probable value (as de- 
duced from the observations) of the sum of squares of real 
errors is the sum of squares of apparent errors ; a statement 
which is quite consistent with the admission that this latter 
value is certainly (infinitely probably) less than the real value. 
The case might be compared to the following : — A random 
selection of an abscissa being made from the group indicated 
1 * a 
by the curve y— — =— e~^, required the most probable value 
* See Phil. Mag. Oct. 1883, p. 306. 
