ERROR TO CASES OF NORMAL DISTRIBUTION AND CORRELATION. 
135 
choose m and /? so as to make the values of E“ and of as small as possible. It is 
obvious that one of the class-indices must be positive and the other negative. 
Suppose a to be negative, and equal to — y ; then it will be found from Kramp’s 
tables that E’ is a minimum when ^ and y are each taken a little greater than ‘459, 
the probable error in the mean being then •74951 aj\/n, which is about the same as 
the probable error due to using the quartiles; and that H" is a minimum when 
/3 and y are each taken a little less than '862, the probable error in the semi-para¬ 
meter being then ‘59055 ajs/n, which is about 25 per cent, less than the probable 
error due to using the quartiles, but nearly 24 per cent, greater than that due to 
the average-and-average-square method. 
(3.) Suppose the values of the mean and of the semi-parameter to be found by the 
extended class-index method of § 22 (3.). Then, with the notation used above, the 
errors in the observefl values of X, Y, U, . . . are of the form w -f- xp (f), oj yp \jj, 
fo lip . where i//, y, . . . are eri'ors whose mean products with w, and also 
with p, are zero. Substituting in (i.) of § 22 (3.), and taking account of (ii.), we see 
that the resulting errors in Li and in a due to this method are respectively 
+ mxb H- px + • • ■)I(J + -f + . . .) 
and 
P + + wi'V' + P'x + • • + •••)• 
Hence if and are the mean squares of 
(l(jj -j- mi//' -|- ^;x + • • p . .) 
and of 
{l'4> + 7 nxp -f 2>'x + • • + ^n'y -b p'u + ...), 
the mean squares of the errors in Lj and in a, due to the use of the class-index 
method, are (a^ and (-| -j- Since these are necessarily greater than 
arjn and ^ cPjn respectively, the probable errors in Li and in a due to this method 
are greater than the probable errors due to the average-and-average-square method. 
In other words, we cannot, by observation of the values corresponding to particular 
class-indices, obtain such good results for Li and a as by calculating the a verage and 
the standard deviation.* 
(4.) Generally, let R be any quantit}’’ which would be known if the true mean and 
mean square of the distribution were known ; let lii be the value obtained by taking 
the mean and mean square as equal to the average and the average square for the n 
observations, and let R, be the value obtained by any other method involving obser¬ 
vation of the class-indices of any finite number of values of L, with or without the 
* Professor Edgeworth’s contrary statement (‘ Phil, Mag.,’ vol, 36, 1893, p. 100) appears to be based 
on neglect of the correlation of errors. 
