! THEORY OF STATISTICS. 
(or median), in a grouped frequency-distribution, is found to be S. Find the 
correction to be applied to this sum, in order to reduce it to the mean (or 
median) as origin, on the assumption that the observations are evenly dis- 
tributed over each class-interval. Take the number of observations below the 
interval containing the mean (or median) to be n;, in that interval =, and 
above it my; and the distance of the mean (or median) from the arbitrary 
origin to be d. 
Show that the values of the mean deviation (from the mean and from the 
median respectively) for Example ii., found by the use of this formula, do not 
differ from the values found by the simpler method of §§ 16 and 17 in the 
second place of decimals. 
8. (W. Scheibner, “Ueber Mittelwerthe,” Berichte der kgl. sdchsischen 
Gesellschaft d. Wissenschaften, 1873, p. 564, cited by Fechner, ref. 2 of 
Chap. VIL : the second form of the relation is given by G. Duncker (Die 
Methode der Variationsstatistik ; Leipzig, 1899) as an empirical one.) Show 
that if deviations are small compared with the mean, so that (2/2/)® may be 
neglected in comparison with z/J, we have approximately the relation 
a? 
e=1(1-1] 7): 
where @ is the geometric mean, J the arithmetic mean, and ¢ the standard 
deviation : and consequently to the same degree of approximation M2 - G2=42 
9. (Scheibner, loc. cit., Qu. 8.) Similarly, show that if deviations are small 
compared with the mean, we have approximately 
. 2 
eri) 
H being the harmonic mean. 
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