Researches into tlie theory of probaljility. 11 



in reliition to the mean. If the excess of the curve is small, it will l)e allowable 

 to use the formula (8*). 



As to the coordinate x.^ of the median, it may obtained in the following 

 manner. 



The median is defined in such a manner that the number of individuals 

 between negative infinity and the median [x.^ is equal to the remaining number of 

 individuals between .r^ and positive infinity. Hence the ordinate corresponding to 

 X — divides the frequency curve into two equal parts. 



We hence have 



\F{x)dr — j F{.r)dx = 0, 

 — CO x.^ 



or, if the expression (1) for F (x) is introduced, 



+ 



(9*) 0 ^„ / Y (,r) dx - A J z (x) dx + 2 A, 'f + 2 A, (-r,) + . . . 



CO X., 



For solving this eijuation wo assume that J.,, aud A^, and in a still higher 

 degree yl. and the following coefficients, are small quantities. As 



b CO 

 j'^[.i)ilx =^ \<^[j:)dx 

 — -Jj b 



it is therefore necessary, that x.^ has a value little different from b. We put 



= b -\- o 



and consider .^^ iis a small (juantity. 



For developing (9*) in powers of we observe, that 



x.^ b 



j (./■) dx = / cp (,-r) dx -\- I 'f (.r) (/.'■ 



— CO — cc b 



X., 



= ^ + |'f ('^i ^^.A 



and also 



so that 



X., 



'f (.'■) d.r = ]^ — 'f (x) d.i 



as. 



X„ -)- CO X., 



f'x, (,/■) dx - / 'f (,/;) dx ^ 2 j'f (,r) dx 

 - CO X., b 



Z.2 



2 a/ 'f {b + a ,c) d.i. 

 0 



