Researches into the tlieory of probability. 



9 



The comparison between the observed and the calculated values of the fre- 

 quency cannot be performed directly with the help of this table. For this purpose 

 it is necessary to make use of the fuller tables at the end of this memoir. The fre- 

 quency curve iiiay, however, be constiucted with the help of the above abridged 

 table and compared with the empirical frequency curve. Compare the examples 

 1 and 4 beneath. 



We write the scries (1) in the form 



(5) 0 F (,r) = [a [r) + a'' (.,■) + a-'' 'f {.r) + . . . J 



or 



(5*) .F(,r) = .J.J'.„(,r) + ß^,'..,(,.) + ß^'., 



where 



and generally 



4„ [i.^ 



■■0 



Using the abbreviation 

 (6) V, = 



we obtain the following simple formuke for the calculation of the cocfHcients 

 ß:.p ß,, • • ■ 



j3ß, = -v,:o^ 

 |4ß.^= v,:o^- 3, 

 ,jô ß. = — v„ : a- + It) v., : a'', 

 |6ß«= v„:a«-lf,v,:a^+ 15, 



(7) 



The functions 'f {.r) are even functions of ./■ — h, if s is an even nuiuber, and 

 change the sign with .'■ — b if s is odd Hence we lind that the functions 'f'" (.'), 

 tp"*' . . . are liable to give to the frequency curve an unsymmetrical form, which 

 is not the case with 'f"' (,/■), /f^' (./■), a. s. o. We find from the diagrams numbered 

 1, 2, 3, 4, 5 some instances of the influence of the first two terms on the form of 

 the frequency curve. 



Fig. 1 is the usual normal-curve. Figures 2 and 3 show the effect of different 

 values of ß., on the frequency curve. It is here supposed that ß^ and all other 

 coefficients in (5) vanish. For great values of ,/■ — h we here obtain negative values 

 of the frec|uency, which is not possible in reality. The neglected terms of higher 

 order must compensate those negative values. If ßj and all following coefficients 

 are small, it is convenient to choose ßg as a measure of the skewness or dissymn:!e- 

 try of the curve. We hence will call ß^ the coefficient of dissymmetry (or skewness) 



Lunds Univ:s Årsskrift. X. F. AM. i. Bd 1 . 2 



