Researches into the theory of prohability. 



13 



to be demonstrated that the formuhi (1) is actually suitable to represent frequency 

 curves, that is, that the number of coefficients iu the series necessary for obtaining 

 a practically sufficient representation is rather small. It will be shown that for most 

 purposes it suffices to know the coefficients ßg and ß^. When the series of obser- 

 vations on which the frequency curve is based is very numerous, it may be de- 

 sirable to know the values of ßr, and ß,, also. This naturally is also the case, if 

 the curve of frequency differs much from the normal curve. 



As to the calculation of the moments of the curve I refer to the researches 

 of Pearson and Sheppard (Proc. Loud. Math. Soc. Vol. XXIX). The methods 

 for obtaining the numerical values of the moments are clearly summarised by Daven- 

 port (»Statistical Methods* P. 19 ff.). In a certain point it will be necessary to 

 complete the numerical methods used by these authors, namely in respect to the 

 checking of the numerical results. It must be considered as a rather laborious and 

 imperfect method to cheek numerical work through double calculation or »calcula- 

 tion in pairs», as is recommended by the last named author. A scheme for nume- 

 rical calculus Luust be so arranged, that errors may be detected by the computor 

 hiüiself, and such arrangements are generally easy to perform. In the first example 

 I have carried out the control in extenso. 



I bring here together the fornmlœ necessary for the calculation of the moments 

 and of the coefficients of skewness and excess (ß^ and ßj. 



ï;c'F(;r). (s = O, ], 2, 3, 4). 



^' : [J-o'- 



(") [K = 



(!') . 



GonffoJ: 



(c) l(x- \rF{:r) = 

 or 



(d) i(xi- \rF{x) = 



(f) 

 (g) 

 (1') 



= v, , 

 a- = = Vj' — 6^ 



v, = Vg' ^ Uv.: + 2P, 

 = v/ — 45V3' + — 



Control: 



(i) 



v, + 4?;v.,, + ()?;^v, + l> 



(3) 



ßg = ~ Vg : 60«, 

 = A (v, : - 3). 



