XIV XV] lnti'u<lin-tinii \c\ 



(6) Wo will consider now the prob.ibl. .11 MIS ..f ^ , ^3, ami m, runiiMiibcrifi^ that 



^2 = 3-821,776 and JV-8689. 



07449 



::vj 1,770 V4-34H.825-1 



= 007,236 x 3-821,776 x 1-829,980 

 = 0500*, 



67449 

 '''''' M3 = V8689 ( 3 ' 821 > 77G )* x ^S'076,648 - 26-092,950 - 1 '033,412 + 9 



= -2959, 



67449 

 P.E. of ^ 4 = - l _ (3'821,776) 2 



V8689 



x \/906-334,872 - 98-627,712 - 18-912,279 + 16-534,592 

 = 2-9993. 



If, instead of using the hyperbolic formula and table, we use the exact values 

 of the /9'sf, we find 



P.E. of ^ 3 = -2957, P.E. of fn = 2-9883 ; 



the differences are of no importance for the purposes to which probable errors are 

 applied. It would seem therefore as if the hyperbolic formula or double linear 

 interpolation into Table XIV is adequate to determine the probable errors of the 

 moment coefficients of large samples. 



TABLE XV. 



Ratio of Standard Error of Mode to that of Mean. On the Probable Error of 

 the Mode. (K. Yasukawa, Biometrika, Vol. xvm. pp. 265 292.) 



The probable error of the mean is 67449<7/v / J\T, where a is the standard deviation 

 of the population and N the size of the sample. 



The distance d between the mode and the mean in the case of a Pearson type 

 curve is given by 



d-. 



Hence, knowing the mean we can find the position of the mode, as soon as the 

 values of /3x and /6? 2 have been determined. 



Table XV gives the probable error of the position of the mode in terms of the 

 probable error of the position of the mean, i.e. it is not the probable error of d, 

 but of the absolute position of the mode, that the table provides. 



67449<r -67449 _ 



* Had we used the formula corresponding to P.E. of a= = , i.e. o>,= _= 2pj, we should 



have found P.E. of ju 2 = '0391, not a very good approximation. 



t #,= 12-333,049, /3 4 = 48-034,051, |3 5 = 199 -076, 421 and 6 = 900-471, 809. 



