xcii Tables for Statisticians and Siometricians [XIV 



of the higher 's and thus of the higher moments and so to avoid the very tedious 

 computation of the latter. They are based on the assumption that, if the sample 

 be of considerable size, it can be adequately described by one of Pearson's Types, 

 and accordingly the higher moments may be deduced from the first four. The 

 fundamental equations are 



2s = M2s+2/y"2 S+1 , 2*+! = l"2s+3 ^3,W +3 , 



and, if a = (2 2 - 3ft - 6)/(ft + 3), 



/j) .\fl/r> /! i i ..\ n 1/fi 1 /o_\ _ I \ J* 



+1 \^ s ~r A) 



It will be seen that the table for 3 , 4 , 5 and 6 is a laborious one to 

 compute, and as each successive depends upon two earlier ones errors are 

 cumulative. Some errors in Yasukawa's original table have been discovered in 

 the use of it and are here corrected, and the Editor will be glad to receive any 

 further emendations. The table cannot be trusted to the last decimal place, as the 

 computer has reduced his proper fractions to decimals before the final stage, and 

 the last decimal in 3 , 4 or 6 may be in error, and in 6 the last two decimals. 

 Thus for 1 = 1, 2 = 6, Yasukawa has 



3 =25, 4 =195, 5 =2279-99999, 6 = 57434-99984, 



instead of 



3 = 25, 4 =195, 5 = 2280, 6 = 57435. 



This is not a matter of much importance because (i) we rarely need 's to more 

 than five significant figures, (ii) the higher moments and therefore the 's are 

 subject to large probable errors, and (iii) for very many distributions relatively 

 large changes in the higher 's appear to have? small influence on the shape of the 

 frequency curve. 



Yasukawa has provided values of the higher 's for 2 1 < 1 ; it is not at 

 present obvious that any use can be made of these values in practical statistics, but 

 those in the neighbourhood of 2 ft 1 = are of service for purposes of inter- 

 polation. The chief value of the table is to obtain approximate values of the 

 probable errors (or standard errors) in large samples, such as those used in social 

 investigations or in anthropometric work. 



The following formulae for large samples are well known* : 



36) ............ (iii), 



= ( 6 - 4 2 4 - 8 3 + 4 2 3 - 2 2 + 16! 2 + 16ft) ...... (iv). 



* Phil. Trans. Vol. 198, A, pp. 274279; Biometrika, Vol. n. pp. 276277; and ibid. Vol. vn. 

 pp. 127147. 



