Editorial 27!) 





whence wo soo: 



Tlius \vu timl : 



. _ .'/.(g-'-x.' ) 



Thus if an error bc made in tho freqiiciicy of a groiip with charactcr Icss tlian o-, the error in 

 tho Standard dcviation will be nej;atively (-orrclated with it. If the error be in a frc(mcin-y 

 group with character grcator than o-, tho error in «• will be positively correlated witli the error in 

 the group. 



We can now return to our original problem. Wc have : 



»((VV = '^'(-<V''ß.'A)i 



S^i, = S/,j-(/S/i/i',;_i + terms in /;, 



»«S/iVV; = ^ (■'•«''■fi'/.S/i',;) - qi^,i - 1 ti {■'■/ Sy,S/i) + terms in /(, 

 ...,,r,v„= ''>-'-''''^'^f-^'-^i -g/,_ /'-'-^>'^-'^' +termsin^. 



^mV^^V^ _g/''»-i^V^i+terms in k. 

 m ni 



Whence putting h = 0, we ha\e f<ir the correlation of errors in the two kinJs of momcnt.s, 

 i.e. those measured from a tixed point and thosc measured from the mean : 



tTu' .(Tu 3 u- .u = 1^11. 



The Chief use of thi.s formula i.s to find tho correlation of errors in the mean and in the 

 moments about the mean. For this purposc put 5' = !, i.e. ii\' = h, and wc have since /Ji=0, on 

 right-hand .side : 



<^k<r^,H>^, = (^")- 



Illzistrations. 



(i) To find the correlation in error between the mean and the Standard deviation. 

 All we need to do is to put q = 2, then : 



But r^,j = rha, therefore 



<rh<rii,r,,„=^Jm (xiii). 



Hence if a random sample be taken which is more variable than the general pojiulation, the 

 probability is that the mean of the sample will be higher than the mean of the general population 

 if the third moment be positive, and lower if it be negative. Thus we cannot by random sampling 

 change the variability without changiug the tyi^e or the type without changing the variability. 

 The only exception occurs when /i3 = 0, for example in the normal curve. In that case errors 

 in fche mean are independent of errors in the variability. 



(ii) To find the correlation in error between tho mean and the third monient. 



Put q = Z in (xii), we have 



M4 ~ •V2" 



This vanishes again in the case of the normal cnrve. It niay be shewn that the coirelation 

 between the mean and all the moments vanishes for normal systems and normal Systems only. 



