EditorUd 275 



Therefore 



inhh = S{xby), 



m^ {8hf = S{x,^ 8i/;i) + 2,S" (.^vr.. Sy, 8_i/,,), 



where S' is a sum for all valucs of .< aiul s' foi- wliich s Im not cqual to s'. This gives dividitig liy 

 the muuber of raiuloni sainpliiig.s after siiiuiiiing U)V all such samplcs : 



in'(Th- = S (.(■;- o-'-„p - 2.S" (.1-, .i-j. <r„^ (r,,^, ;v^ „^,), 

 or, using (i) and (ii) 



M m 



= Hi(;i2'-/(2), 



where M/i/ i« the secoud mouient about the fixod poiut. Hut fi2'-/r = (T^ = Square of Standard 

 deviation of sample. Hence : 



a-^ = cr/-Jin (iii). 



Problem IV. To find the stamhird deriation »/' the (/*' iiioment = mfi,/ of t/ic sample about 

 a fixed point. 



ot^.,' = Ä(.iV'.'a)- 

 Therefore 



and i"-<'\', = * (■«•.'■"o-^.) + 2^" (^'.'.'V'"-!,, o-i,,, ^\,y,) 



as before. CJsing (i), and (ii), we have 



»-vr -^ (■'■.-» 





(iv). 



Problem V. To find the correlation nf en-o/s of the 7"' and 5'"» moments, both moments 

 being taken about the same fixed point. 



As in Problem I\^. : 



«iS/,. = .S'(.-!-/''Sy,). 

 Multiplying these together : 



or, using the defiuitions of correlation and Standard deviation : 



Hence by (i) and (ii) 



Thu.s 



V, %>">.' m ^ '^' 



(iv) and (v) thus give the Standard deviation of errors in any nioment and the correlation in errors 

 between two moments, when the momcntnare taken about a fixed point. We now require to find 



35—2 



