278 On the Probable Errors of Frequencji Constants 



Lastly, to find certfiin probable errors we reqwiro : 



Problem IX. Tofind the correlation between errors in the j"" moment about the mean and the 

 j'"> iHoment about aßxed point, in terms ofthese moments. 



In Order to work out tlüs problcni we take a preliniinary proposition : 



To find the correlation between au error in the group y, and an error in the moment /i',- about a 

 fixed point. 



We have : 



m^\.^S{x,fy,), 

 therefore : 



mhy.\.hj,=x,'> {by.f+S' (x,.«'8y.8y^), 



y.(^.'''-MV) /• V 



or: «^"V ""/"-(,•''.= m ^ '■ 



This is the correlation between an error in y^ and one in tlie j'"' nioiuent al>out a fixed 

 point. 



To find the correlation between an error in y, and one in the j"" moment about the mean, we 

 have as before : 



8/x^ = 6/,- yS/i/i',_i + tenn.s in h, 



Bli^&y, = &li qdy,— q8hSy,ii',,. j + tcrms in h, 



"■»",, "■i',''''^i/,, = '^»'','^i/,'V,,!/,-</'fATi/,'"/.v,M'.(-i + t«.'rms in h, 



^y,(£,l:V,) _ gi/.i^.-h ) ^,^_^^ ^^^^ .^ ,^_ 



7)1 »n 



since /(=/*i' and we can use (i.\). Hence p\itting A = 0, we have 



^M^:^!^:j^si:S!^o^ ^^^ 



X being here measured from the mean. 

 Ultutrations. 



(i) Tu find the correlation between an error in any .single group and an error in the mean. 

 We have from (ix) when 5'=!, A=/i,', 



y, (x,-h) 



= ^-!-^ , if .r, be measured from mean. 

 m 



(ii) To lind tiie correlation between tlio error in the standai"d deviation and an error 

 in any single group. 



We have from (x) if j = 2, 



But/jj = (r* 8^ = 2o-6(T, 



therefore : 8^28^« - 2(rSy,8(T, 



and <^Mj<^»,''M.»,=2<f<»',,<r(7'",,<r- 



