27<; <)/i thc Probable Errors of Frequency Constants 



tbe Standard deviation in errors iu any uionieiit, and the correlation of errore in mouicuts, when 

 these moments are taken, not about a fixed point, but about the mean, which is a point varying 

 with tlie errors of randoni sampling. 



Problem VI. To find the correlation between an error in the mean and an error in the 

 j"* momenl both determined from a fij-ed point. 



This is given at ouce by (v) siuce h=iij'. We have only to put 9'= 1, aud \ve have 



V/^'''"V,- m ^ ■'■ 



Now it naay be noted once and fcjr all that afler these expressions like (iv), (v) and (vi) have 

 been found we can caleulatc tho riglit-hand sido takiiig for oiu* fixcd ixjint any origin we please, 

 i.e. we can take the point which actnally coincides with the mean iu the sample, i.e. write /i = 0, 

 and /i'„ = /i,, where /i, is the moment about the observed mean. 



Problem VII. To find the Standard deviation for errors in the 5"' moment /i^ taken about 

 the mean. 



Vi'e have : 



m^^ = S{(.v.-h)''}/.} 



{M',-?V,-. + '-fi^^AV,-. + ...)- 



Or: ;j„ = /g-(yV,_, + A-x, 



where x is a function of q, h and /i',_2i h'<i-3' '^^'^• 



Hence : 



8/i,=8/i', -gA8/x',_j — 8/( {<jfi\_i + 2h)() + tervaü in h- and higher powere. 



Thu.s if we are going to put /(=0 tinally or measure oiu' monicnts from a fixed point, which 

 coincides with the actual position of thc mean in the sami)le, we may write : 



8/t5=S;i'3-(;S/(/i',;_ , + terms which vanish with h. 

 Thus : 



Sfig- = 8fi\- + (/-8h- /»'-^_ 1 - 2(jii\ _ j 6/( 8/i', + terms vanishing with h. 



Or: 



o'>,=<f>„ + '?-o-A'>"-,^i-2y/ii',j-, (T^<T^.^j-j^.^ + terms in /(, 



and using (iv), (iii) aud (vi) : 



/x'äj-Vj +£VVj - 1 - 25/i', . 1 m', + 1 + terms in h 



""■> " m • 



Put /i = 0, and n\ = in, we have: 



''« V m, ^ '' 



niustrations. 



(i) Put q = 2, and remember that ;i, = 0. 



Probable error of ii.,= -67449 \/ '^*~'^- . 



For example, in the case of a normal curve this = •67449 . / — /'«i sinco /»4=3/ij'. 

 Further we have, since <7 = \//'2, f'>i' t'ie normal curve 



Probable error of 0- = •67449 a / „ - <r. 

 A' 2m 



(ii) Put 5 = 3. Probable error of ^3= 67449 ^ It^ZJ^lz^ftl+M . 



