25G 
Miscellanea 
But : S (n Xp a/)/iV= x S (n Xp )jiY= <r v \ 
= a/, if rj be zei'o, 
S (a s Vx ) = <T 2 yjl . i + <r\ (ll .^+a 2 y Xa +...+tr i y x , if there be k arrays, 
= Ka-y 2 , if 7] be zero, 
for all the arrays will be samples of the general population. 
Thus, for a system without correlation 
JYx mean value of 8 (o-j,, 1 ) = k oy + o-,, 2 - 2a-/ 
=(k-1)ov 2 . 
K - 1 
Mean value of 1 2=2 — ) f° r true ?; = 0. 
Or, mean value of r)= \f K ^r' • 
Hence no value of r) is really significant if it does not differ sensibly from 
— = + -67449 -L . 
If we do not suppose ij zero, the mean addition to the value of rf 1 will be 
WO i 
jV x b (»*) = K --+1-2(1- n 2 ). 
The question arises as to what value should be given to the mean of the unweighted standard 
deviations squared of the arrays. We shall not err much in this small order expression, if we 
give it the value it would take if the arrays were weighted, or if the distribution were homo- 
scedastic, i.e. (1 — 17 2 ) a/. Accordingly we have : 
8 (,*)={(« -2) (W)+l}/tf 
Or : Mean observed value of rj- = true value of + — — — ^ ^ — - . 
M... 2 -2 (*-2)(W)+l . 
or. >? -r ^ ; 
we see therefore that the correction for ^ rapidly diminishes as q approaches unity. Of course 
we have, if we prefer, instead of writing rf 1 on the left, the observed value 
^ = {^_ (k _ 1)/#}/{ i_ (k _ 2 )/^}, 
where we shall put for ij 2 , the mean value, the observed value as most probably coincident with 
that mean value. 
Hence, substituting we tind : 
, N — ^ — = - ir^'xp + w v. 
which had to be proved. 
+ S, <3tt**xp»j ~ S s ]y 
= % ( - y% + y Xp y- Ux p y) + s s (i/ s 2 %■/.,) 
= s« I % ,v s ( .'/ s 2 - F%) } = s s { % Vs ( y s - y* p ) 2 } 
