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Miscellanea 
probable error of p is -67449 (1 - p 2 ))^ N= •67449/v' N if p be really zero ; then if p be the mean 
value of p we should expect p to be 
p±-67449/VF 
if p be truly zero. In other words we must consider the question of whether the observed p is 
significant compared with this. 
I have found the value of 8p 2 , i.e. the mean increment of p 2 due to errors of random sampling, 
but I postpone its consideration in the hope of still further reducing its determinantal ex- 
pression in the general case. 
Let us now apply these results to the general theory of selection. Suppose we have m 
variates x x , x 2 , ... x m , with means I\,...x m , standard deviations <t 1 , a. 2 , ... <r m and correlations 
given by It the determinant 
1 ) *'l2 ) r 13 > • • • *lm 
fu > I > r 23 , ... r 2m 
1*m\ ) ? 'm2 ) ^m3 > > ' • 1 
in the usual way. 
Now we may suppose a selection to be made out of this variate complex of a subpopulation 
%i, x 2 ... x n , 11 < m, given by the means : 
h u h 2 , ... h n , 
the standard deviations 
s l ) s 2 ) • ■ • s ll 
and the correlations : 
1 , Pn > Pi3 > ••• pm , 
Pnl, P«2> P«3) ••• 1 
the selected population having values consistent with those of the unselected population. 
We can then ask : 
(i) How will this modify the mean and standard deviation of a non-selected variate 
ss p , p > n < = m ? 
(ii) How will this modify the correlation r pq between two non-selected variates x p and x q , 
p and q > n and < = m 1 
(iii) How will this modify the correlation r pt of a non-selected and a selected variate 
p>n and <=m 3 while t< = n1 These are the fundamental problems of the influence of 
selection on variation and correlation. 
(i) Let us take x n + i as the non-selected organ and let the characters of one of the selected 
group be given by x t = k~ t + £ t . 
Then .r ll + l will differ from its probable mean value by some quantity rj n + 1 and by (vii) 
we have 
aW^KH-i+Wi-'S j /*'"* 1 — (ht+Zt-Xp)} ■ 
Or taking the mean value, x n + i, S {rj n+ \) = 0 and S(tj t ) = 0, and 
Sn + i*r* n+ i-S (Jh - xA (viii). 
