Miscellanea 
255 
positive or negative, i.e. the mean value of 8(r„ 2 must be a positive quantity or a negative 
quantity and cannot be zero. 
Let y Xp , n x and y represent the values in an indefinitely large population. Let n Xp +8n Xp , 
Vxp+byxpi y + ^j/ represent the values in any random sample, then 
NdaJ = S {(n^+dn^) {y Xp + 8y Xp -y- h~yf) - 8 {n Xp (y Xp - j,f\ 
= S {n Xp (by Xp - byf) + 2S {n Xp {y Xp -y) (by Xp - by)\ 
+ S [8u Xp (y Xp - yf\ + 2S {Bu Xp (8y Xp - by) (y Xp - y)\. 
Sum for all values and divide by number m of such values, then remembering : 
(i) That the sum of all linear variations is zero, i.e. 2 (8n Xp )lm=1 (8y Xp )/m=2 (by)/m = 0, and, 
(ii) That there is no correlation between the number in any array and the mean of that 
array or between the mean of all the y's and the number in any .v-array. 
2(S^/, p -^) 2 } 
i z{oy x -$y)'\ 
We have : Ny. mean value of 5 (a 
No„: 
m n. 
* m ~ N ' 
~ = N 
The first two simply represent the square of the standard deviations of the mean of an 
array of y's at x p containing n Xp individuals, and of the y mean of the whole population con- 
taining N individuals. The last result is proved in the footnote below*. Hence substituting 
these values we have : 
/ „ 2\ ( n x p o- 2 „ \ 
N x mean value of b (a- J) =S(<r* y ) + S( n x ^)-2S 
* We have: n x p y Xp ^n XpvVl + » v , ;/,+ ... + » Jp „ s ;/ s + ... , 
Ny = n v 2/1 + //-+••• + »;/, y, + •••• 
Hence : %^ p =5'V 1 f' + 5 'V.,-' /2 + -+ S "-W» +•••- ^p 5 " 
N6y=8n,y y 1 + 5n y y., + ... + 5% s ?/ s + .... 
Of if m be the number of samples : 
_ ( 
** N m = "•"V s - \Vi 
| s (5n„ Sn xy )\ 
+ ~ [ 
88 ( 111 I 
But + : 
m 
fSn y Sn x v \ ( a 
t See Drapers' Research Memoirs, " Skew Correlation," p. 12, and Biometrika, Vol. v. pp. 191, 192. 
