J. A. Harris 
405 
females, /, from each district. Then for a district, 2(s')=100 s, the population 
means are given by $[2 (s')]/iV, the population standard deviations* by 
s.d. 2 = S [100 (s 2 + <r»)]/N - [S (100 s)/iV]'-, 
where s represents stature in males or females as the case may be, the bars and o-'s 
indicate county means and standard deviations and N= S(n) = mw = 22 x 100. 
The population constants are : 
For Males : 
S(l00s) = 145080, lY = 2200, 
S(l00s-) = 9567888, s = 65-945, 
S (100 ay*) = 17335-08, s.d. 2 = 8-116546, 
For Females : 
S (100 s) = 134760, N = 2200, 
S (100 s J ) = 8255232, I = 61-255, 
8 (100 a-/) = 13992-05, s.d. 2 = 6618921. 
These will suffice for getting the correlations, since all classes comprise the 
same number of individuals and no change in the s or a s is introduced by 
weighting. 
The totals of the correlation surface are 22 x 100 x 99 = 217800 for the intra- 
patrial correlation for stature of random pairs of males and random pairs of females, 
and the rough product moment coefficients are given by 
{100 8 (100 s 2 ) - [S (100s- 2 ) + #(100 ov)]}/£ [n (n - 1 )], 
whence, for males, r = *019, 
for females, r = "029. 
But for the inter-sexual correlation between random pairs of men and women for 
the several counties, pooled together, there are vm-, not m [n(n — 1)] combinations, 
since each male may be compared with all the females of the district, while in 
intra-sexual correlations the individual's own measure must be deducted. For 
random pairs of men and women we have, therefore, for the unadjusted product 
moment 
S [t (s' m ) 2 (s'f)] = 888716500, N = 220000, 
whence 
r= (4039-620454 - 65 945455 x 61-254545)/2\848955 x 2-572726 = -022. 
The result just deduced taken in comparison with that secured in a similar 
manner for the several parishes of the Wensleydale district-]-, seems to indicate 
that the geographical differentiation in the population plays a very small part in 
the production of the correlation between husband and wife. 
* In getting »S [100 (s 2 + <r s 2 )] for the standard deviations we can save the labour of mental additions 
of large squares by taking separately S (100s 2 ) and S (lOOir, 2 ). We have then only to multiply 
S (100s 2 ) by 100 to get ,S'[100s] 2 = S[S (s')] 2 . S2 (s' 2 ) is of course given by the second moment calculated 
for the population s.d. Thus the squaring and summation can be quickly done simultaneously on any 
of the usual types of calculators. 
t Biometrika, Vol. n. p. 485. 
