252 
Miscellanea 
and having determined its value use Palin Elderton's Tables in Biometrika, Vol. I. p. 155. 
Clearly 
iV iV'/ J/ 
/p _ fp 
N A 7 ' 
N't M 
.(xiv). 
It remains, exactly as in my paper referred to (pp. 164 — 6), to select the most reasonable 
value for fip/M, the proportional value of the ^>th frequency class in the population from which 
both samples are by hypothesis selected. 
Now the best hypothesis as to the constitution of this population, on the assumption that 
both frequencies are random samples of it, will be that its pth frequency class is that indicated 
by the combined two samples, i.e. that 
fp+fp is proportional to fi p , or (f p +f p ')l(N+ i\") = fi p /M. 
Substituting this value we find 
Ni\' 
AT N' 
fp+fp 
.(xv). 
The calculation of x 2 now presents no difficulties in any actual case. 
(3) Illustration 1. Let us inquire whether hair colour exercises a differential selection 
with regard to the incidence of scarlet fever and measles. 
The following data are provided by Dr Macdonald, Biometrika, Vol. VIII. p. 28, for all 
scarlet fever and measles cases : 
Hair Colour 
Black 
Dark 
Medium 
Fair 
Red 
Totals 
Scarlet Fever 
Measles 
(i) 
(ii) 
12 
0 
289 
85 
1109 
367 
360 
184 
94 
25 
/ 
1864 
661 
(i)+(ii) ... 
(i) /1864 
(ii) /661 
(iv)-(v) ... 
Square of (vi) 
(vii)-(iii) ... 
(iii) 
(iv) 
(v) 
(vi) 
(vii) 
(viii) 
12 
•0064 
'0000 
+ -0064 
•000,041 
•000,0034 
374 
•1551 
•1286 
+ -0265 
■000,702 
•000,0019 
1476 
•5950 
•5552 
+ 0398 
•001,584 
■000,0011 
544 
•1931 
■2784 
- -0853 
•007,276 
•000,0134 
119 
•0504 
•0398 
+ -0126 
•000,159 
■000,0013 
/+/' 
f/N 
fW 
flN-f'IN' 
{fIN-f'jNf 
f+f 
2525 
1-0000 
roooo 
•000,0211 
Therefore 
x * = A T A T 'x -000,021 I 
= 1864 x 661 x -000,0211 
= 26-00. 
