XIIL—SIMPLE SAMPLING OF ATTRIBUTES. ) 
albinos were obtained in 121 litters from hybrids of Japanese 
waltzing mice by albinos, crossed inter se (A. D. Darbishire, 
Biometrika, iii. p. 30): — 
Percentage. Frequency. Percentage. Frequency. 
L 40 2 
LE 43 2 
WI 50 16 
20 oi 
22 60 
25 - ul 
29 &0 
33 100 
The distribution is very irregular owing to the small numbers in 
the litters, and the standard-deviation is 23-09 per cent. The 
numbers of litters of different sizes were given in § 27 of Chap. 
VII. p. 128, and the harmonic mean size of litter was found to be 
3:03. The expected proportion of albinos is 25 per cent., and 
hence the standard-deviation of sampling is 
25 x T5\! 
(S55) =205 
in very close agreement with the actual value. The proportion 
of albinos amongst all the offspring together was 24-7 per cent. 
12. If one of the two proportions » and ¢ become very small, 
equation (1) may be put into an approximate form that is very 
useful. Suppose p to be the proportion that becomes very small, 
so that we may neglect p2 compared with p: then 
Pq =p —p*=p approximately, 
and consequently we have approximately 
on= nnp= JM (4) 
That is to say, if the proportion of successes be small, the 
standard-deviation of the number of successes is the square root of 
the mean number of successes. Hence we can find the standard- 
deviation of sampling even though p be unknown, provided only 
we know that it is small. 
Thus (ref. 15) in 10 Prussian army corps in 20 years (1875- 
1894) there were 122 men killed by the kick of a horse, or, on an 
average, there were 0-61 deaths from that cause in each army 
corps annually. From equation (4) we accordingly have for the 
standard-deviation of simple sampling 
o=(061)!=0178, 
26%