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 standarddeviation
 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,

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