XIIL.—SIMPLE SAMPLING OF ATTRIBUTES. 245] 
would not apply if the samples were compounded by always 
taking one person from district 4, another from district B, and 
so on, these districts not being similar as regards the distribution 
of hair-colour. 
The above conditions were only tacitly assumed in our previous 
work, and consequently it has been necessary to emphasise them 
specially. The third condition was explicitly stated: (c) The 
individual *‘events,” or appearances of the character observed, 
must be completely independent of one another, like the throws 
of a die, or sensibly so, like the drawings of balls from a bag 
containing a number of balls that is very large compared with 
the number drawn. Reverting to the illustration of a death-rate, 
our formule would not apply even if the sample populations 
were composed of persons of one age and one sex, if we were 
dealing, for example, with deaths from an infectious or contagious 
disease. For if one person in a certain sample has contracted 
the disease in question, he has increased the possibility of others 
doing so, and hence of dying from the disease. The same thing 
holds good for certain classes of deaths from accident, e.g. railway 
accidents due to derailment, and explosions in mines: if such an 
accident is fatal to one person it is probably fatal to others also, 
and consequently the annual returns show large and more or 
less erratic variations. 
When we speak of simple sampling in the following pages, the 
term is intended to imply the fulfilment of all the conditions (a), 
(8), and (ec), all the samples and all the individual contributions to 
each sample being taken under precisely the same conditions, 
and the individual “events” or appearances of the character being 
quite independent. It may be as well expressly to note that we 
need not make any assumption as to the conditions that determine 
p unless we have to estimate i/mpg a priori. If we draw a 
sample and observe in it the actual proportion of, say, 4’s: 
draw another sample under precisely the same conditions, and 
observe the proportion of 4’s in the two samples together: add 
to these a third sample, and so on, we will find that p approaches 
—not continuously, but with some fluctuations—closer and closer 
to some limiting value. Tt is this limiting value which is to be 
used in our formulee—the value of » that would be observed in 
a very large sample. The standard-deviation of the number of 
sixes thrown with » dice, on this understanding, may be «/npg, 
even if the dice be out of truth or loaded so that pis no longer 
1/6. Similarly, the standard-deviation of the number of black 
balls in samples of » drawn from an infinitely large mixture of 
black and white balls in equal proportions may be «npg even 
at