XIV.—REMOVING LIMITATIONS OF SIMPLE SAMPLING. 287 
It should also be noted that the case when r is positive covers 
the departure from the rules of simple sampling discussed in 
$ 9-10: for if we draw successive samples from different records, 
this introduces the positive correlation at once, even although the 
results of the events at each trial are quite independent of one 
another. Similarly, the case discussed in §§ 11-12 is covered by 
the case when 7» is negative : for if the chances are not the same 
for every event at each trial, and the chance of success for some 
one event is above the average, the mean chance of success for the 
remainder must be below it. The cases (a), (6) and (c) are, how- 
ever, best kept distinct, since a positive or negative correlation 
may arise for reasons quite different from those discussed in 
§ 9-12. 
3 14. As a simple illustration, consider the important case of 
sampling from a limited universe, e.g. of drawing n balls in 
succession from the whole number in a bag containing pw white 
balls and gw black balls. On repeating such drawings a large 
number of times, we are evidently equally likely to get a white 
ball or a black ball for the first, second, or nth ball of the sample : 
the correlation-table formed from all possible pairs of every sample 
will therefore tend in the long run to give just the same form of 
distribution as the correlation-table formed from all possible pairs 
of the w balls in the bag. But from Chap. XI. § 11 we 
know that the correlation-coefficient for this table is — 1 [(w—1), 
whence 
n-1 
0? =n.pq(1 oo 1) 
w—n 
= 9% 5 
If n=1, we have the obviously correct result that o = (p9)}, as 
in drawing from unlimited material : if, on the other hand, n=w, 
o becomes zero as it should, and the formula is thus checked for 
simple cases. For drawing 2 balls out of 4, ¢ becomes 0-816 
(npg); for drawing 5 balls out of 10, 0-745 (npg)t; in the case 
of drawing half the balls out of a very large number, it approxi- 
mates to (0-5.npq)}, or 0-707 (npg)t. 
In the case of contagious or infectious diseases, or of certain 
forms of accident that are apt, if fatal at all, to result in whole- 
sale deaths, r is positive, and if n be large (as it usually is in such 
cases) a very small value of » may easily lead to a very great increase 
in the observed standard-deviation. It is difficult to give a really 
good example from actual statistics, as the conditions are hardly 
ever constant from one year to another, but the following will