XIII.—SIMPLE SAMPLING OF ATTRIBUTES. ) 
nation 4 than for nation B, we cannot necessarily conclude that 
the real mean stature is greater in the case of nation 4 : possibly 
if the observations were repeated on different samples of 1000 
men the ratio might be reversed. 
2. The theory of such fluctuations may be termed the theory 
of sampling, and there are two chief sections of the theory corre- 
sponding to the theory of attributes and the theory of variables 
respectively. In tossing a coin we only classify the results of the 
tosses as heads or tails; in drawing balls from a mixture of black 
and white balls, we only classify the balls drawn as black or as 
white. These cases correspond to the theory of attributes, and 
the general case may be represented as the drawing of a sample 
from a universe containing both 4’s and o’s, the number or 
proportion of 4’s in successive samples being observed. If, on the 
other hand, we put in a bag a number of cards bearing different 
values of some variable X and draw sample batches of cards, we 
can form averages and measures of dispersion for the successive 
batches, and these averages and measures of dispersion will vary 
slightly from one batch to another. If associated measures of 
two variables X and Y are recorded on each card, we can also form 
correlation-coefficients for the different batches, and these will vary 
in a similar manner. These cases correspond to the theory of 
variables, and it is the function of the theory of sampling for such 
cases to inform us as to the fluctuations to be expected in the 
averages, measures of dispersion, correlation-coefficients, ete, in 
successive samples. In the present and the three following 
chapters the theory of sampling is dealt with for the case of 
attributes alone. The theory is of great importance and interest, 
not only from its applications to the checking and control of 
statistical results, but also from the theoretical forms of frequency- 
distribution to which it leads. Finally, in Chapter XVII. one or 
two of the more important cases of the theory of sampling for 
variables are briefly treated, the greater part of the theory, owing 
to its difficulty, lying somewhat outside the limits of this work. 
3. The theory of sampling attains its greatest simplicity if 
every observation contributed to the sample may be regarded as 
independent of every other. This condition of independence 
holds good, e.g., for the tossing of a coin or the throwing of a die : 
the result of any one throw or toss does not affect, and is un- 
affected by, the results of the preceding and following tosses. 
It does not hold good, on the other hand, for the drawing of balls 
from a bag: if a ball be drawn from a bag containing 3 black 
and 3 white balls, the remainder may be either 2 black and 3 
white, or 2 white and 3 black, according as the first ball was 
black or white. The result of drawing a second ball is therefore 
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