THEORY OF STATISTICS. 
more strictly not-tall) on the basis of this classification. Similarly, 
the methods that are specially adapted to the treatment of 
statistics of variables, making use of each value recorded, are 
available to a greater extent than might at first sight seem possible 
for dealing with statistics of attributes. For example, we may 
treat the presence or absence of the attribute as correspending to 
the changes of a variable which can only possess two values, say 
0 and 1. Or, we may assume that we have really to do with a 
variable character which has been crudely classified, as suggested 
above, and we may be able, by auxiliary hypotheses as to the 
nature of this variable, to draw further conclusions. But the 
methods and principles developed for the case in which the observer 
only notes the presence or absence of attributes are the simplest 
and most fundamental, and are best considered first. This and 
the next three chapters (Chapters I.-IV.) are accordingly devoted 
to the Theory of Attributes. 
3. The objects or individuals that possess the attribute, and 
those that do not possess it, may be said to be members of two 
distinct classes, the observer classifying the objects or individuals 
observed. In the simplest case, where attention is paid to one 
attribute alone, only two mutually exclusive classes are formed. 
If several attributes are noted, the process of classification may, 
however, be continued indefinitely. Those that do and do not 
possess the first attribute may be reclassified according as they do 
or do not possess the second, the members of each of the sub- 
classes so formed according as they do or do not possess the 
third, and so on, every class being divided into two at each step. 
Thus the members of the population of any district may be 
classified into males and females; the members of each sex into 
sane and insane; the insane males, sane males, insane females, 
and sane females into blind and seeing. If we were dealing with 
a number of peas (Pisum sativum) of different varieties, they 
might be classified as tall or dwarf, with green seeds or yellow 
seeds, with wrinkled seeds or round seeds, so that we would have 
eight classes—tall with round green seeds, tall with round yellow 
seeds, tall with wrinkled green seeds, tall with wrinkled yellow 
seeds, and four similar classes of dwarf plants. 
4. It may be noticed that the fact of classification does not 
necessarily imply the existence of either a natural or a clearly 
defined boundary between the two classes. The boundary may 
be wholly arbitrary, e.g. where prices are classified as above or 
below some special value, barometer readings as above or below 
some particular height. The division may also be vague and 
uncertain : sanity and insanity, sight and blindness, pass 
into each other by such fine gradations that judgments may 
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