te THEORY OF STATISTICS.
Arithmetical work, however, should be executed from first
principles, and not by quoting formule like the above.
Example ii.—Check the work of Example i., § 13, by finding the
frequencies of the ultimate classes from the frequencies of the
positive classes.
i = (4B) - (4BC)=338 —57=281
Ay) = (dy) - (ABy)= (4) - (40) ~ (4By)
=877 - 143 — 281 =453
(aBy) = (By) - (4By) =H = (B) - (C) + (BO) - (43)
=10,000 — 1086 — 286 +135 — 453
=10,135 — 1825 = 8310
and so on.
18. Examples of statistics of precisely the kind now under
consideration are afforded by the census returns, e.g., of 1891 or
1901, for England and Wales, of persons suffering from different
“infirmities,” any individual who is deaf and dumb, blind or
mentally deranged (lunatic, imbecile, or idiot) being required to
be returned as such on the schedule. The classes chosen for
tabulation are, however, neither the positive nor the ultimate
classes, but the following (neglecting minor distinctions amongst
the mentally deranged and the returns of persons who are deaf
but not dumb) :—Dumb, blind, mentally deranged ; dumb and
blind but not deranged; dumb and deranged but not blind;
blind and deranged but not dumb ; blind, dumb, and deranged.
If, in the symbolic notation, deaf-mutism be denoted by 4, blind-
ness by B, and mental derangement by C, the class-frequencies
thus given are (4), (B), (C), (4By), (480), (aBC), (ABC) (cf.
Census of England and Wales, 1891, vol. iii., tables 15 and 16,
p. vii. Census of 1901, Summary Tables, table xlix.). This set of
frequencies does not appear to possess any special advantages.
19. The symbols of our notation are, it should be remarked,
used in an inclusive sense, the symbol 4, for example, signifying
an object or individual possessing the attribute 4 with or without
others. This seems to be the only natural use of the symbol,
but at least one notation has been constructed on an exclusive
basis (cf. ref. b), the symbol 4 denoting that the object or in-
dividual possesses the attribute 4, but not B or C or D, or what-
ever other attributes have been noted. An exclusive notation is
apt to be relatively cumbrous and also ambiguous, for the reader
cannot know what attributes a given symbol excludes until he
has seen the whole list of attributes of which note has been
taken, and this list he must bear in mind. The statement that
the symbol A is used exclusively cannot mean, obviously, that the
object referred to possesses only the attribute 4 and no others
yd