L—NOTATION AND TERMINOLOGY. 11
gate of frequencies of the second order, and the twelve classes of
the second order which can be formed where three attributes
have been noted may be grouped into three such aggregates.
12. Class-frequencies £ ay in tabulating, be arranged so that
frequencies of the same order and frequencies belonging to the
same aggregate are kept together. Thus the frequencies for the
case of three attributes should be grouped as given below ; the
whole number of observations denoted by the letter I being
reckoned as a frequency of order zero, since no attributes are
specified :—
Order 0. WN
Order 1. (4) (B) (7
(a) (B) i
Order 2. (4B) (40) «
4p) (dy)
(aB) (aC) | . ()
(a3) (a7) (,
Order 3. (ABC) (a BC)
(4By) (aBy)
(ABC) (afC)
(487) (apy)
13. In such a complete table for the case of three attributes,
twenty-seven distinct frequencies are given :—1 of order zero, 6
of the first order, 12 of the second, and 8 of the third. It
is, however, in no case necessary to give such a complete
statement.
The whole number of observations must clearly be equal to the
number of 4’s together with the number of a’s, the number of
4’s to the number of 4’s that are B together with the number of
4’s that are not B ; and so on,—i.e. any class-frequency can always
be expressed vn terms of class-frequencies of higher order. Thus—
N=(4)+(a)=(B)+(B)=ete.
= (LR) + (4B) + (aB) + (af3) = ete. @)
(4)= (4B) + (48) = (40) + (47) =eto. |
(4B) = (4BC) + (4 By) = ete. )
Hence, instead of enumerating all the frequencies as under (1),
no more need be given, for the case of three attributes, than
the eight frequencies of the third order. If four attributes had
been noted it would be sufficient to give the sixteen frequencies of
the fourth order.
The classes specified by all the attributes noted in any case,
t.e. classes of the nth order in the case of n attributes, may be