110 THEORY OF STATISTICS.
small (¢f. Chap. VL. § 5). In this process each class-frequency
is multiplied by the mid-value of the interval, the products added
together, and the total divided by the number of observations.
If f denote the frequency of any class, X the mid-value of the
corresponding class-interval, the value of the mean so obtained
may be written—
1
M= = yates : . ot 140)
8. But this procedure is still further abbreviated in practice
by the following artifices:—(1) The class-interval is treated
as the unit of measurement throughout the arithmetic; (2) the
difference between the mean and the mid-value of some arbitrarily
chosen class-interval is computed instead of the absolute
value of the mean.
If 4 be the arbitrarily chosen value and
X=41¢&. . . (3)
ther
3(fX) =3(f.4) +3(f.9),
or, since 4 is a constant,
U=d+33(78) .®
The calculation of 3(f.X) is therefore replaced by the calculation
of 3(f.£). The advantage of this is that the class-frequencies
need only be multiplied by small integral numbers; for 4
being the mid-value of a class-interval, and X the mid-value of
another, and the class-interval being treated as a unit, the E's
must be a series of integers proceeding from zero at the arbitrary
origin 4. To keep the values of £ as small as possible, 4 should
be chosen near the middle of the range.
It may be mentioned here that 3(£), or 3(f.¢) for the grouped
distribution, is sometimes termed the first moment of the distribution
about the arbitrary origin 4: we shall not, however, make
use of this term.
9. The process is illustrated by the following example, using
the frequency-distribution of Table VIIL, Chap. VI. The
arbitrary origin 4 is taken at 3-5 per cent., the middle of the
sixth class-interval from the top of the table, and a little nearer
than the middle of the range to the estimated position of the
mean. The consequent values of ¢ are then written down as in
column (3) of the table, against the corresponding frequencies, the
values starting, of course, from zero opposite 3:5 per cent. Hach
frequency f is then multiplied by its £ and the products entered
Nr
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