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 arbi- 
trarily 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 calcula- 
tion 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 distribu- 
tion 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 
CC aoil