VIL.—AVERAGES. 115 
(8) If a series of IV observations of a variable X consist of, say, 
two component series, the mean of the whole series can be 
readily expressed in terms of the means of the two components. 
For if we denote the values in the first series by X; and in the 
second series by X,, 
3(X) = 3(X) + 3(Xy), 
that is, if there be NV, observations in the first series and %, in 
the second, and the means of the two series be M/;, J, respectively, 
NM=N.M +N, M, . (5) 
For example, we find from the data of Table VI., Chap. VI, 
Mean stature of the 346 men born in Ireland =67-78 in. 
gy 2 2 741 % y Wales=6662 in. 
Hence the mean stature of the 1087 men born in the two countries 
is given by the equation— 
1087. M = (346 x 67-78) + (741 x 66-62). 
That is, #/=66'99 inches. It is evident that the form of the 
relation (5) is quite general : if there are » series of observations 
xX, X, .... KX, the mean M of the whole series is related to 
the means M;, M, ... . M, of the component series by the 
equation 
NM=N.M, +N. M+ .... +N. M, . (6) 
For the convenient checking of arithmetic, it is useful to note 
that, if the same arbitrary origin 4 for the deviations ¢ be taken 
in each case, we must have, denoting the component series by the 
subscripts 1, 2, . . . r as before, 
(fH) =3(fp&)+3(fpb)+ . .. - +3(ME) (7) 
The agreement of these totals accordingly checks the work. 
As an important corollary to the general relation (6), it may 
be noted that the approximate value for the mean obtained from 
any frequency distribution is the same whether we assume (1) 
that all the values in any class are identical with the mid-value 
of the class-interval, or (2) that the mean of the values in the 
class is identical with the mid-value of the class-interval. 
(¢) The mean of all the sums or differences of corresponding 
observations in two series (of equal numbers of observations) is 
equal to the sum or difference of the means of the two series. 
This follows almost at once. For if 
X=X, +X, 
3X) =23(X)) + 3(X,) 
a