142 THEORY OF STATISTICS. 
give a more definite and concrete meaning to the standard 
deviation, and also to check arithmetical work to some extent— 
sufficiently, that is to say, to guard against very gross blunders. 
It must not be expected to hold for short series of observations : 
in Example i., for instance, the actual range is a good deal less 
than six times the standard deviation. 
11. The standard deviation is the measure of dispersion which 
it is most easy to treat by algebraical methods, resembling in this 
respect the arithmetic mean amongst measures of position. The 
majority of illustrations of its treatment must be postponed to a 
later stage (Chap. XI.), but the work of § 3 has already served as 
one example, and we may take another by continuing the work of 
§ 13 (0), Chap. VII. In that section it was shown that if a series 
of observations of which the mean is M/ consist of two component 
series, of which the means are J; and J/, respectively, 
NM=N.M +N, M, 
XN; and XN, being the numbers of observations in the two com- 
ponent series, and N=, +, the number in the entire series. 
Similarly, the standard deviation o of the whole series may be 
expressed in terms of the standard deviations o; and o, of the 
components and their respective means. Let 
M-M=d, 
M,- M=d, 
Then the mean-square deviations of the component series about 
the mean JM are, by equation (4), 32 +d;% and o,%+d,? respec: 
tively. Therefore, for the whole series, 
N.o%=N (2 +d2) + Noa 2 +45) o =D) 
If the numbers of observations in the component series be equal 
and the means be coincident, we have as a special case— 
0? = (oy? + 07?) 6) 
go that in this case the square of the standard deviation of the 
whole series is the arithmetic mean of the squares of the standard 
deviations of its components. 
It is evident that the form of the relation (5) is quite general : 
if a series of observations consists of » component series with 
standard deviations oy, oy, . . . 0, and means diverging from the 
general mean of the whole series by d,, dy, . . . d,, the standard 
deviation o of the whole series is given (using m to denote any 
subscript) by the equation— 
N.o?=3(N,.0,% + 2 N,.d,2%) (7) 
is