VIIL.—MEASURES OF DISPERSION, ETC. 143 
Again, as in § 13 of Chap. VIL, it is convenient to note, for the 
checking of arithmetic, that if the same arbitrary origin be used 
for the calculation of the standard deviations in a number of 
component distributions we must have 
(LE) =2NED +3) + LL +3(ED. (8) 
12. As another useful illustration, let us find the standard 
deviation of the first #7 natural numbers, The mean in this case 
is evidently (¥+1)/2. Further, as is shown in any elementary 
Algebra, the sum of the squares of the first VV natural numbers is 
NHN +1)(2N +1) 
La 2 4 
The standard deviation o is therefore given by the equation— 
C=F(NV+1)2N+1)-L(N +1) 
that is, ol=7(N2-1) . (9) 
This result is of service if the relative merit of, or the relative 
intensity of some character in, the different individuals of a series 
is recorded not by means of measurements, e.g. marks awarded on 
some system of examination, but merely by means of their 
respective positions when ranked in order as regards the character, 
in the same way as boys are numbered in a class. With & 
individuals there are always N ranks, as they are termed, 
whatever the character, and the standard deviation is therefore 
always that given by equation (9). 
Another useful result follows at once from equation (9), namely, 
the standard deviation of a frequency-distribution in which all 
values of X within a range +7/2 on either side of the mean are 
equally frequent, values outside these limits not occurring, so that 
the frequency-distribution may be represented by a rectangle. The 
base / may be supposed divided into a very large number & of equal 
elements, and the standard deviation reduces to that of the first NV 
natural numbers when & is made indefinitely large. The single 
unit then becomes negligible compared with J. and consequently 
12 
l= 12 : . (10) 
13. Tt will be seen from the preceding paragraphs that the 
standard deviation possesses the majority at least of the properties 
which are desirable in a measure of dispersion as in an average 
(Chap. VIL § 4). It is rigidly defined ; it is based on all the 
observations made ; it is calculated with reasonable ease ; 1t lends 
itself readily to algebraical treatment ; and we may add, though the 
student will have to take the statement on trust for the present, 
that it is, as a rule, the measure least affected by fluctuations of