VIIL.—MEASURES OF DISPERSION, ETC. 135 
But the sum of the deviations from the mean is zero, therefore 
the second term vanishes, and accordingly 
2=c2+d2. (4) 
Hence the root-mean-square deviation is least when deviations 
are measured from the mean, 7.e. the standard deviation is the least 
possible root-mean-square deviation. 
3(&2), or 3(f.&) if we are dealing with a grouped distribution 
and f is the frequency of & is sometimes termed the second moment 
of the distribution about 4, just as 3(¢) or 3(f.§) is termed 
the first moment (¢f. Chap. VII. § 8): we shall not make use 
of the term in the present work. Generally, 3(f.£") is termed 
the nth moment. 
4. If o and d are the two sides of a right-angled triangle, s is 
a—— 
Fic. 2.. 
the hypotenuse. If, then, #/H be the vertical through the 
mean of a frequency-distribution (fig. 25), and AS be set off 
equal to the standard deviation (on the same scale in which the 
variable X is plotted along the base), S4 will be the root-mean- 
square deviation from the point 4. This construction gives a 
concrete idea of the way in which the root-mean-square deviation 
depends on the origin from which deviations are measured. It 
will be seen that for small values of d the difference of s from o 
will be very minute, since 4 will lie very nearly on the circle 
drawn through A/ with centre .S and radius SJ/: slight errors 
in the mean due to approximations in calculation will not, there- 
fore, appreciably affect the value of the standard deviation. 
5. If we have to deal with relatively few, say thirty or forty, 
ungrouped observations, the method of calculating the standard 
deviation is perfectly straightforward. It is illustrated by the 
figures given below for the estimated average earnings of