1° THEORY OF STATISTICS. 
chapter, on all the observations made, so that no single observation 
can have an unduly preponderant effect on its magnitude ; indeed, 
the measure should possess all the properties laid down as desir- 
able for an average in § 4 of Chap. VII. There are three such 
measures in common use—the standard deviation, the mean 
deviation, and the quartile deviation or semi-interquartile range, 
of which the first is the most important. 
2. The Standard Deviation.—The standard deviation is the 
square root of the arithmetic mean of the squares of all deviations, 
deviations being measured from the arithmetic mean of the 
observations. If the standard deviation be denoted by o, and a 
deviation from the arithmetic mean by z, as in the last chapter, 
then the standard deviation is given by the equation 
N70 
of = 2(27) : : : a (TY 
To square all the deviations may seem at first sight an artificial 
procedure, but it must be remembered that it would be useless to 
take the mere sum of the deviations, in order to obtain a measure 
of dispersion, since this sum is necessarily zero if deviations be 
taken from the mean. In order to obtain some quantity that 
shall vary with the dispersion it is necessary to average the 
deviations by a process that treats them as if they were all of the 
same sign, and squaring is the simplest process for eliminating 
signs which leads to results of algebraical convenience. 
3. A quantity analogous to the standard deviation may be 
defined in more general terms. Let 4 be any arbitrary value of 
X, and let & (as in Chap. VIL. § 8) denote the deviation of X 
from 4 ; <.e. let 
E=X-4. 
Then we may define the root-mean-square deviation s from the 
origin 4 by the equation 
1 
See ro RUE, . 2 
= 3(8) (2) 
In terms of this definition the standard deviation is the root- 
mean-square deviation from the mean. There is a very simple 
relation between the standard deviation and the root-mean-square 
deviation from any other origin. Let 
M-4=d. ‘3) 
so that E=x +d. 
Then £2=0p2 + 2x.d + d?, 
2(£2) = 3(«?) + 2d.3(x) + N.dA. 
34 
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