144 THEORY OF STATISTICS. 
sampling. On the other hand, it may be said that its general 
nature is not very readily comprehended, and that the process of 
squaring deviations and then taking the square root of the mean 
seems a little involved. The student will, however, soon surmount 
this feeling after a little practice in the calculation and use of the 
constant, and will realise, as he advances further, the advantages 
that it possesses. Such root-mean-square quantities, it may be 
added, frequently occur in other branches of science. The 
standard deviation should always be used as the measure of disper- 
sion, unless there is some very definite reason for preferring another 
measure, just as the arithmetic mean should be used as the measure 
of position. It may be added here that the student will meet with 
the standard deviation under many different names, of which we 
have adopted the most recent (due to Pearson, ref. 2): many of 
the earlier names are hardly adapted to general use, as they bear 
evidence of their derivation from the theory of errors of observation. 
Thus the terms “mean error” (Gauss), “error of mean square” 
(Airy), and “mean square error” have all been used in the same 
sense. The standard deviation multiplied by the square root of 
2 has been termed the “modulus” (Airy),—the student will see 
later the reason for the adoption of the factor—-and the reciprocal 
of the modulus the “precision” (Lexis). For the square of the 
standard deviation, often required, R. A. Fisher has suggested 
the term variance.” 
14. The Mean Deviation.—The mean deviation of a series of 
values of a variable is the arithmetic mean of their deviations 
from some average, taken without regard to their sign. The 
deviations may be measured either from the arithmetic mean or 
from the median, but the latter is the natural origin to use. Just 
as the root-mean-square deviation is least when deviations are 
measured from the arithmetic mean, so the mean deviation is 
least when deviations are measured from the median. For 
suppose that, for some origin exceeded by m values out of &, the 
mean deviation has a value A. Let the origin be displaced by 
an amount ¢ until it is just exceeded by m — 1 of the values only, 
t.e. until it coincides with the mth value from the upper end of 
the series. By this displacement of the origin the sum of devia- 
tions in excess of the origin is reduced by m.c, while the sum of 
deviations in defect of the mean is increased by (& —m)c. The 
new mean deviation is therefore 
N —m)c — me 
apliiopne 
=A ! N-2 
=A + 7 - 2m)e.