XVIL—SIMPLER CASES OF SAMPLING FOR VARIABLES. 351 
Similarly, to express the standard error of the standard-deviation 
we_ require to know, in the general case, the mean (deviation)? 
with respect to the mean. Either, then, we must find this quantity 
for the given distribution—and this would entail entering on a 
field of work which hitherto we have intentionally avoided—or we 
must, if that be possible, assume the distribution to be of such a 
form that we can express the mean (deviation) in terms of the 
mean (deviation)?. This can be done, as a fact, for the normal 
distribution, but the proof would again take us rather beyond 
the limits that we have set ourselves. To deal with the standard 
error of the correlation coefficient would take us still further 
afield, and the proof would be laborious and difficult, if not 
impossible, without the use of the differential and integral cal- 
culus. We must content ourselves, therefore, with a simple 
statement of the standard errors of some of the more important 
constants, 
Standard-deviation.—]If the distribution be normal, 
standard error of the o 
standard-deviation in \ = i (12) 
a normal distribution Van 
This is generally given as the standard error in all cases: it is, 
however, by no means exact : the general expression is 
standard error of the standard- 1 
deviation in a dein = J fy 14 (13) 
of any form py. m 
where pu, is the mean (deviation)*—deviations being, of course, 
measured from the mean—and Py the mean (deviation)? or the 
square of the standard-deviation: n is assumed sufficiently large 
to make the errors in the standard-deviation small compared with 
that quantity itself. Equation (13) may in some cases give 
values considerably ‘oreater—twice as great or more—than (12). 
(Cf. ref. 17.) If, however, the distribution be normal, equation 
(12) gives the standard error not merely of standard-deviations of 
order zero, to use the terminology of Chap. XII, but of standard- 
deviations of any order (ref. 33). It will be noticed, on reference 
to equation (4) above, § 8, that the standard error of the standard. 
deviation is less than that of the semi-interquartile range for a 
normal distribution. 
For a normal distribution, again, we have— 
standard error of the co- 24 v \2)1# 
efficient of variation a Bo) 1+ (150) } - (14)