XVIL.—SIMPLER CASES OF SAMPLING FOR VARIABLES. 343 
the standard error of the semi-interquartile range 1/,/3 times 
the standard error of a quartile. Taking the value of the 
standard error of a quartile from the table in § 4, we have, finally, 
standard error of the semi- | 7 
interquartile range in a - =0-78672——=, (4) 
normal distribution J X# 
Of course the standard-deviation of the inter-quartile, or semi- 
interquartile, range can readily be worked out in any particular 
case, using equation (2) and the value of the correlation 
given above: it is best to work out such standard errors 
from first principles, applying the usual formula for the standard 
deviation of the difference of two correlated variables (Chap. XI. 
§ 2, equation (1)). 
9. If there is any failure of the conditions of simple sampling, 
the formule of the preceding sections cease, of course, to hold 
good. We need not, however, enter again into a discussion of 
the effect of removing the several restrictions, for the effect on 
the standard error of p was considered in detail in § 9-14 of 
Chap. XIV., and the standard error of any percentile is directly 
proportional to the standard error of p (¢f. § 3). Further, the 
student may be reminded that the standard error of any per- 
centile measures solely the fluctuations that may be expected in 
that percentile owing to the errors of simple sampling alone: it 
has no bearing, therefore, save on the one question, whether an 
observed divergence of the percentile, from a certain value that 
might be expected to be yielded by a more extended series of 
observations or that had actually been observed in some other 
series, might or might not be due to fluctuations of simple 
sampling alone. It cannot and does not give any indication of 
the possibility of the sample being biassed or unrepresentative of 
the material from which it has been drawn, nor can it give any 
indication of the magnitude or influence of definite errors of 
observation—errors which may conceivably be of greater im- 
portance than errors of sampling. In the case of the distribution 
of statures, for instance, the standard error almost certainly gives 
quite a misleading idea as to the accuracy attained in determining 
the average stature for the United Kingdom : the sample is not 
representative, the several parts of the kingdom not contributing 
in their true proportions. The student should refer again to the 
discussion of these points in §§ 4-8 of Chap. XIV. Finally, we 
may note that the standard error of a percentile cannot be 
evaluated unless the number of observations is fairly large—large 
enough to determine f, (eqn. 2) with reasonable accuracy, or