XVIL.—SIMPLER CASES OF SAMPLING FOR VARIABLES. 349 
«+ . . d, from the mean for a large sample from the entire record, 
we have 
3" dogy igi d 
ga = R20 ) + JAP). 
Henra 
1 
or m= (0 7 
To _ Sm 
= e  {10) 
The last equation again corresponds precisely with that given for 
the same departure from the rules of simple sampling in the case 
of attributes (Chap. XIV. § 11., eqn. 4). If, to vary our previous 
illustration, we had measured the statures of men in each of » 
different districts, and then proceeded to form a set of samples 
by taking one man from each district for the first sample, one 
man from each district for the second sample, and so on, the 
standard-deviation of the means of the samples so formed would 
be appreciably less than the standard error of simple sampling 
ao/s/n. Asa limiting case, it is evident that if the men in each 
district were all of precisely the same stature, the means of all the 
samples so compounded would be identical : in such a case, in fact, 
oy =8,, and consequently o,,=0. To give another illustration, if 
the cards from which we were drawing samples had been arranged 
in order of the magnitude of X recorded on each, we would get 
a much more stable sample by drawing one card from each 
successive nth part of the record than by taking the sample 
according to our previous rules—e.g. shaking them up in a bag 
and taking out cards blindfold, or using some equivalent process. 
The result is perhaps of some practical interest. It shows that, 
if we are actually taking samples from a large area, different 
districts of which exhibit markedly different means for the 
variable under consideration, and are limited to a sample of =n 
observations ; if we break up the whole area into n sub-districts, 
each as homogeneous as possible, and take a contribution to the 
sample from each, we will obtain a more stable mean by this 
orderly procedure than will be given, for the same number of 
observations, by any process of selecting the districts from which 
samples shall be taken by chance. There may, however, be a 
greater risk of biassed error. The conclusions seem in accord 
with common-sense. 
(c) Finally, suppose that, while our conditions (a) and (3) of § 2 
hold good, the magnitude of the variable recorded on one card 
drawn is no longer independent of the magnitude recorded on 
RAG N