345 THEORY OF STATISTICS. 
standard-deviation, and so on, or if we draw the successive 
samples from essentially different parts of the same record, the 
standard error will be greatly increased. For suppose we draw 
k, samples from the first record, for which the standard-deviation 
(in an indefinitely large sample) is oy, and the mean differs by 
d, from the mean of all the records together (as ascertained by 
large samples in numbers proportionate to those now taken); &, 
samples from the second record, for which the standard-deviation 
is o,, and the mean differs by d, from the mean of all the records 
together, and so on. Then for the samples drawn from the first 
record the standard error of the mean will be o/,/n, but the 
distribution will centre round a value differing by d; from the 
mean for all the records together: and so on for the samples 
drawn from the other records. Hence, if 0, be the standard error 
of the mean, XV the total number of samples, 
Ji 
Nat, = (iD) + 3k). 
But the standard-deviation o, for all the records together is given 
by 
N.o2 = 2(ka®) + Z(kd?). 
Hence, writing 2(kd?) = N.s;,, 
oh mir iol (9) 
n n 
This equation corresponds precisely to equation (2) of § 9, Chap. 
XIV. The standard error of the mean, if our samples are drawn 
from different records or from essentially different parts of the 
entire record, may be increased indefinitely as compared with the 
value it would have in the case of simple sampling. If, for 
example, we take the statures of samples of # men in a number 
of different districts of England, and the standard-deviation of all 
the statures observed is o,, the standard-deviation of the means 
for the different districts will not be a,/x/n, but will have some 
greater value, dependent on the real variation in mean stature 
from district to district. 
(b) If we are drawing from the same record throughout, but 
always draw the first card from one part of that record, the 
second card from another part, and so on, and these parts differ 
more or less, the standard error of the mean will be decreased. 
For if, in large samples drawn from the subsidiary parts of the 
record from which the several cards are taken, the standard- 
deviations are oy, 0, . . . . On and the means differ by d;, dos 
LK