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 standarddeviations
 are oy, 0, . . . . On and the means differ by d;, dos

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