2x) THEORY OF STATISTICS. 
another card, e.g. that if the first card drawn at any sampling 
bears a high value, the next and following cards of the same 
sample are likely to bear high values also. Under these circum- 
stances, if 7, denote the correlation between the values on the 
first and second cards, and so on, 
ES 
5 == +25(rp +r + gC pe El Ba 0 
There are n(n —1)/2 correlations; and if, therefore, r is the 
arithmetic mean of them all, we may write 
: Or 
on=—{1+7r(n-1)] . . (LLY 
As the means and standard-deviations of #;, x, . . . . #, are all 
identical, » may more simply be regarded as the correlation 
coefficient for a table formed by taking all possible pairs of the 
n values in every sample. If this correlation be positive, the 
standard error of the mean will be increased, and for a given 
value of » the increase will be the greater, the greater the size of 
the samples. If » be negative, on the other hand, the standard 
error will be diminished. Equation (11) corresponds precisely to 
equation (6), § 13, of Chap. XIV. 
As was pointed out in that chapter, the case when » is positive 
covers the case discussed under (a): for if we draw successive 
samples from different records, such a positive correlation is at 
once introduced, although the drawings of the several cards a? 
each sampling are quite independent of one another. Similarly, 
the case discussed under (6) is covered by the case of negative 
correlation, for if each card is always drawn from a separate and 
distinct part of the record, the correlation between any two «’s will 
on the average be negative : if some one card be always drawn 
from a part of the record containing low values of the variable, 
the others must on an average be drawn from parts containing 
relatively high values. It is as well, however, to keep the cases 
(a), (8), and (c) distinct, since a positive or negative correlation 
may arise for reasons quite different from those considered under 
(a) and (b). 
15. With this discussion of the standard error of the arithmetic 
mean we must bring the present work to a close. To indicate 
briefly our reasons for not proceeding further with the discussion 
of standard errors, we must remind the student that in order to 
express the standard error of the mean we require to know, in 
addition to the mean itself, the standard-deviation about the mean, 
or. in other words, the mean (deviation)? with respect to the mean. 
2h