THEORY OF STATISTICS. 
were discussed very fully for the case of attributes (Chap. XIII 
§ 8), and we would refer the student to the discussion then given. 
ere it is sufficient to state the assumptions briefly, using the 
etters (a), (6) and (c) to denote the corresponding assumption 
indicated by the same letters in the section cited. 
(a) We assume that we are drawing from precisely the same 
record throughout the experiment, so that the chance of drawin 
a card with any given value of X, or a value within any assigned 
limits, is the same at each sampling. 
(6) We assume not only that we are drawing from the same 
record throughout, but that each of our cards at each drawin 
ay be regarded quite strictly as drawn from the same record (or 
rom identically similar records): e.g. if our card-record is con- 
ained in a series of bundles, we must not make it a practice to 
ake the first card from bundle number 1, the second card from 
undle number 2, and so on, or else the chance of drawing 
card with a given value of X, or a value within assigned limits 
may not be the same for each individual card at each drawing. 
(c) We assume that the drawing of each card is entirely 
independent of that of every other, so that the value of X recorde 
on card 1, at each drawing, is uncorrelated with the value of 
recorded on card 2, 3, 4, and so on. It is for this reason that w 
spoke of the record, in § 1, as containing a practically infinit 
umber of cards, for otherwise the successive drawings at each 
sampling would not be independent: if the bag contain te 
ickets only, bearing the numbers 1 to 10, and we draw the car 
bearing 1, the average of the following cards drawn will be higher 
han the mean of all cards drawn ; if, on the other hand, we dra 
he 10, the average of the following cards will be lower than the mea 
f all cards—.e. there will be a negative correlation between th 
umber on the card taken at any one drawing and the card taken 
at any other drawing. Without making the number of cards i 
he bag indefinitely large, we can, as already pointed out for th 
ase of attributes (Chap. XIII. § 3), eliminate this correlation b 
replacing each card before drawing the next. 
Sampling conducted under these conditions we shall, as before 
speak of as simple sampling. We do not, it should be noticed 
make the further assumption that the sample is unbiassed, 7.e. 
hat the chance of inclusion in the sample is independent of the 
value of X recorded on the card (cf. the last paragraph in § 8, 
hap. XIII, and the discussion in §§ 4-8, Chap. XIV.). This 
assumption is unnecessary. If it be true, the interpretation o 
our results becomes simpler and more straightforward, for we 
can substitute for such phrases as ‘the standard-deviation of 
'n a very large sample,” “the form of the frequency-distributio 
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